ABSTRACT: Shape theory is an exciting extension of homotopy theory for general topological spaces. In this talk, some of the foundations of shape theory will be introduced. The notion of a pro-category will be defined and the concept of strong monomorphisms and strong epimorphisms in these pro-categories will be examined. Further characteristics of these strong monomorphisms and strong epimorphisms will then be explored.
ABSTRACT: We investigate the asymptotic properties of the likelihood ratio test statistic for testing homogeneity in normal mixture models with possibly three components in the presence of a structural parameter. We show that the maximum likelihood estimator of the structure parameter is consistent. The likelihood ratio statistic converges in distribution to the supremum of a quadratic form of a Gaussian process. Based on this result, we show that the distribution of the modified likelihood ratio statistic is well approximated by the chisquared distribution with 3 degrees of freedom.
ABSTRACT: We determine the optimal investment strategy of an individual who targets a given rate of consumption and who seeks to minimize the probability of going bankrupt before she dies, also known as ‘lifetime ruin'. We assume that the financial market is a Black-Scholes market with a single riskless asset with constant return and a single risky asset following geometric Brownian motion. We impose two types of borrowing constraints: First, we do not allow the individual to borrow money to invest in the risky asset nor to sell the risky asset short. Second, we allow the individual to borrow money but only at a rate that is higher than the rate earned on the riskless asset. We solve the two minimization problems via associated differential equations. For this talk, I will assume that the individual consumes at a constant (real) dollar rate and will include a numerical example to illustrate our results.
This is joint work with Erhan Bayraktar.
ABSTRACT: This talk will give a general overview of Thomas Hales 1998 proof of the Kepler Conjecture wherein he resolves the 3-demensional sphere packing problem. This result is a veritable tour-de-force of mathematical ingenuity; it includes over 250 pages of mathematics and 3 gigabytes of computer code and data.
ABSTRACT: It will be shown how compact Riemann surfaces appear in the theory of pseudo-differential operators. More precisely, we will consider isospectral deformation of the corresponding pseudo-differential operators and show how the integrability of a wide class of nonlinear differential equations corresponds to the linearization through the appropriate theta functions. If time allows, we will represent these results in terms of infinite Grassmannians. This is the first of a series of lectures.
Beverages will be served in Grad Lounge after the talk.
This is joint work with B.B. Banaschewski and M.M. Ebrahimi.
ABSTRACT: The objects of study here are two-dimensional lattice walks, with a fixed set of step directions, restricted to the first quadrant. These walks are well studied, both in a general context of probabilistic models, and specifically as particular case studies for fixed direction sets, notably the so-called Kreweras' walks defined by the direction set {NE, W, S}.
The goal here is to examine two series associated to these walks: a simple length generating function, and a complete generating function which encodes endpoints of walks, and to determine combinatorial criteria which decide when these series are algebraic, D-finite, or none of the above. (Indeed we have examples that we believe to be non-D-finite) We shall present an (almost) complete classification of all nearest neighbour walks where the set of directions is of cardinality three, and discuss how this leads to a natural, well supported, conjecture for the classification of nearest walks with any direction set.
Work in progress with M. Bousquet-Melou.
ABSTRACT: Unshifted and shifted multiscaling functions are used as mathematical models for curve fitting of irregularly sampled data. A pre-processing design for the discrete multiwavelet transform based on this curve fitting method is proposed. This pre-processing procedure combined with multiwavelet neural networks for data-adaptive curve fitting is shown to perform well in the case of high resolution. In the case of low resolution, it is more accurate than numerical integration and cheaper than matrix inversion. This is joint work with Akira Morimoto and Remi Vaillancourt.
Beverages will be served in Grad Lounge after the talk.
ABSTRACT: Seismic images of the earth's subsurface are created by processing the gigabytes of data recorded from geophones planted on the earth, which measure the vibrations caused by seismic waves reflected off geological structures deep under the surface. An accurate image is crucial to the commercial detection and recovery of hydrocarbons.
Useful seismic images rely on accurate mathematical models of the propagation and reflection of seismic energy, and accurate numerical methods to implement and invert the physical model. Typically these involve linear operators on a Hilbert space. We discuss the use of pseudodifferential operators to model the physics of wave propagation, and implementation of robust numerical techniques derived from time-frequency analysis using Gabor multipliers.
ABSTRACT: Fraud occurs in all areas of the financial services industry including banking, insurance, investments, brokerage, and securities and commodity exchanges. The cumulative effects are enormous. For example, approximately 10% to 20% of all insurance claims in the United States are though to be fraudulent, resulting in losses of approximately $20 billion per year. A few data mining techniques have such as outlier analysis, cluster analysis and neural nets, have been applied to detect fraud. This talk will briefly review these available techniques and discuss their relative strengthens and weaknesses in practice. Following, the focus of this talk will shift to Norkom's implementation of fraud detection -- the Alchemist (TM) platform.
The first important component of the platform is Norkom Alchemist's rules engine. The rules engine is used to monitor data and detect specific behaviour patterns within mission critical high transaction volume environments. Based on the detected patterns the rules engine can identify and communicate the recommended best course of action. In Alchemist's rules engine, efficient rules evaluation is based on an optimised version of the RETE algorithm. The RETE algorithm is widely recognized as by far the most efficient algorithm for the implementation of rule based expert systems.
The second important part is robust regression technique used to support the development and refinement of business rules. Robust regression technique is one variation of Support Vector Regression. This technique can limit the generalization error on future points not in the training data. Robust regression can give insights into the characteristics of suspicious activity. These characteristics can be used to refine the existing business rules. They can also be codified and used as input to the building of additional business rules.
The talk finishes with discussion of successful deployment of Alchemist platform at a major financial institution.
ABSTRACT: The ideology of the theory of fewnomials is that a real variety defined by a "simple" system of equations should have a "simple" topology. This theory is applicable for a large class of systems of transcendental equations. It gives new information on levels of real elementary functions and even on algebraic equations. Arnold has suggested a linearization of Hilbert's 16th problem in a neighbourhood of the Hamiltonian systems. The theory of fewnomials allows a proof of the existence of a uniform bound in Arnold's problem. No special knowledge is required for this talk.
Beverages will be served in Grad Lounge at 3:30 p.m.
Organizer's Notes: Askold Khovanskii is Professor of Mathematics at the University of Toronto. He obtained the Russian doctoral degree in mathematics from the Steklov Mathematical Institute in Moscow in 1988 with a thesis entitled "Newton Polyhedra and Fewnomials". His book "Fewnomials" was translated into English and published by the American Mathematical Society in 1991. This talk runs hand in hand with Huaiping's talk on Hilbert's 16th problem. It shows how ideas and techniques from algebra, geometry and the theory of singularities may bear on an important problem, which has been studied by many experts in ODEs for many years.
ABSTRACT: The talk will give a survey on some important developments of the high-order finite element methods in recent years. These developments involve in the mathematical framework for a-priori and a-posteriori error estimation, parallel and iterative algorithms and applications to engineering and sciences.
The talk will discuss some of current research topics in modeling and computing, which are new challenges to today's mathematics, computational mathematics and applied mathematics.
ABSTRACT: Let L be a finite lattice, that is, L is a finite set
equipped with a partial order, such that every pair of elements x,y
has a least upper bound x v y and a greatest lower bound xy . A real
valued function f on L is said to be supermodular if for all
x,y in
L
f(x v y) + f(xy) >= f(x) + f(y)
If equality holds for all x and y, f is said to be modular
The problem we consider is that of determining the extreme rays of the quotient S/M where S is the cone of supermodular functions, and M is the vector space of modular functions. ( A ray R in a cone K is said to be extreme if a =b+c where a is in R and b,c are in K implies that b,c are in R.)
This was motivated by problems in probability theory dealing with stochastic orderings.( The particular application is to determine dominance for the supermodular ordering on multivariate distributions).
We are able to solve this problem completely for the simplest type of lattices, namely those which are the disjoint union of chains.
For the particular application, we are interested in the lattice Z_N ^k consisting of all k -tuples with entries from the set {0,1, ... N-1}, equipped with the usual pointwise order. We are able to get complete answers for the cases : k = 2; N=2 , k= 3 or 4; N=3, k = 3. We present some conjectures for the general case.
ABSTRACT: Metapopulation models consider a division of space into patches, between which individuals migrate. I will justify the use of a metapopulation approach in epidemiological modelling. Then, I will present some of the challenges that arise when dealing with the resulting large systems of differential equations, and some of the solutions that were given to these problems.
ABSTRACT: In this talk, an introduction to Hilbert's 16th problem and its development will be given. There is an ongoing project aiming at proving the finiteness part of the problem for quadratic vector fields, namely, the existence of a uniform bound for the number of limit cycles for quadratic vector fields. This project and its latest progress will be presented.
Beverages will be served in Grad Lounge at 3:30 p.m.
ABSTRACT: Starting with the basic family of "Rooted Plane Trees", by assigning "weights" to the different trees , one obtains (infinitely many) different tree families (for example: binary trees, labelled trees). In applications involving tree-like structures the choice of weights is determined by the problem and its desired parameters.
ABSTRACT: Recently I emailed Nantel a short applied algebra question (asked jointly with Dylan Thurston), and he (along with Christophe Hohlweg) quickly solved it. In return, I have to explain why is it interesting.
After quickly stating the question I'll tell you about categorification (a bold suggestion of I. Frenkel, that much of math is the Euler characteristic of some "higher math", much like much of algebra is q-algebra at q=1). I'll then define traces and trace groups, which allow Euler characteristics to take values in objects more interesting than merely numbers. Finally I'll introduce the category of matrix factorizations, which is the core of a surpising new method for constructing homological theories from local data.
The ideas to be introduced in my talk (categorifcation, trace groups and matrix factorizations) are all conceptual and foundational and worthy of your time, definitely more than the incomplete (though possibly valid) logic that lead us to our question to Nantel. So assuming some luck, I'll only have time to tell the latter part of the story over coffee after my talk. I hope there's good coffee up there north of 401.
ABSTRACT: Existing clustering algorithms fall mainly into two categories, model-based (parametric) and non-model-based (nonparametric) methods. Parametric methods perform well when the specific model approximately fit data, but not so when there is non-negligible deviation between them. Nonparametric methods are robust, but efficiency loss may become an important issue in some situations. We propose a semi-parametric method where the distribution of data is modeled by a mixture model. The mixture proportions are unknown parameters while the sub-distribution of each cluster is modeled nonparametrically. The EM-algorithm along with a classification step is used to cluster data, and the BIC is used to determine the optimal number of clusters. We apply the proposed method to analyze various types of microarray data. As an illustration, the proposed method is applied to a real microarray data set. Simulation studies show the proposed method performances well and are more robust than the commonly used parametric methods (Fraley and Raftery, 2002).
ABSTRACT: Time-frequency analysis, rooted in the short-time Fourier transform, provides a flexible tool with which to analyze linear operators. In this talk we will focus on the representation of general linear operators in terms of their action on phase space, using the continuous version of the Gabor transform and its adjoint. Formulae relating the pseudo-differential symbol of an operator to its so-called Gabor symbol will be discussed, as will various open problems and directions for future research.
Beverages will be served in Grad Lounge after the talk.
ABSTRACT: The Principle of Continuity says, broadly speaking, that what holds in a given case also holds in what appear to be like cases. It has played a significant role in the evolution of mathematics in the seventeenth, eighteenth, and nineteenth centuries-in algebra, analysis, geometry, and number theory. At various times it implied one or another of the following assertions: What is true up to the limit is true at the limit; what is true for finite quantities is true for infinitely small and infinitely large quantities; what is true for polynomials is true for power series; what is true for circles is true for other conics; what is true for positive numbers is true for negative numbers; what is true for real numbers is true for complex numbers; and what is true for ordinary integers is true for (say) Gaussian integers. Such analogies, even when they failed to materialize, were often starting points of fruitful theories.
In the main part of the talk I will focus on examples of the Principle of Continuity in its historical context. I will conclude with several observations with implications for teaching.
ABSTRACT: Louis Solomon showed that the group algebra of the symmetric group has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. For any Weyl group, Paola Cellini proved the existence of a different, commutative subalgebra of the group algebra. We derive the existence of such a commutative subalgebra for the case of the symmetric group and of the hyperoctahedral group using a variation on Richard Stanley's theory of $P$-partitions.
ABSTRACT: In this talk, we will first present some global asymptotic stability results for a chemostat model with general nonmonotone response functions and delays in the nutrient conversion process. Under certain conditions, when $n$ species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations, even if time lags in the nutrient conversion process are taken into consideration, our results show the competitive exclusion principle holds. However, it is also shown that the time lags have an important effect on the outcome of competition, and that initial condition dependent outcome is possible. Then, by investigating a special case (n=1), we provide an analytic method to account for the transient oscillations observed in experiments.
ABSTRACT: Measured signals are typically of finite duration. They are dynamic and non-stationary processes whose frequency characteristics vary over time or space. This often requires algorithms capable of locally analyzing and processing signals. The recently developed Stockwell transform or S-transform in short (ST) combines the time-frequency representation of the windowed Fourier transform with the multi-scale analysis of the wavelet transform. Applying this transform to a temporal signal reveals information on what and when frequency events occur. In addition, its multi-scale analysis allows more accurate detection of subtle signal changes while interpretation in a time-frequency domain is easy to understand. In this talk, we give an overview of the theory of the ST and illustrate its effectiveness in biomedical applications. This is joint work with Dr Ross Mitchell.
Beverages will be served in Grad Lounge at 3:30 p.m.
ABSTRACT: What happens if we continue counting after reaching the infinity, and why would we want to do that? I will give some answers and examples.
ABSTRACT: In Part I we have introduced the theory of languages and finite automata. We talked about the Chomsky heirarchy of langauages and discuss the relationship between classes of languages and their generating functions.
Part II: Classes of permutations which do not contain certain patterns arise in several areas of algebraic combinatorics (e.g. in Schubert calculus vexillary permutations avoid the pattern 2143). The problem of enumerating permutations which avoid any given pattern has become a small subfield of combinatorics in recent years. There are very few general techniques to approach these enumeration problems and we will discuss a new way of looking at pattern avoiding permutations which will help to understand some of the results that are known.
In the second part we will apply last week results to pattern avoiding permutations and show that the class of permutations with a fixed number of descents that avoid any fixed pattern has a rational generating function.
A permutation, pi in S_n, is said to contain a pattern, X in S_m with m<=n, if there is a subword of pi, pi_{i_1} p_{i_2} ... p_{i_m} where i_r < i_{r+1} such that pi_{i_r} < pi_{i_s} if and only if X_r < X_s. e.g. 6132754 contains the pattern 4132 since the subword 6254 has the same relative order 6132754 avoids the pattern 1234 since there is no increasing subsequence of 4 numbers.
ABSTRACT: In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress levels such as temperature, voltage, and load, on the lifetimes of experimental units. Accelerated testing allows the experimenter to increase these stress levels to obtain information on the parameters of the life distributions more quickly than would be possible under normal operating conditions. A special class of accelerated tests are step-stress tests that allow the experimenter to increase the stress levels at fixed times during the experiment. In this talk, I will consider the simple step-stress model under Type-II censoring. I will derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters will be obtained through the use of conditional moment generating functions. I will also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions and the parametric bootstrap method, and assess their performance through a Monte Carlo simulation study. Finally, I will also propose an alternate (and simpler) step-stress model and discuss the inferential issues connected with that model, and make some comparative comments between the two step-stress models.
ABSTRACT: Magnetic Resonance Imaging (MRI) is a modern modality for visualizing internal organs of human body non-invasively. It has been widely used for disease diagnosis, treatment, and follow-ups. There are millions of MRI examinations performed each year. However, there is no such thing as a perfect image in MR. Often numerous artifacts could occur in MR images. In this talk, we will briefly overview the principles of MRI and discuss a number of commonly seen artifacts in MR.
ABSTRACT: In this talk I will describe the structure of the semi-classical short-range scattering amplitude. In particular, I will prove that the scattering amplitude at suitable energies $\lambda>0$ quantizes, in the sense of semi-classical Fourier integral operators, the scattering relation. I will provide all the necessary definitions during the talk.
Refreshments will be provided in the Grad Lounge after the talk.
ABSTRACT: Classes of permutations which do not contain certain patterns arise in several areas of algebraic combinatorics (e.g. in Schubert calculus vexillary permutations avoid the pattern 2143). The problem of enumerating permutations which avoid any given pattern has become a small subfield of combinatorics in recent years. There are very few general techniques to approach these enumeration problems and we will discuss a new way of looking at pattern avoiding permutations which will help to understand some of the results that are known.
This talk will include an introduction to the theory of languages and finite automata. We will talk about the Chomsky heirarchy of langauages and discuss the relationship between classes of languages and their generating functions. Finally, we will apply these results to pattern avoiding permutations and show that the class of permutations with a fixed number of descents that avoid any fixed pattern has a rational generating function.
A permutation, pi in S_n, is said to contain a pattern, X in S_m with m<=n, if there is a subword of pi, pi_{i_1} p_{i_2} ... p_{i_m} where i_r < i_{r+1} such that pi_{i_r} < pi_{i_s} if and only if X_r < X_s. e.g. 6132754 contains the pattern 4132 since the subword 6254 has the same relative order 6132754 avoids the pattern 1234 since there is no increasing subsequence of 4 numbers
ABSTRACT: Sequential designs have been widely used in clinical trials to address the ethical and efficiency issues for experiments with human volunteers. The conduct and analysis of a sequential clinical trial is relatively more involved than that for a fixed sample size trial. In this presentation, we discuss the bias in point estimate introduced by the possibility of early stopping and the overrunning data generated by the momentum in the system, and propose some suitable methods to deal with those problems.
ABSTRACT: In the fluorescence problem one tries to determine the source of light passing through an absorbing and scattering tissue, from far away radiation or, equivalently, from boundary measurements. When there is no scattering, inversion of the attenuated X-ray transform does the job. In the presence of small enough (anisotropic part of) scattering, this can still be done. For three dimensional models, only scatterings in the directions parallel to a fixed plane need be small. This is joint work with G. Bal.
Beverages will be served in the Grad Lounge after the talk.
ABSTRACT: A coupled cell system is a collection of individual, but
interacting, dynamical systems. Coupled cell models assume that the
output from each cell is important not just the dynamics considered as a
whole. In these systems the signals from two or more cells can be compared
and patterns of activity can emerge. We ask when can the cell dynamics in
a subset of cells be identical (synchrony) or differ by a phase shift. In
particular: How much of the qualitative dynamics observed in coupled cells
is the product of network architecture and how much is related to the
specific dynamics of cells and the way they are coupled?
We illustrate the ideas through a series of examples and discuss three
theorems. The first theorem classifies spatio-temporal symmetries of
periodic solutions; the second gives necessary and sufficient conditions
for synchrony in terms of network architecture and its symmetry groupoid;
and the third shows that synchronous dynamics may itself be viewed as a
coupled cell system through a quotient construction.
Refreshments will be served in N620 Ross Building at 2:45p.m.
ABSTRACT: Fitts' law is a robust model of psychomotor behavior developed by applying the information theory of physical communication systems to the human sensory-motor system. However, the information-theoretic development of Fitts' law is incomplete: the experimental observation of non-zero and possibly negative Y-intercepts in the linear relationship between movement time and Index of Difficulty (ID) can not be explained within the theory. Furthermore, this law is known to breakdown when ID is small. We show that both of these phenomenon can be explained as consequences of delay within the nervous system. By introducing delayed feedback into the Vector Integration to Endpoint (VITE) circuit of Bullock and Grossberg, we show that the Shannon formulation of Fitts' law is only an approximation to a more general relationship in which an approximately linear relationship with non-zero Y-intercept holds between movement time and ID when movement times are large relative to the delay. The slope and Y-intercept are determined by two parameters: the delay $\tau$, and the relaxation rate $\alpha$ of the circuit's negative feedback loop. As the movement time approaches the scale of the delay, a non-linear breakdown occurs and the movement time approaches a limiting value of $2\tau$. A re-analysis of data from the literature suggests this model is at least as good as, or better than, linear regression in Shannon Index of Difficulty. Furthermore, it provides an indirect way to measure delay within the nervous system from the speed-accuracy trade-off alone.
ABSTRACT: An outstanding open problem in the theory of partial differential equations is the well-posedness of initial-value problems for nonlinear hyperbolic equations in more than one space dimension. This talk will set a context for the problem: Why are mathematicians interested in partial differential equations, what are the differences between the way pure and applied mathematicians approach the subject, and how can different approaches reinforce each other?
The talk is intended for an audience which is not expert in partial differential equations, and will begin by explaining why the division of equations into "hyperbolic" and "elliptic" is natural mathematically as well as being grounded in applications. We will describe briefly the analysis used to prove existence theorems for linear equations of both types.
Generalizing the elliptic theory to quasilinear and nonlinear elliptic equations has been largely achieved, but the corresponding theory for hyperbolic equations is still being developed. Some simple examples serve to show the sorts of obstructions we may expect. A number of routes through these challenges seem ready to be explored. Finally, I will describe a new approach that I, along with co-workers and others, are pursuing, which exploits the better-developed theory of quasilinear elliptic equations to study multidimensional quasilinear hyperbolic equations.
Refreshments will be served in N620 Ross Building at 2:45p.m.
ABSTRACT: We begin by looking at definitions and some properties of the Maslov index and the H\"ormander index. Then we explain how to generalize them to infinite-dimensional settings using functional analytic methods. We conclude with an example of the infinite-dimensional H\"ormander index, which expresses the asymmetry of Cauchy data spaces of an elliptic PDE when we decompose a manifold into two components by a hypersurface.
Beverages will be served in the Grad Lounge after the talk.
ABSTRACT: I will talk about the development of Fitts law from an information-theoretic and experimental point of view, and neuraldynamic models of motor control.
SYLLABUS: Neural Networks; Dynamical Systems and Delay Dynamical Systems; Oscillation Theory of Linear Systems; Information Theory; Fitts Law.
ABSTRACT: The descent algebra has been introduced by Solomon, with an homomorphism which takes values on the algebra of characters, to solve a problem on characters theory of finite Coxeter groups.
We introduce two new properties of this homomorphism and give applications on characters theory.
ABSTRACT: A crystal for a representation of a semisimple Lie algebra is a combinatorial object which encodes the structure of the representation. There is an interesting tensor product on these crystals. We give a construction of a commutor (natural isomophisms A x B -> B x A) for the category of crystals of a semisimple Lie algebra. This commutor is symmetric but does not satisfy the usual hexagon axiom. Instead it obeys a different axiom which makes the category of crystals into a coboundary category. Motivated by the above construction, we investigate the structure of coboundary categories. Just as the braid group acts on repeated tensor products in a braided category, the fundamental group of the moduli space of stable real genus 0 curves with n marked points acts on repeated tensor products in a coboundary category.
ABSTRACT: A large class of combinatorial models of lattice walks,
including the self-avoiding walk and a variety of directed walk models,
have been studied as models that include the conformational entropy in
long linear molecules with conformational degrees of freedom (polymers).
These models have also become interesting from the combinatorial point of
view. I shall make some introductory comments about these models, and talk
about some of the mathematical and computational techniques commonly used
to analyse them. The physical relevance of the models suggests that a
statistical mechanics approach be used to describe the properties of such
models, and I shall also briefly explain the connection between
statistical mechanics concepts such as the partition function, and
combinatorial concepts such as the generating function.
ABSTRACT: Credit risk arises due to the possibility of a change in the credit quality of a counterparty. In extreme cases it is the risk that a counterparty will be unable to meet its obligations, also called default. Because there are many types of counterparties and many different types of obligations-from auto loans to derivatives transactions-credit risk takes many forms. So there are many ways to manage it. For the purpose of this talk, I will focus on rating-based models in the context of trading derivatives. This presentation explains how to construct the forward distribution of the values, how to use the resulting distribution to estimate risk and how to compare the calculated results with the observed results within a back testing methodology. This work arises from an internship in the financial engineering diploma program.
ABSTRACT: Concerning to synchronization, in this talk it will be presented mathematical methods and some interesting examples. In the first part of the lecture the author will discuss the concept of synchonization and will show simulations envolving many examples of coupled systems, namely: Lorenz Equations, Duffing Equations, Chua systems, Hodgkin-Huxley equations, Power systems, etc..Applications to communication systems will be emphasized. In the second part, the author will present some mathematical methods. Some theorems that provide uniform estimates of attractors and synchronization, by using Liapunov like functions, will be analysed.
Refreshments will be served in N620 Ross at 3:00p.m.
ABSTRACT: Andrews and Curtis conjectured in 1965 that every balanced presentation (i.e. with an equal number of generators and relations) of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. Their conjecture was originally motivated by topological questions. Some recent computational work by Miasnikov and Myasnikov on this problem is based on genetic algorithms. We show that a computational attack based on a breadth-first search of the tree of equivalent presentations is also viable, and seems to outperform that based on genetic algorithms. It allows us to extract shorter proofs (in some cases, provably shortest). We discuss short potential counterexamples.
The talk will be accessible to a fairly wide audience of those interested in algebra or computer science.
SCHEDULE:
10:00-10:10 -- Jianhong Wu -- Opening Remarks
10:10-11:50 -- Hildebrando Munhoz Rodrigues, University of Sao Paulo, will speak on "Smooth Linearization in Infinite Dimensional Banach Spaces"
11:50-11:30 -- Pietro-Luciano Buono, University of Ontario Institute of Technology, will speak on "Linear and nonlinear unfoldings of delay-differential equations"
11:30-12:10 -- Yuming Chen, Wilfrid Laurier University, will speak on "Global Asymptotic stability of delayed Cohen-Grossberg neural networks"
12:10-2:00 -- Lunch Break
2:00-2:40 -- Jifa Jiang, University of Science and Technology of China, will speak on "Saddle-Point Behavior for Monotone Semiflows and Reaction-Diffusion Models"
2:40-3:20 -- Greg Lewis, University of Ontario Institute of Technology, will speak on "Bifurcations in models for large-scale geophysical fluids"
3:20-4:00 -- Xinfu Zou, University of Western Ontario, "Dynamics in numerics: on two different difference schemes for ODEs."
4:00-4:20 -- Coffee Break
4:20-5:00 -- Dong Liang, LIAM, York University, will speak on "Mathematical Modelling of Population Growth with Delayed Nonlocal Reaction and of a Marine Bacteriophage Infection"
5:00-5:40 -- Yi Zhang, University of Waterloo, will speak on "Stability of switched delay system"
Light lunch, coffee and beverages will be provided.
ABSTRACT: Stem cells have the potential to serve as renewable
source of tissue-specific cells in clinical tissue-replacement therapies.
The key limitation to the widespread use of stem cells in such
applications is the lack of understanding of precise mechanisms driving
proliferation and differentiation properties in vitro of the stem cell
compartment.
Although a large number of deterministic and stochastic models of stem
cell development appeared in the literature over the past 40 years, none
remained unquestionable and there is still need for careful modeling
addressing particular issues and hypothesizes.
Typical population of stem cells is highly heterogeneous, consisting
apart from stem cells of cells passing the differentiation stage as well
as differentiated towards various lineages. Biologically, it is fairly
difficult to disentangle properties of different cell types in the dynamic
mixed population.
We propose a multitype Markov branching process model to imitate
temporal development of a general stem cell population. Fundamental
parameters such as rates of proliferation and differentiation are obtained
by maximum likelihood estimation. Being a relatively crude approximation
of real processes, the model could be extended to address intrinsic
complexity of a biological system and potentially account for events on
molecular level.
The syllabus of the exam is available for perusal in N519 Ross.
ABSTRACT: I will talk about Species giving main definitions and examples; the structures of species and the way the structures are transported by bijections between their underlying sets; generating function and connection between species and generating functions.
ABSTRACT: I will talk about gene expression data processing from the viewpoint of a graduate student in mathematics: Topics include what gene expression data are, how these data are processed and what the challenges are. The focus of this talk is applications of gene expression data and the relevant methods and literatures, with emphasis on clustering analysis.
ABSTRACT: State-space models have been a powerful tool for modeling and forecasting serially correlated data, because they are based on a structural analysis of the data. The components that contribute tomodeling different aspects of the data, such as trend, seasonal,together with the effects of explanatory variables andinterventions, can be specified separately before being assembledinto one state-space model. Because of the presence of latent (orstate) variables, integration evaluation is generally inevitablein likelihood-based statistical inference. This thesis consistsprimarily two parts: the first concentrating on computationalaspects and the other on statistical aspects.
In the first part, we develop a new numerical approximationmethod, called the Best Quadrature Formula (BQF), for integrationevaluation. Adapted from the BQF to accommodate different featuresof integrands, we propose a smoothed variant of BQF, called Smoothed Best Quadrature Formula (SBQF). Motivated by the SBQF, we further develop an algorithm in the form of mixture of nomalsapproximation (MoNA), which is particularly suitable to deal with the calculation of density functions, such as the Kalman filter density, at different smoothing levels.
ABSTRACT: Regression clustering is an important model-based clustering tool with wide applications in a variety of disciplines. It discovers and reconstructs the hidden structure for a data set which is believed to be a random sample from a population comprising of a fixed, but unknown, number of sub-populations each of which is characterized by a class-specific regression hyperplane. A fundamental problem, as well as a preliminary step in most of clustering techniques including regression clustering is to determine the underlying ``true'' number of clusters in the data set.
We attempt to tackle this problem using model selection techniques, in particular the information-based approach. Thus model-selection based procedures are proposed to estimate the number of regression hyperplanes for data with either continuous or binary response variable respectively. And for the former case, we also propose an M-estimator based robust-augmented criterion to deal with abnormality of the data. We present the asymptotic results of the proposed procedures, which are obtained under the framework of classification likelihood approach. Finally their small sample performance is illustrated by examples from simulation studies and data analysis.
ABSTRACT: We consider nonlinear elliptic systems with Dirichlet boundary condition on a bounded domain in R^n which is invariant with respect to the action of some group G of orthogonal transformations. For every subgroup K of G, we give a simple criterion for the existence of infinitely many solutions which are K-invariant but not G-invariant. We include a detailed discussion of the case N=3.
ABSTRACT: In classical model selectionn procedure, one "best" model is selected according to some model selection criterion such as MSE or AIC. However, the underlying assumption is that the true model is included in the set of all candidate models. Therefore, the model uncertainty is not addressed. To overcome this, model averaging methods are proposed and studied in the literature.
In this presentation, Bayesian model averaging methods and stacking methods are reviewed. Bayesian model averaging(BMA) is proposed in (Leamer,1978) and several different to realize it will be discussed in the presentation. Compared to BMA, stacking is new and the complete theory has not been established. It is based on cross-validation. We will review the stacking technique in the presentation as well.
ABSTRACT: Polya Theory is based on a simple idea to count collections of objects possessing some symmetry, for example counting the number of necklaces with n beads of c possible colors. I will speak about group acting on a set, provide necessary definitions and theorem.
ABSTRACT: In this study we consider the quantum mechanical problem of scattering of a charged particle by a potential field, we look at the partial wave analysis of scattering by a short-ranged potential and a long-ranged known as Coulomb potential.
ABSTRACT: We look at this integral equation methods which is a general technique used to solve linear partial differential equations in the presence of a source term , and the important approximation that can be made in order to find a solution valid in the asymptotic regime for relatively weak Interactions .
ABSTRACT: The study of human nature had traditionally been the realm of novelists, philosophers, and theologians, but has recently been studied by cognitive science, neuroscience, research on babies and on animals, anthropology, and evolutionary psychology. In this talk I will show--by surveying relevant research and by analyzing some mathematical "case studies"--how different parts of mathematical thinking can be either enabled or hindered by aspects of human nature. This new theoretical framework can add an evolutionary and ecological level of interpretation to empirical findings of math education research, as well as illuminate fundamental classroom issues. Consider, for example, the well-documented phenomenon that students tend to confuse between a theorem and its converse. This phenomenon can now be understood as resulting from a clash between Mathematical Logic and the Logic of Social Exchange--a fundamental part of human nature.
This example points to the general thesis of this talk: people fail in some mathematical tasks not because of a weakness in their mental apparatus, but because of its strength! This seeming paradox stems from the evolutionary origins of our brain and mind, and the selection pressures that influenced their "design" over millions of years. The point is, what may have been adaptive in the ancient ecology in which our stone-age ancestors lived (and is still adaptive today under similar conditions), may often clash with the requirements of modern civilization.
ABSTRACT: This system of equations describes the time evolution of the quantum mechanical behaviour of a large ensemble of particles in a vacuum where the long range interactions between the particles can be taken into account. The model also facilitates the introduction of external classical effects. As tunneling effects become more pronounced in semiconductor devices, models which are able to bridge the gap between the quantum behaviour and external classical effects become increasingly relevant. The WP system is such a model.
Local existence is shown by a contraction mapping argument which is then extended to a global result using macroscopic control (conservation of probability and energy). Asymptotic behaviour of the WP system and the underlying SP system is established with a priori estimates on the spatial moments.
Finally, conditions on the energy are given which a) ensure that the solutions decay and b) ensure that the solutions do not decay.
ABSTRACT: Regression clustering is an important model-based clustering tool with wide applications in a variety of disciplines. It discovers and reconstructs the hidden structure for a data set which is believed to be a random sample from a population comprising of a fixed, but unknown, number of sub-populations each of which is characterized by a class-specific regression hyperplane. A fundamental problem, as well as a preliminary step in most of clustering techniques including regression clustering is to determine the underlying ``true'' number of clusters in the data set.
We attempt to tackle this problem using model selection techniques, in particular the information-based approach. Thus model-selection based procedures are proposed to estimate the number of regression hyperplanes for data with either continuous or binary response variable respectively. And for the former case, we also propose an M-estimator based robust-augmented criterion to deal with abnormality of the data. We present the asymptotic results of the proposed procedures, which are obtained under the framework of classification likelihood approach. Finally their small sample performance is illustrated by examples from simulation studies and data analysis.
ABSTRACT: The science of Geometry originated in India in connection with the construction of fire altars (Yajna vedis) needed for the performance of Vedic Yajnas. According to the strict injunctions laid down in the Vedic texts, each Yajna must be performed in an altar of prescribed size and shape. The slightest departure from the prescribed injunction or the slightest irregularity was supposed to nullify the very object of the Yajna. Thus the greatest care was required to strictly ensure the right shape and size of the altar. This Yajna, in fact, was a central theme around which the entire Vedic culture and civilization developed in the earliest times. It was this Yajna only that led to the origin of Vedas and the ensuing Vedic literature. So in order to understand the origin of geometry in the Vedic age, it is essential to understand the nature and types of Yajna as well as the nature and types of fire-altars required for them. It will also be interesting to know about the Vedic texts dealing with the various aspect of Yajna, particularly the construction of altars and various geometric operations carried out in the concerned Vedic texts in connection with the construction of altars of various geometric shapes and with relative dimensions.
In view of the above, the focus of the talk will be to discuss the origin of the concept of Yajna, the origin of Vedic literature centered around Yajna, fire altars with various geometric shapes and dimensions prescribed for the Yajna, geometric operations laid down for the construction of altars in the concerned Vedic texts, and geometric technical terms used therein.
ABSTRACT: This is the last of a series of lectures on characterizing harmonic functions on symmetric spaces in terms of an invariantaverage mean value property. The ubiquitous heat kernels on suchspaces, which play a fundamental role in this research, will be explained and developed from first principles.
ABSTRACT: This is the second of a series of lectures on characterizing harmonic functions on symmetric spaces of rank one in terms of an invariant average mean value property. The ubiquitous heat kernels on such spaces, which play a fundamental role in this research, will be explained and developed from first principles.
ABSTRACT: This is the first of a series of lectures on characterizing harmonic functions on symmetric spaces of rank one in terms of an invariant average mean value property. The ubiquitous heat kernels on such spaces, which play a fundamental role in this research, will be explained and developed from first principles.
ABSTRACT: The lecture will address the fundamental role of almost automorphic dynamics played in differential systems involving the interaction of several frequencies especially when they are close to resonance. It will show that almost automorphic solutions, representing somewhat irregular multi-frequency oscillations or oscillations covered with noise, exist in a wide class of forced monotone systems, (forced or free) damped oscillatory systems, and conservative systems, whereas regular multi-frequency (such as quasi-periodic) ones may fail to exist. Dynamical complexity of these systems caused by almost automorphic dynamics will also be discussed.
Refreshments will be served in N620 after the colloquium.
ABSTRACT: Case-cohort designs are widely used in epidemiologic studies and preventive trials with time-to-event data. These designs may be viewed as two-phase designs, a rich family of cost-effective biased sampling designs corresponding to a class of missing data models, where measurements of some important covariates are only available for a small subsample. Commonly used estimators of the finite dimensional parameter in a Cox-type regression model for these designs are obtained by maximizing pseudo-likelihood functions. An interesting question is: How efficient are the pseudo-likelihood estimators? In this talk, starting from some general facts about missing data, I will present basic ideas involved in deriving efficient scores and results of information bound calculations for the Cox model under these designs. Loss of efficiency for pseudo-likelihood estimators will be illustrated through simple examples.
Joint work with Mary Emond and Jon Wellner.
ABSTRACT: We all know that in the three centuries before Euclid mainstream Greek culture (with, of course, some dissenters) assigned to mathematics a unique significance and prestige of several kinds -- ontological, epistemological, methodological. How and why did this happen? My attempted overview will set our subject's internal progress against a social and intellectual background that includes the philosophical watershed defined by Parmenides, the birth of democracy, the development of rhetoric, the moral and epistemological relativism of the sophists, and more. If time allows I shall venture a few remarks on the subsequent legacy of this rise of mathematics to special status.
ABSTRACT: We propose and analyze a discrete time model for metapopulation on two patches with local logistic dynamics. The model carries a delay in the dispersion terms to account for long distance dispersion. Our results on this models shows that the impact of the dispersion on the global dynamics of the metapopulation is complicated and interesting: it can affect the existence of a positive equilibrium; it can either drive the metapopulation to global extinction, or prevent the metapopulation from global extinction and stabilize a positive equilibrium; it can also destabilize a positive equilibrium or a periodic orbit.
ABSTRACT: The notion of a "time scale" or "measure chain" was introduced by S. Hilger in 1988 to extend and unify discrete and continuous analysis. Although many results from the theory of ordinary differential equations carry over quite easily to difference equations, there are some instances where the results for discrete equations are quite different from their continuous counterparts. The introduction of a 'time scale' (which is any closed subset of the reals) permits the simultaneous treatment of discrete and continuous problems. An introduction to the theory of time scales will be presented along with a discussion of some properties of exponential functions, linear equations, and nonlinear boundary value problems to illustrate the applicability of the ideas.
Refreshments will be served at 3:00p.m. in N620 Ross.
ABSTRACT: A single gene auto-regulatory network is analyzed. The main goal is to investigate the effects of the negative and positive feedbacks on the intrinsic and external noises. The central finding in this research is that: for the intrinsic noise, both the negative and positive feedback regulations increase the fluctuation strength of mRNA levels (where the fluctuation strength is measured by the Fano factor for both the fluctuations of mRNAs and proteins), and the negative feedback Decreases, but the positive feedback increases, the fluctuation strength of proteins; for the external noise, the negative feedback not only increase the fluctuation strength of mRNA levels but also the fluctuation strength of proteins, and though the effect of the positive feedback on the fluctuation strength of mRNA levels depends on the size of positive feedback parameter k, the positive feedback must decrease the fluctuation strength of proteins.
ABSTRACT: In 1983, Deutsch and Kenderov give some necessary and sufficient conditions for a convex-valued multifunction to have continuous approximations. Inspired by Deutsch and Kenderov's result, we introduce and characterize coherent multifunctions. We investigate the relationship between lower semicontinuity and coherence, providing several examples. We then interpolate the lemmas behind the well-known Michael results on continuous selections. In doing so, we define a suitable and quite natural convex structure on every topological space, not just on metrizable ones. We produce a selection theorem stronger than Michael's selection theorem, both the convex-valued versions and the zero-dimensional one, in general considered as two independent cases in the literature.
In 1998, Bertacchi and Costantini obtain some conditions equivalent to the existence of continuous selections for the Wijsman hyperspace of ultrametric Polish spaces. We introduce a new class of hyperspace topologies, the macro-topologies. Both the Wijsman topology and the Vietoris topology belong to this class. We show that subject to natural conditions, the base space admits a closed order such that the minimum map is a continuous selection for every macro-topology. In the setting of Polish spaces, these conditions are substantially weaker than Bertacchi and Costantini's ones. In particular, we conclude that Polish spaces satisfying these conditions can be endowed with a compatible order and that the minimum function is a continuous selection for the Wijsman topology, just as it is for [0, 1]. This also solves a problem implicitly raised in Bertacchi and Costantini's paper.
Finally, we introduce and study some completeness properties for systems of open coverings of a given topological space. A Hausdorff space admitting a system of cardinality k satisfying one of these properties is of type G_k.In connection with these properties, we define several new variants of the Cech number and use elementary submodels to determine further results. As an application we give estimates for both the tightness and the Lindelof number of a generic upper hyperspace. Two recent results of Costantini, Hola and Vitolo on the tightness of co-compact hyperspaces follow from ours as corollaries.
ABSTRACT: The original definition of the FDR can be understood as the frequentist risk of false rejections conditional on the unknown parameter $\btheta$, while the Bayesian posterior FDR is conditioned on the data $\bT$, a particular realization of an experiment. From a Bayesian point of view, it seems natural to take into account the uncertainty in both the parameter and the data. In this spirit, we propose the Average FDR (AFDR) and Average FNR (AFNR) approaches in which the frequentist risks of false rejections and false non-rejections are averaged out with respect to some prior distribution of parameter $\btheta$. A linear combination of the AFDR and AFNR, called the Average Bayes Error Rate (ABER), is considered as an overall risk. Some useful formulas for the AFDR, AFNR and ABER are developed for normal samples with hierarchical mixture priors. The idea of finding threshold values minimizing the ABER is illustrated using a gene expression data and a simulation study. Keywords: Average false discovery rate, Average false non-discovery rate, Average Bayes error rate, Hierarchical mixture model, Microarray experiment.
ABSTRACT: The paper shows the close connection between queueing theory and random walks. The most important feature of this connection is the Wiener-Hopf factorization, which led to the simplification of queueing theory. The factorization is applied to the M/M/1 queue.
ABSTRACT: A very important class of Hausdorff topological groups is
represented by the locally compact abelian groups (LCA). The most
important theorem in the theory of LCA groups is Pontryagin's Duality
which states that any LCA group is isomorphic in the category of Hausdorff
topological groups with its bidual when the dual groups are the groups of
continuous group morphisms into the unitary circle with the compact-open
topology. Associated to any abelian topological group is the weak
topology induced.
By its characters, known as the Bohr topology. Relating an LCA group
with its Bohr topology is a very important theorem known as Glicksberg's
Theorem that states that the LCA topology and the Bohr topology associated
to it determine the same compact subsets of the considered group. Any
group topology on an abelian group that determines the same compact
subsets of the group as the Bohr topology associated to it will be said to
respect compactness. Using Pontrygin's Duality and Glicksberg's Theorem
as our main ingredients we look at the following problem: given an abelian
group G and a group of morphisms of groups from G into the unitary circle
T, when is the considered group of morphisms the group of all characters
for an LCA topology on G? The solution to this problem will naturally
lead to considering certain types of refinements of topologies. A general
study of these refinement will be presentes both in the category of
topological spaces and topological groups.
For the refinements considered in the category of topological groups,
duality type theorems and Ascoli type theorems will be given. The Duality
theorems prezented are satisfied by very important classes of groups that
respect duality and satisfy Pontryagin's Duality, among these classes are
products of LCA topologies and complete abelian groups that have a
neighbourhood base around the identity consisting of open subgroups.
Compactness in the category of topological groups is studied and a new
proof of a theorem known as Goto's Theorem is given. Also, a topological
proof of Glicksberg's Theorem is given. Another important problem
discussed is known as the cowellpowerdness problem for Hausdorff
topological groups. It is known that epimorphisms in the category of
Hausdorff topological groups need not have dense image and the
cowellpowerdness problem asks wether the class of extensions of a
topological group for which the embeding is epimorphism is a set or a
proper class.
We will present new examples of epimorphisms that don't have dense
image and generalizations of a few related problems are presented.
ABSTRACT: We study the trajectory dynamical systems approach to study the positive solutions of semilinear elliptic equations on unbounded domains. The existence of the global attractor for the trajectory dynamical systems associated with this problem is proved.The symmetrization and stabilization properties of positive solutions are established.
ABSTRACT: I will describe some recent work that aims to generalize a well-known lower central series formula for the fundamental group of a hyperplane arrangement complement. The main object of interest here is a "homotopy Lie algebra" which arises in commutative algebra and rational homotopy theory. I will show that one can describe the algebra explicitly in terms of combinatorial ingredients in certain cases, then give some applications.
ABSTRACT: Traditionally, financial losses from natural disasters such as hurricanes, earthquakes, floods, etc. are covered by the insurance industry. Several disasters that happened in the last decade, however, have exposed the vulnerability of the insurance industry to catastrophic losses. Hurricane Andrew in 1992 and the Northridge Earthquake in 1994 resulted in almost $31 billion in insured industry losses and caused the failure of more than ten insurance companies. The estimated financial losses from the tragic event on September 11, 2001 are more than $50 billion. The US insurance industry now regularly discuss potential catastrophic losses of $50-$100 billion.
As large as these losses are, they are trivial compared to the aggregate income and financial wealth in the US economy. For example, a mere half percent drop in the stock market would imply about $80 billion financial losses, much larger than any catastrophic losses in history. Furthermore, catastrophe risk has found to be uncorrelated with financial market risk. According to the standard risk-sharing theory, then, there should be plenty room for the insurance industry to diversify the catastrophic risk in capital markets. However, this has not been the case empirically. Froot (2001) examines the reinsurance market for catastrophic event risk and finds that most insurers purchase relatively little reinsurance against large loss event, and that premiums are very high relative to expected losses. To explain these findings, Froot argues that the lack of demand and high premiums of insurance against large loss events are due to the limited supply of capital to the reinsurance market.
Sun (2002) examines the futures and options market for catastrophe risk at the Chicago Board of Trade. Because in principle any investor can participate in trading these futures and options, limited capital supply should not be a serious problem in this market. However, she finds similar behavior as that in the reinsurance market: low demand for securities that give protection against large loss events, and very high premiums relative to expected losses.
In this paper, we offer an alternative explanation for the low volume and high prices of catastrophic event contingent claims based on model uncertainty and agents' aversion to uncertainty. By nature, there is a high degree of uncertainty about the probabilities of most catastrophic events. There is also considerable uncertainty about the magnitude of losses of any given catastrophic event. As pointed out by Froot and Posner (2001), model uncertainty itself cannot explain the high prices in the catastrophe insurance market. For model uncertainty to have a significant impact on market prices, investors have to be averse to ambiguity or uncertainty.
To formalize our argument, we apply Epstein and Wang (1994)'s uncertainty-averse preference to a risk-sharing model when agents are ambiguous about probabilities of catastrophic events. We show that (1) there is generally a significant uncertainty premium associated with catastrophe risk, and (2) when the degree of uncertainty of investors is significantly higher than that of agents who demand for insurance, there will be no trade.
The main contribution of the paper, however, is by introducing learning and updating into a dynamic risk-sharing/asset pricing model. Since we do not have independent evidence on agent's degree of uncertainty aversion, we cannot tell quantitatively whether the high premium in the market can be justified by agents' aversion to uncertainty. To generate empirically testable implications, we examine how the uncertainty premium varies across events for which the amount of historical information varies. Intuitively, the more historical data one has about a particular event, the less uncertainty one should have about the probability of the event, and therefore, the lower the uncertainty premium associated with insurance against the event risk. Using the learning and updating rule under ambiguity introduced by Mariacci (2002) and Epstein and Schneider (2002), we show that this is exactly what our model predicts. Furthermore, we show that the implication of our model is consistent with many phenomenon observed in financial and insurance markets, including the following:
In summary, our paper shows that even when capital markets are complete and frictionless, the catastrophe insurance market may still be thin and the market prices of catastrophe insurance contracts may still be extremely high relative to expected losses. This is so because there is high degree of uncertainty about catastrophic event probabilities and losses, and because investors are uncertainty-averse.
References:
Epstein, L. G. and Wang, T. (1994). Intertemporal Asset Pricing under
Knightian Uncertainty. Econometrica 62, 283-322.
Epstein, L. G. and Schneider, Martin (2002). Learning under Ambiguity.
Working paper, University of Rochester
Mariacci, Massimo (2002). Learning from Ambiguous Urns. Statistical
Paper 43. 145-151.
Froot, Kenneth. (2001) The Market for Catastrophe Risk: A Clinical
Examination. Journal of Financial Economcis 60. 529-571.
Froot, Kenneth. And Steven Posner (2001) The Pricing of Event Risks
with Parameter Uncertainty. Working Paper.
Sun, Y. (2002) An Empirical Analysis of Catastrophe-linked Securities.
University of Toronto
ABSTRACT: Subspace clustering is a useful technique in data mining for identifying hidden patterns in high dimensional data. Generalized projective clustering technique has been developed which able to construct clusters in arbitrarily aligned subspace of lower dimensionality.
This talk will focus on ORCLUS(arbitrarily ORiented projected CLUSter generation) algorithm and present some comparisons with other clustering techniques.
ABSTRACT: We consider the situation of incomplete rankings in which n judges are presented with k(i) objects which are ranked independently by the judges. We wish to test the null hypothesis that each judge, when presented with the specified k(i) objects, picks the ranking at random from the space of k(i) permutations of (1,2,...,k(i)). The statistic considered is a generalization of the Friedman test in which the ranks assigned by each judge are replaced by real valued functions of the ranks. We define a measure of pairwise similarity between complete rankings based on such functions, and use averages of such similarities to construct measures of the level of concordance of the judges' rankings. In the complete ranking case, the resulting statistics coincide with those previously defined defined in the literature. We extend these measures of similarity in the complete case to the situation of incomplete rankings, and derive a statistic in this more general situation and investigate its properties.
ABSTRACT: The short-time Fourier transform and its various discretizations lie at the heart of so-called Gabor theory, or Gabor analysis. This is a relatively young branch of harmonic analysis that has a wide range of potential applications, from signal processing to operator theory. In this talk a brief introduction to Gabor analysis will be given, after which we will focus on a particular topic: the diagonalization of linear operators by means of the short-time Fourier transform. Some recent results that have helped to establish the basic theory will be presented; in this context the extreme value distribution emerges unexpectedly as an advantageous windowing function for the short-time Fourier transform. Time permitting, we will discuss in addition some promising avenues of current research aimed at solving linear equations, and in particular PDE with badly behaved coefficients.
Notes: Dr Gibson is a candidate for the Analysis Position in the Department of Mathematics and Statistics, Faculty of Arts.
Refreshments will be served in N620 Ross Building at 3:00p.m.
ABSTRACT: Eugen Lukacs in 1956 has observed that $\mathbb{E}(X|X+Y)=a(X+Y)$ and $ \mathbb{E}(X^2|X+Y)=b(X+Y)^2$ for two real independent random variables (rv) implies that they are gamma distributed. Fred Wang in 1982 has provided an extension to rv of $\mathbb{R}^2$ with replacement of the second condition by $ \mathbb{E}(X_1^2|X+Y)=b(X_i+Y_i)^2$ for $i=1,2.$ We consider the problem for rv in $\mathbb{R}^n$ in a more geometric manner (ie without coordinates) with a second condition of the form $\mathbb{E}(q(X)|X+Y)=bq(X+Y)$ for all quadratic forms orthogonal to a given quadratic form $v.$ We identify the corresponding distributions in $\mathbb{R}^n$ and we relate them to the classification of natural exponential families by their variance functions. This is joint work with Jacek Weso{\l}owski.
ABSTRACT: We are going to talk about chaos theory in simple and easy to understand terms without going into mathematical detail. What is chaos? What does it mean for something to be "chaotic"? If the universe is governed by simple and elegant mathematical principles, whyis it so unpredictable?
Pizza lunch will be served after the talk in N537 Ross.
ABSTRACT: This talk will be a survey, from the plannar isoperimetric inequality perspective, of conformally invariant function spaces and their applications to Complex Variables, Harmonic Analysis, Operator Theory, PDEs and Differential Geometry. The focus will be on the most interesting results.
ABSTRACT: The work of Koike, Terada and Kleber defines three distinguished bases of the symmetric functions related to the characters of the Lie groups of type $B_n$, $C_n$ and $D_n$. We give formulas for these bases and find a uniform way to express them and manipulate them. Using techniques developed to analyze Hall-Littlewood and Macdonald symmetric functions we consider the q-deformation of these bases and find that they are related to the X=M conjecture of Hatayama, et al. allowing us to give a formula for coefficients K which we conjecture are equal to X and M (X=M=K conjecture).
This is joint work with Mark Shimozono.
ABSTRACT: To estimate the tree structure for a set of taxa, we typically use a statistical model for evolution and compute the maximum likelihood estimate. Molecular Biologists recognize that the model is a rough approximation to reality and there is considerable literature on the effects of model deviations. In this talk, I will develop the standard Markov models used for phylogeny and then examine some of the likely deviations from the model. Will illustrate in a couple of settings that the deviations do have an effect on the results.
ABSTRACT: We introduce a noncommutative binary operation on matroids, called the free product, and discuss some of its properties. In particular, free product is characterized by a certain universal property, is associative, and respects matroid duality. We characterize matroids that are irreducible with respect to free product and show that, up to isomorphism, every matroid factors uniquely as a free product of such matroids. We use these results to prove an inequality involving the numbers of nonisomorphic matroids on n elements which was conjectured by Welsh, and to show that the Hopf algebra of matroids with restriction-contraction coproduct is cofree.
ABSTRACT: The Heisenberg group is the simplest, non-commutative, nilpotent Lie group. It arises in two fundamental but different settings in analysis. On the one hand, it can be realized as the boundary of the unit ball in several complex variables. On the other hand, there is its genesis in the context of quantum mechanics. In this talk, I will introduce the Heisenberg group and its Lie algebra from the setting of complex analysis. The Laguerre calculus is the symbolic tensor calculus induced by the Laguerre function on the Heisenberg group. Then I will use the Laguerre calculus to solve the $\bar\partial$-Neumann problem in the non-isotropic Siegel Domain. I will also talk about the non-solvability of the Hans Lewy operator. If time permits, I will also talk about my recent joint work with Chang and Greiner about the embedding problems of the Heisenberg group.
Refreshments will be served in N620 Ross Building at 10:00 a.m.
ABSTRACT: We consider a class of nonparametric marginal models in which the regression coefficients are assumed to be time-varying smooth functions. Such models are appealing in longitudinal data analysis to characterize the time-dependent effects of covariates on the expected value of the response variable. A local quasi-likelihood method is employed to estimate the coefficient functionals, based on the nonparametric technique of local polynomial kernel regression. We discuss the issue of locality for the proposed estimation method associated with traditional kernels, and establish asymptotic distribution theory for the estimators considered. The proposed models are illustrated on two Monte Carlo studies and three data sets from a clinical trial of multiple sclerosis, a quality of life study in chemotherapeutic treatments on breast cancer and a genomic fine-scale mapping association study on chromosomal region 5q31 for Crohn's disease, respectively.
ABSTRACT: Numerical notation systems are graphic systems for representing numbers, such as the Roman and Hindu-Arabic numerals. Historians of mathematics have traditionally classified these systems on the basis of only one criterion: whether or not they use place-value (positional notation) to express numbers. Such typologies inadequately represent the various structural features of systems, and place excessive weight on positionality as the only relevant or interesting issue. In their place, I propose a two-dimensional typology that better reflects the variety of systems encountered historically. This new typology emphasizes that all numerical notation systems have advantages and disadvantages, and allows interesting historical questions to be posed regarding the evolution of numerical notation from antiquity to the present.
ABSTRACT: Studies on the combinatorics of descents in permutations led to the discovery of a pair, (QSym,NSym), of mutually dual graded Hopf algebras. Here, QSym is the graded Hopf algebra of quasi-symmetric functions, and its graded dual, NSym, is the graded Hopf algebra of noncommutative symmetric functions. Recent investigations on the combinatorics of peaks in permutations resulted in the discovery of an interesting new pair, (Peak,Peak*), of graded Hopf algebras. The first one, Peak, originally due to Stembridge, is a subalgebra of $QSym$. Its graded dual, Peak*, can therefore be identified as a homomorphic image of NSym. Our main result is to provide a representation theoretical interpretation of (Peak,Peak*) as Grothendieck rings of the tower of Hecke-Clifford algebras at $q=0$.
This is joint work with F. Hivert and J.Y. Thibon (Marne la vallee, France).
ABSTRACT: This paper combines the effects on asset price dynamics of two groups of traders: feedback traders who mechanically respond to price changes and bounded rationality traders who learn from lagged values of prices and dividends. First, we find that in the weak limit, as the trade interval goes to zero, the asset price is described by a mean reverting process around the level given by the forecasted price. Second, we show how feedback trading and learning effects on asset price dynamics may explain the empirical finding of long run dependencies on dividend yields in financial time series.
ABSTRACT: The first Fields medals were awarded in 1936 at the
International Congress of Mathematicians in Oslo to Professors Lars
Ahlfors (Harvard University) and Jesse Douglas (Massachusetts Institute of
Technology). Thus began the tradition of awarding the Fields medals to
mathematicians of the highest calibre at the International Congress of
Mathematicians, held every four years (except during the years of World
War II) .
This talk will be divided into three parts, roughly as follows:
(i) Who was Professor Fields, after whom the medal was named?
(ii) Why, when, and how was the medal established?
(iii) Who were the Fields medalists, and what was the work of some of
them
about?
ABSTRACT: A very important class of Hausdorff topological groups
is
represented by the locally compact abelian groups (LCA). The most
important theorem in the theory of LCA groups is Pontryagin's Duality
which states that any LCA group is isomorphic in the category of Hausdorff
topological groups with its bidual when the dual groups are the groups of
continuous group morphisms into the unitary circle with the compact-open
topology. Associated to any abelian topological group is the weak
topology induced.
By its characters, known as the Bohr topology. Relating an LCA group
with its Bohr topology is a very important theorem known as Glicksberg's
Theorem that states that the LCA topology and the Bohr topology associated
to it determine the same compact subsets of the considered group. Any
group topology on an abelian group that determines the same compact
subsets of the group as the Bohr topology associated to it will be said to
respect compactness. Using Pontrygin's Duality and Glicksberg's Theorem
as our main ingredients we look at the following problem: given an abelian
group G and a group of morphisms of groups from G into the unitary circle
T, when is the considered group of morphisms the group of all characters
for an LCA topology on G? The solution to this problem will naturally
lead to considering certain types of refinements of topologies. A general
study of these refinement will be presentes both in the category of
topological spaces and topological groups.
For the refinements considered in the category of topological groups,
duality type theorems and Ascoli type theorems will be given. The Duality
theorems prezented are satisfied by very important classes of groups that
respect duality and satisfy Pontryagin's Duality, among these classes are
products of LCA topologies and complete abelian groups that have a
neighbourhood base around the identity consisting of open subgroups.
Compactness in the category of topological groups is studied and a new
proof of a theorem known as Goto's Theorem is given. Also, a topological
proof of Glicksberg's Theorem is given. Another important problem
discussed is known as the cowellpowerdness problem for Hausdorff
topological groups. It is known that epimorphisms in the category of
Hausdorff topological groups need not have dense image and the
cowellpowerdness problem asks wether the class of extensions of a
topological group for which the embeding is epimorphism is a set or a
proper class.
We will present new examples of epimorphisms that don't have dense
image and generalizations of a few related problems are presented.
ABSTRACT: In the computation of exterior scattering problems (acoustic, electromagnetic or elastic) we run across a basic problem- how do we truncate the infinite exterior region in order to compute? Mathematically, this often reduces to the problem of computing a Dirichlet-to-Neumann (DtN) map. This talk will introduce these ideas, and provide a brief survey of truncation techniques, illustrated by means of a model problem. A simple series can be used to construct this DtN map under special circumstances, an idea exploited by Givoli and Keller. I then describe a new perturbative method for constructing a DtN map on boundaries which are perturbations of simple geometries, allowing us to extend the ideas of Givoli and Keller. I end with some computational experiments.
This is joint work with Dave Nicholls, Univ. Notre Dame.
ABSTRACT: We present a general risk model where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a pure jump Levy process into the model. This seeks to account for the discrete nature of claims and asset prices. We illustrate this model with Normal Inverse Gaussian (NIG) and Generalized Inverse Gaussian (GIG) Levy processes. These allow us to incorporate aggregate claims and premium fluctuations in the same process. We discuss important features of such processes and their relevance to risk modelling.
ABSTRACT: In 1983, Deutsch and Kenderov give some necessary and
sufficient conditions for a convex-valued multifunction to have continuous
approximations. Inspired by Deutsch and Kenderov's result, we introduce
and characterize coherent multifunctions. We investigate the relationship
between lower semicontinuity and coherence, providing several examples.
We then interpolate the lemmas behind the well-known Michael's results on
continuous selections. In doing so, we define a suitable and quite natural
convex structure on every topological space, not just on metrizable ones.
We produce a selection theorem stronger than Michael's selection theorem,
both the convex-valued versions and the zero-dimensional one, in general
considered as two independent cases in the literature.
In 1998 Bertacchi and Costantini obtain some conditions equivalent to
the existence of continuous selections for the Wijsman hyperspace of
ultrametric Polish spaces. We introduce a new class of hyperspace
topologies, the macro-topologies. Both the Wijsman topology and the
Vietoris topology belong to this class. We show that subject to natural
conditions, the base space admits a closed order such that the minimum map
is a continuous selection for every macro-topology. In the setting of
Polish spaces, these conditions are substantially weaker than the ones
given by Bertacchi and Costantini. In particular, we conclude that Polish
spaces satisfying these conditions can be endowed with a compatible order
and that the minimum function is a continuous selection for the Wijsman
topology, just as it is for $[0, 1]$. This also solves a problem
implicitely raised in Bertacchi and Costantini's paper.
Finally, we introduce and study some completeness properties for
systems of open covering of a given topological space. An Hausdorff space
admitting a system of cardinality k satisfying one of these properties is
of type G_k. In connection with these properties, we define several new
variants of the Cech number and use elementary submodels to determine
further results. In particular, we introduce the notions of M-hull and
M-network, where M is an elementary submodel. As an application of the
results obtained, but again using the technique of elementary submodels,
we give estimates for both the tightness and the Lindelof number of a
generic upper hyperspace. Two recent results of Costantini, Hola and
Vitolo on the tightness of co-compact hyperspaces follow from ours as easy
corollaries.
ABSTRACT: We consider a Sparre Anderson risk process that is perturbed by an independent diffusion process, in which claim waiting times have a generalized Erlang(n) distribution (i.e. as the sum of n independent exponentials, with possibly different means). This leads to a generalization of the defective renewal equations for the expected discounted penalty function at the time of ruin given by Tsai and Willmot (2002a, b) and Gerber and Shiu (2003, 2004) . The limiting behavior of the expected discounted penalty function is studied, when the dispersion coefficient goes to zero. Finally, explicit results are given for the case where n=2.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: A group is Hopfian if each surjective endomorphism of the group has trivial kernel. Zlil Sela proved that torsion-free hyperbolic groups are Hopfian. His proof is based on properties of JSJ decompositions of groups, Rips' theory of groups acting on real trees and theorems due to Bestvina and Paulin on degeneration of hyperbolic structures. I will explain how Sela's proof works and why it extends to subgroups of torsion-free hyperbolic groups. If time permits, I will also explain how the techniques used in this proof apply to algebraic geometry over groups.
Refreshments will be served at 10:30a.m. in N620 Ross.
ABSTRACT: E. coli 0157 is transmitted between cattle and from cattle to humans via contaminated faeces. A model for E. coli concentrations has been developed to ascertain the relative importance of various components of variation and assess the influence of several covariates. Each data point consists of counts of colonies, Poisson distributed but for an abundance of zeros, and more sensitive binary presence/absence measure. The data are modeled as a Zero-Inflated Poisson process where both the Poisson mean and binary probability are influenced by random effects. The two sets of random effects are potentially correlated. The model is fit using Markov chain Monte Carlo with the software package Winbugs.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: Given a locally compact group G, the Fourier and
Fourier-Stieltjes algebra are important objects in abstract harmonic
analysis. They are defined as spaces of coefficient functions associated
with continuous unitary representations of G. In the same time, they can
also be looked at as preduals of certain von Neumann algebras and,
consequently, they have a natural operator space structure.
It is known that, for abelian groups, there is a deep connection
between Fourier algebra homomorphisms and piecewise affine maps between
their underlying groups. We explore this fact from the point of view of
operator spaces, for general locally compact groups.
Refreshments will be served before the talk in N620Ross. Everybody welcome!
ABSTRACT: In this talk, the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process is introduced, and the time of ruin and the severity of ruin are defined. Then the probability of ruin, the expectation of the present value of the time of ruin, the (discounted) moment of the severity of ruin caused by a claim, the joint moment of the severity of ruin and the time of ruin caused by a claim, the moments of the time of ruin due to oscillation and claim, respectively, and the (discounted) defective joint/marginal distribution and probability functions of the surplus immediately before and at ruin are studied.
Refreshments will be served in N620 Ross at 3:30p.m.
ABSTRACT: The search for invariant subspaces of continuous
operators on Banach spaces has long been one of the most exciting topics
of Functional Analysis. In this talk, I am going to describe several
directions where this search has recently produced interesting and
important results. In particular, the following topics will be discussed:
extensions and limitations of Lomonosov's Theorem;
As for the algebraic version of the Invariant Subspace Problem, some
recent results on transitive and strictly semitransitive algebras will be
presented.
examples of operators with no invariant subspaces;
invariant subspaces of adjoint operators on dual Banach spaces.
Notes: Dr Troitsky is a candidate for the Analysis Position in the Department of Mathematics and Statistics, Faculty of Arts.
Refreshments will be served in N620 Ross Building at 10:00a.m.
Pizza lunch will be served in N537 Ross after the talk.
ABSTRACT: The results about geometrical properties [completeness and Riesz basisness] of the system of exponential solutions for functional differential equations of neutral type will be presented. The results about asymptotic behaviour and sharp estimates of the solutions of above mentioned equations will be also formulated.
ABSTRACT: We will discuss constructions of Banach spaces having
"few" symmetries. We will present results showing that such constructions
can be achieved as subspaces of general Banach spaces, or at least as
subspaces of Banach spaces from certain natural (and large) classes of
spaces. This supports the idea that phenomena of this type are not merely
accidents but they reflect a common behavior.
Results will include a new isomorphic characterization of a Hilbert
space in terms of unconditionality.
Refreshments will be served in N620 Ross at 3:30p.m.
ABSTRACT: Steinberg unitary Lie algebras were introduced by Allison and Faulkner which is a unitary version of Steinberg Lie algebras. We will use twisted vertex operators over Fock space to construct modules for a class of Steinberg unitary Lie algebras. This is a joint work with Naihuan Jing.
ABSTRACT: Results on solutions of heat equations, integral
representations of distributions and hyperfunctions are given on compact
Lie groups, Heisenberg groups, Riemannian manifolds and homogeneous
spaces. For compact Lie groups, we have the following results:
For Heisenberg groups, we give an integral representation of positive
definite solutions of heat equations. Uniqueness of solutions of heat
equations on complete Riemannian manifolds with Ricci curvature bounded
below, hyperbolic spaces and unit spheres are presented.
1. Hyperfunctions are boundary values of heat equations satisfying
some exponential growth conditions.
2. Solutions of heat equations satisfying some exponetial growth
conditions are shown to be unique.
3. Integral representations of positive solutions of heat equations
are given.
4. A characterization of central hyperfunctions is given. (This is the
analogue of the Schwartz-Godement theorem for compact Lie groups.)
5. An analogue of Schwartz' kernel theorem for bilinear hyperfunctions
is established.
6. An integral representation of translation-invariant positive
definite bilinear hyperfunctions is obtained.
A copy of the syllabus of the exam is available in Primrose's office.
ABSTRACT: We apply the modified likelihood ratio test to two binomial mixture models arising in genetic linkage analysis. The limiting distribution of the test statistic for both models is shown to be a mixture of chi-squared distributions. A consideration of random family sizes for both models gives similar results. We also explore the power properties under local alternatives. Simulation studies show that the modified likelihood ratio test is more powerful than other methods under a variety of model specifications.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: Standing quietly appears for the eye as a simple matter
of being inert or at rest. However precise measurements reveal that it
actually involves complex dynamical motions, which are too small for the
eye to see. Thus the simple act of standing is revealed to be a complex
control problem, involving both physiological and psychological processes.
The standing capability may be learned and it may be affected by disease.
The study of quiet standing may also be possible to help diagnose some
disease.
This talk will introduce the experimental data analysis and some
phenomenological models, and focus on our simple physiological model which
turns out to be a delayed stochastic differential equation. The numerical
simulation and Hopf bifurcation analysis on the model give us an
interesting explanation how people stand stilly and easily.
ABSTRACT: Self-dual regular polytopes posseses a polarity, that is, an involutory duality. We´ll discusse two families of self-dual chiral maps and show that self-dual chiral polytopes of odd rank posseses a polarity. We´ll also give an example of a self-dual chiral 4-polytopes that doesn´t posses a polarity.
ABSTRACT: Given a partially ordered set (poset) P and a labelling of its vertices, we will give a definition of a P-partition, as introduced by Richard Stanley in his Ph.D. thesis. In this thesis, Stanley made a conjecture concerning a certain quasi-symmetric generating function for the set of P-partitions of a labelled poset. This conjecture, which remains open, says that the generating function is a symmetric function if and only if our labelled poset is a "Schur labelled skew shape poset." In 1995, Claudia Malvenuto reformulated the conjecture so that the symmetry of the generating function needs to be related only to the local structure of the labelled poset, rather than its global structure. We will discuss a generalization of the idea of a P-partition, an appropriate extension of Stanley's conjecture, and an extension of Malvenuto's reformulation. We will also explain how Stanley's conjecture is almost always true and discuss several open problems concerning these quasi-symmetric generating functions.
ABSTRACT: Entropy provides a numerical measure of the complexity of a dynamical system. Of particular interest is whether or not it is positive, i.e., whether the system is chaotic or deterministic. We introduce a notion of dynamical entropy for Banach spaces and show that chaotic behaviour in this case can be described both geometrically and topologically. This leads to a geometric characterization of positive entropy in topological dynamics as well as to applications in C*-algebra structure theory and the study of noncommutative geodesic flows. This is joint work with Hanfeng Li.
Refreshments will be served after the talk in N620Ross. Everybody welcome!
The speaker is a candidate for a position in the Department.
ABSTRACT: Of various quantitative measures for the transmission
of infectious diseases, such as SARS, much interests are in the
following aspects. One is the transmission potential: when a new agent
invades a host population, what is the probability that it will become
naturally extinct before manifesting itself into a large outbreak. Once
a major outbreak does happen, a major concern is the effectiveness of
control measures, such as isolation of symptomatic cases and quarantine
of exposed but not yet symptomatic cases. Once data are collected,
trends interpretation becomes essential. Most of the infectious disease
models have an implicit assumption that the transmission pattern is
homogeneous, that all individuals follow similar contact patterns and
transmission probability per contact. In this presentation, the focus is
on transmission heterogeneity, as documented by super-spreading events
in SARS outbreaks, and how it affects the three aspects.
Stochastic models with random effects are applied to the contact
frequency and probability of transmission by extension of the work by
Crump and Mode (J. Appl. Math. Analis. Applic., 24, 25, 1968,1969) and
Jagers (Skand. Akturarietidskift, 1969). Combined with statistical
methods one can achieve the following results.
(1) Transmission heterogeneity makes it more difficult for an
infectious agent to invade and establish itself in a host population,
and start a major outbreak. Even when the basic reproductive number is
greater than 1, the probability of the infection become naturally
extinct (before becoming an outbreak) is always greater than that in
homogeneous transmission settings when all other parameters are assumed
the same. This may explain why approx. 30 countries reported SARS cases
in spring 2003, but only very few cities reported major outbreaks.
(2) If a major outbreak does happen, transmission heterogeneity makes
it more difficult to put the outbreak under control, than that in
homogeneous transmission settings. Here "control" refers to isolation of
symptomatic cases. In homogeneous transmission settings, average time to
isolation needs to be shorter than average time to infection produced by
infectious individuals as "race against time", both measured from onset
of symptoms. This race becomes more difficult to win if there is
heterogeneity. The isolation speed needs to be much faster. In high
heterogeneous situations with large extra-Poisson variance, isolation
alone may not be sufficient. Control measures must include "quarantine"
of exposed but not yet infectious cases, through contact tracing.
(3) With further information on incubation period and published
"epidemic curves" as trends by date of onset over time, combined with
linked data by "who infects whom", such as that from Singapore, one can
use statistical methods to establish trends by time of infection and
pin-point the timing of the super-spreading events. This helps the
interpretation of SARS trends over time in heterogeneous transmission
settings, with the potential to evaluate successfulness of control
measures implemented over time.
ABSTRACT: A problem identified in the recent SARS outbreaks in
Toronto was the absence of useful models to help answer the questions that
arose in choosing and executing the governmental and health systemic [and
societal] response. Indeed, that is a lack affecting any outbreak of a new
disease, or of a familiar one before it can be identified.
We are developing "rough and ready" dynamic models to use in the event
of an outbreak of a disease, in order to answer policy related questions
in the heat of an ongoing infectious disease outbreak, starting before the
disease is well characterized or the causal agent found, and continuing
throughout the outbreak lifespan. Critical questions we aim at answering
include whether to quarantine victims, isolate potential victims,
vaccinate susceptible individuals [if a vaccine in available], setting the
relative priority of finding vaccine or diagnostic test, among others. We
have model results for SARS and smallpox, and we need now to make the
model more generic.
ABSTRACT: The talk will begin with a survey of some of the main enumerative results in the subject of restricted (or pattern-avoiding) permutations. Next, recent developments and new directions will be discussed, including simultaneous avoidance of several patterns, enumeration of occurrences of a particular pattern in permutations, and generalized patterns (i.e., with the requirement that some elements occur in adjacent positions). The second part of the talk will focus on the study of statistics in restricted permutations, in which bijections to Dyck paths play an important role. We give a new bijection between 321-avoiding permutations and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. We will discuss recent work with Emeric Deutsch, Toufik Mansour, Marc Noy and Igor Pak.
ABSTRACT: As witnessed during the recent outbreak of SARS, quarantine and other transmission-blocking disease control measures such as the use of surgical masks are primary tools for halting the spread of newly emerging infectious agents. Such interventions are clearly desirable from an epidemiological standpoint, but they can also have unintended evolutionary effects on the pathogen population. I will discuss a possible approach towards a general theory for determining how these control measures might be altered to minimize the risks associated with evolutionary change during the initial emergence of any new disease.
ABSTRACT: The Kostka numbers $K_{\lambda\mu}$ appear in
combinatorics when expressing the Schur functions in terms of the monomial
symmetric functions, as $K_{\lambda\mu}$ counts the number of semistandard
Young tableaux of shape $\lambda$ and content $\mu$. They also appear in
representation theory as the multiplicities of weights in the irreducible
representations of type $A$.
Using a variety of tools from representation theory (Gelfand-Tsetlin
diagrams), convex geometry (vector partition functions), symplectic
geometry (Duistermaat-Heckman measure) and combinatorics (hyperplane
arrangements), we show that the Kostka numbers are given by polynomials in
the cells of a complex of cones. For fixed $\lambda$, the nonzero
$K_{\lambda\mu}$ consist of the lattice points inside a permutahedron. By
relating the complex of cones to a family of hyperplane arrangements, we
provide an explanation for why the polynomials giving the Kostka numbers
exhibit interesting factorization patterns in the boundary regions of the
permutahedron. We will consider $A_2$ and $A_3$ (partitions with at most
three and four parts) as running examples, with lots of pictures.
I will also say a few words as to how some of the techniques used
generalize to the case of Littlewood-Richardson coefficients.
This is joint work with Sara Billey and Victor Guillemin.
ABSTRACT: Viral kinetics have been extensively studied in the past through the use of spatially homogeneous ordinary differential equations describing the time evolution of the diseased state. However, spatial characteristics such as localized populations of dead cells might adversely affect the spread of infection, similar to the manner in which a counter-fire can stop a forest fire from spreading. In order to investigate the influence of spatial heterogeneities on viral spread, a simple 2-D cellular automaton (CA) model of a viral infection has been developed. In this initial phase of the investigation, the CA model is validated against clinical immunological data for uncomplicated Influenza A infections. Our results will be shown and discussed.