ABSTRACT: We will state the problem and show how it can be reduced to the analytic conjugacy problem for germs diffeomorphisms of (C,0) with multiplier on the unit circle and satisfying a symmetry condition. We will describe the formal and analytic invariants. In the particular case of a horn (two tangent curves), we will present the necessary and sufficient conditions for the bisection of the angle. We will explain the geometric meaning of all these conditions by unfolding the situation.
Refreshments will be served in N620 Ross Building at 3:30p.m.
ABSTRACT: There exist several algorithms for the classical problem of computung the eigenvalues of a real symmetric tridiagonal matrix. We first try to gain an algebraic and geometric understanding of the space of such matrices with prescribed spectrum. We then study simple shift strategies, a class of such algorithms, with an emphasis on understanding their speed of convergence. In particular, we shall see that Wilkinson's shift strategy exhibits cubic convergence for generic initial conditions and strictly quadratic for a class of initial conditions parametrized by a thin Cantor set.
Organizer's Note: Professor Saldanha was a former PhD student of Professor William Thurston at Princeton.
Refreshments will be served in N620 Ross Building at 3:30p.m.
ABSTRACT: We give a general introduction to inverse problems and in particular to Calder\'on's inverse problem. This problem consists in finding the electrical conductivity of a medium by making voltages and current measurements at the boundary. In mathematical terms one tries to determine the coefficient of a partial differential equation by measuring the corresponding Dirichlet-to-Neumann map. This problem arises in geophysical prospection and it has been proposed as a diagnostic tool in medical imaging, particular early breast cancer detection. We will also describe some recent progress on this problem in particular in the case that the voltage and current measurements are made on part of the boundary.
Refreshments will be served in N620 Ross Building at 3:30p.m.
ABSTRACT: We provide a model (for both continuous and discrete time) describing the evolution of a flock. Our model is parameterized by a constant $\beta$ capturing the rate of decay--which in our model is polynomial--of the influence between birds in the flock as they separate in space. Our main result shows that when $\beta<1/2$ convergence of the flock to a common velocity is guaranteed, while for $\beta\geq 1/2$ convergence is guaranteed under some condition on the initial positions and velocities of the birds only.
Refreshments will be served in N627 Ross Building at 3:30 p.m.
ABSTRACT: After a quick review of some of the basic ideas of Alain Connes' noncommutative geometry, I shall focus on possible approaches to define holomorphic structures in a noncommutative context. I will show that much of the structure of the 2-sphere as complex curve survives the q-deformation and generalizes to the quantum 2-sphere. Notably among these is the identification of the quantum homogeneous ring with the coordinate ring of the quantum 2 plane. I shall also indicate the importance of positive Hochschild cocycles in the study of holomorphic structures in a noncommutative setting. This is joint work with G. Landi and W. Van Suijlekom and a preprint is available online at http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.0154v2.pdf.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: Beginning with the classical work of Euler on the Riemann zeta function, the special values of L-functions have attracted much attention in number theory. In general, such values are expected to be transcendental numbers, perhaps related to a "period" of a "motive". In this talk, we shall discuss the theme of transcendental values of L-functions and report on some recent joint work with Ram Murty on L-functions that arise from characters of class groups.
Refreshments will be served in N620 Ross Building at 3:30 p.m.
ABSTRACT: We will report recent development in fast multiscale methods for solving operator equations including both differential equations and integral equations. In particular, we will present multiscale augmentation methods based on multiscale wavelet-like bases. Such methods are designed based on the principle that basis functions for solving an operator equation are chosen to optimize the approximation accuracy of the approximate solution and computational efficiency in solving the resulting discrete system. We will present both theory and numerical examples.
Refreshments will be served in N620 Ross Building at 3:30 p.m.
ABSTRACT: In this talk, I will give a description on how to give an estimate of the first two eigenvalues of the Schrodinger operator. The estimate does not depend on perturbation analysis.
Refreshments will be served in N620 Ross Building at 3:30 p.m.
ABSTRACT: In recent years the technique of multiplier ideal sheaves has been developed into a very powerful tool in algebraic geometry and in the application of algebraic geometric methods to estimates in analysis. This talk is a survey on its historic development and most recent results obtained from it with a discussion on open problems.
Refreshments will be served in N620 Ross Building at 3:30 p.m.
ABSTRACT: In recent years, spectral methods have become increasingly popular among computational scientists and engineers because of their superior accuracy and efficiency when properly implemented. In this talk, I shall present fast spectral-Galerkin algorithms for some prototypical partial differential equations. These spectral-Galerkin algorithms have computational complexities which are comparable to those of finite difference and finite element algorithms, yet they are capable of providing much more accurate results with a significantly smaller number of unknowns. A key ingredient for the efficiency and stability of the spectral-Galerkin algorithms is to use (properly defined) generalized Jacobi polynomials as basis functions.
I shall illustrate applications of these fast spectral-Galerkin algorithms to a number of scientific and engineering problems, including, nonlinear wave equations, acoustic scattering and mutli-phase flows.
Refreshments will be served in N620 Ross at 3:30p.m.
ABSTRACT: It is well known that looking at equivalence classes may result in pathology--both topological and Borel. In fact, in the context of a category, with isomorphism as equivalence, there may already be a purely algebraic pathology. (The isomorphism classes may not form a category.) In the theory of operator algebras, this difficulty has been circumvented by the use of a classification functor--by definition mapping from a given category to a second, classifying, category. The reason this can exist (even in non-trivial cases) is that one does not insist that two isomorphic objects be given the same value of the invariant, but only that these values be isomorphic.
While such a functor cannot in general be constructed by ignoring arbitrary automorphisms, it turns out that inner automorphisms (or an abstract analogue of these) can be safely ignored. In settings where such automorphisms exist--not only in the case of an algebra!--a classification functor can be constructed--either naively, by simply dividing out by the (groups of) inner automorphisms, or in a more interesting way by looking at the closures of the resulting equivalence classes of maps, in a topology with suitable properties. As an example of a category for which this construction of a classification functor works, one has that of countable groups. As well as having historical roots in the (analogous) theory of operator algebras, it appears that the construction in this case has a precedent to some extent in the theory of groups.
Refreshments will be served in N620 Ross Building at 3:30p.m. and in Grad Lounge after the talk.
ABSTRACT: The problem of densely packing non-overlapping congruent spheres in Euclidean space of various dimensions N has a long history and rich connections with various aspects of modern pure and applied mathematics. For example, the Leech lattice yields a remarkably dense and symmetrical packing in dimension N=24, conjectured to be the densest possible in this dimension. In this packing, each sphere touches 196560 others. It has been known for some time that this "kissing configuration" is optimal. We review the proof of this and related facts, and explain why the method cannot be used directly to study the sphere-packing problem. We then outline recent work on the sphere-packing problem, culminating in the proof by H.Cohn and A.Kumar that the conjectured optimality of the Leech packing holds under the condition that the centers of the spheres lie on a lattice (previously the densest lattice packing was known only for N=1 through N=8). Without the lattice condition, they show that any packing cannot improve on the Leech density by more than 1 part in 10^30. They also give strong evidence for the existence of mysterious functions of a positive real variable that would prove the optimality of the Gosset (E_8) and Leech packings of spheres for N=8 and N=24.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: Joint distributions for non-commuting variables, and the associated ideas such as the Weyl transform, were introduced about seventy years ago in quantum mechanics and, independently, in engineering. In quantum mechanics they can be thought of as representations of the density matrix, and are used to facilitate calculations and to formulate quantum mechanics in phase space. In engineering they arose out of the need to describe how the spectral content of a function is changing in time. Time-varying spectra is one of the most primitive sensations we experience since we are surrounded by light of changing color, by sounds of varying pitch, and by many other phenomena whose periodicities change with time. Hence, the need to develop the physical and mathematical ideas required to understand what a time-varying spectrum is. We will describe why this is an elegant and challenging problem, the immense strides that have recently been made, and how these methods impinge on issues in probability theory, differential equations, approximation methods, and other aspects of mathematics.
Refreshments will be served in N620 Ross at 3:30p.m.
ABSTRACT: A unital C*-algebra can be thought of as a noncommutative version of a compact space (more accurately, as a noncommutative version of the algebra of continuous functions on a compact space). Noncommutative C*-algebras can sometimes be used as substitutes for compact spaces in situations in which no reasonable space exists. One example is the orbit space of a minimal homeomorphism, such as the rotation by an irrational multiple of 2 pi on the circle. The orbit space itself is an uncountable set with the indiscrete topology. For irrational rotations, the C*-algebras, called irrational rotation algebras, preserve information about the rotation angle, and this information is visible in their algebraic topology.
An irrational rotation algebra can also be obtained as the C*-algebra generated by two unitaries which commute up to a scalar (exp (i r) for rotation by r). It is a simple C*-algebra. Thought of this way, it is a noncommutative deformation of (the continuous functions on) the 2-torus. A higher dimensional noncommutative torus is a C*-algebra obtained as an analogous deformation of a higher dimensional torus. Most of these algebras are simple. Our main result is a structure theorem for the simple case, generalizing the Elliott-Evans theorem for irrational rotation algebras, which allows one to determine exactly when two of them are isomorphic. Surprisingly, actions of finite groups are used in the proof.
ABSTRACT: In 1904, Prandtl introduced the boundary layer method (matched asymptotics) which has become one of the most important tools in applied mathematics. Despite its usefulness, there still does not seem to have a rigorous analysis to validate the accuracy of Prandtl's treatment. In this lecture, we will point out that errors can occur even in very well-known formulas in boundary layer theory, and present some rigorous results for two non-linear two-point boundary value problems, involving Carrier-Pearson equations.
Refreshments will be served in N620 Ross Building at 3:30p.m.
ABSTRACT: In the 1830s and 1840s, John Scott Russell observed and studied a solitary surface water wave that he called the "great wave of translation". In 1895, Korteweg and de Vries (KdV) derived their equation that describes these solitary waves. Seventy years later, in 1965, Kruskal and Zabusky discovered that the solitary wave solutions of the KdV equation have the remarkable property of retaining their identities after collisions with other solitary waves. They gave these special waves the name "solitons." This discovery motivated a more detailed mathematical study of the KdV equation, including a search for conservation laws for the KdV equation that eventually led to devising the "inverse scattering method" for exact determination of the N-soliton solutions. In this talk, I will describe some of these discoveries and their histories. One feature I will demonstrate is that progress in science can be strongly influenced by non-scientific events and circumstances. These discoveries now have been extended and generalized to many different equations and applications, and has led to the new mathematical field of integrable systems.
ABSTRACT: Long-range aperiodic order, as it is presently understood, refers to discrete structures in space (for example, things like Penrose tilings and quasicrystals) that carry very high internal order--notably in the form of pure point diffraction--but lack global translational symmetry. Somehow there is still a lot of inherent symmetry, but the question is how to express it mathematically.
In this talk we will introduce these ideas along with examples, and show how we can use the theory of dynamical systems and their spectral theory along with some ideas from the theory of point processes to get a better understanding of this phenomenon. The talk is intended for a general mathematical audience.
Refreshments will be served in N620 Ross Building at 3:30p.m.
ABSTRACT: Many vector-valued functions, representing expensive computations, are also structured computations. In this case the calculation of the Newton step can be greatly accelerated by exploiting this structure. It is often not necessary, nor economic, to form the true Jacobian in the process of computing the Newton step; instead, a more cost-effective auxiliary Jacobian matrix is used. This auxiliary matrix can be sparse even when the true Jacobian matrix is dense; consequently, sparse matrix technology can be used, to great speed advantage, both in forming the matrix and in solving the auxiliary linear system.
Refreshements will be served at 3:30p.m. in N620 Ross.
ABSTRACT: The Clay Mathematics Institute offers a million dollars to anyone solving the question of whether P = NP. We discuss the importance of the problem and explain why most complexity theorists believe P not= NP, despite the dramatic success of programs solving huge instances of the satisfiability problem. We show how this question is related to other fundamental problems in complexity theory, including the entangled problems of whether NP has polynomial size circuits, and whether some problems are inherently easier to solve using a source of random bits. The question of whether NP = coNP motivates the important field of propositional proof complexity.
Refreshments will be served at 3:30p.m. in N620 Ross.
Please refer to this link for abstract.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: We are interested in the general problem of detecting connectivity, or high correlation, between pairs of pixels or voxels in two sets of images. To do this, we set a threshold on the correlations that controls the false positive rate, which we approximate by the expected Euler characteristic of the excursion set. An exact expression for this is found using new results in random field theory involving Lipschitz-Killing curvatures and Jonathan Taylor's Gaussian Kinematic Formula. The first example is a data set on 425 multiple sclerosis patients. Lesion density was measured at each voxel in white matter, and cortical thickness was measured at each point on the cortical surface. The hypothesis is that increased lesion density interrupts neuronal activity, provoking cortical thinning in those grey matter regions connected through the affected white matter regions. The second example is an fMRI experiment using the "bubbles" task. In this experiment, the subject is asked to discriminate between images that are revealed only through a random set of small windows or "bubbles". We are interested in which parts of the image are used in successful discrimination, and which parts of the brain are involved in this task.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: One approach to the classification of spaces (Riemannian manifolds, complex manifolds) is to study deformations of the structure up to the action of the symmetry group. In this talk, we will describe the general approach and then show how one needs to deform the analysis in order to apply it when the symmetry group is the group of contact diffeomorphisms.
Refreshments will be served in N620 Ross Building at 3:30p.m.
ABSTRACT: We will analyze the familiar Sudoku puzzle and consider its generalizations from a mathematical standpoint. We will also discuss some open problems concerning the puzzle. This is joint work with Agnes Herzberg.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: Available upon request.
The Natural Sciences and Engineering Research Council support this work.
Richard Frayne, PhD, Canada Research Chair in Image Science/AHFMR Senior Scholar/Associate Professor, Departments of Radiology and Clinical Neurosciences, and Hotchkiss Brain Institute, University of Calgary; and Seaman Family MR Research Centre, Foothills Medical Centre, Calgary Health Region.
Refreshments will be served at 3:00p.m. in N620 Ross.
ABSTRACT: This talk will describe several inverse problems in partial differential equations arising in medical imaging. These problems have a rich and beautiful mathematical structure, leading to nonlinear harmonic analysis as well as hard questions in differential geometry.
Refreshments will be served in N620 Ross Building at 3:00p.m.
ABSTRACT: There have been extensive investigations on traveling waves and the long-time behavior of solutions in terms of asymptotic speeds of spread for various evolution systems. The concept of asymptotic speeds of spread (in short, spreading speeds), introduced by Aronson and Weinberger in 1975 for reaction-diffusion equations, is a fundamental one in the study of biological invasion and disease spread. There is an intuitive interpretation for the spreading speed in a spatial epidemic model: if one runs at a speed greater than it, then one will leave the epidemic behind; whereas if one runs at a speed less than it, then one will eventually be surrounded by the epidemic.
In this talk, we first give a brief review of spreading speeds, traveling waves and global stability in evolution systems. Then we present a general theory for monotone semiflows, which shows that the spreading speed coincides with the minimal wave speed. Finally we discuss the applications of this theory to a nonlocal lattice population model, a multi-type SIS epidemic model, and a vector disease model with spatial spread.
Refreshments will be served in N620 Ross Building at 3:00p.m.
ABSTRACT: The KAM theory due to Kolmogorov, Arnold, and Moser is a fundamental theory in Hamiltonian systems which is closely related to important dynamical issues such as the existence of quasi-periodic motions and stabilities. The lecture will give a survey on the history and recent development of the theory for nearly integrable Hamiltonian systems and Poisson-Hamilton systems.
The host for Yingfei Yi is Huaiping Zhu.
Refreshments will be served at 3:00p.m. in N620 Ross.
ABSTRACT: This talk will discuss data that is collected by placing sensors on a subject's body as he walks a treadmill. In this case we will consider exploratory data analysis for the knee joint. Considering the upper and lower legs as rigid bodies, the motion of one relative to the other can be described by a path in the rotation group in three dimensions SO(3). Our approaches to exploratory data analysis will be deeply rooted the geometry of SO(3).
The talk will be highly preliminary in that it is based upon a few days of conversations between the speaker, Peter Kim of Univ. of Guelph Statistics, and Michael Pierrynowski of McMaster Univ. Rehabilitation Sciences. Any suggestions from the audience will be highly welcome.
Refreshments will be served at 3:00p.m. in N620 Ross.
ABSTRACT: It is well known that the standard model of asset price processes, Geometric Brownian Motion, is not capable of reproducing the fat tails in observed price distributions. From a risk management point of view, the most troubling aspect of commonly used models is their inability to provide a useful hedging strategy in the presence of jumps.
If the price process follows a jump diffusion, it is well known that a perfect hedge is not possible with a finite number of hedging instruments. It is also conventional wisdom that hedging with options is too expensive, due to the large transaction costs typical of the option market.
In this study, we suggest a dynamic hedging strategy based on hedging with the underlying and liquid options. We solve an optimization problem at each hedge rebalance date. We minimize both the "jump risk" and the transaction cost.
Simulation studies of this strategy, using typical market bid-ask spread data for the options, shows that even using the underlying and three options in the hedge portfolio does an excellent job of minimizing jump risk, as well as being not too costly in terms of total transaction costs.
Refreshments will be served in N620 Ross at 3:00p.m.
ABSTRACT: As soon as Paul Cohen showed in 1964 that the continuum hypothesis is not provable in ZFC, it was evident that many questions in abstract analysis were likely to be undecidable. In this talk I will discuss some of the topics in measure theory, both on the real line and in abstract measure spaces, in which such questions have led to substantial new theorems of ZFC, and in turn to new problems.
Refreshments will be served at 3:00p.m. in N620 Ross.
ABSTRACT: We survey different types of asymptotic results for orthogonal polynomials including the asymptotics of the polynomials around their largest zeros (Plancherel-Rotach asymptotics) which give rise to the Airy function and Airy kernels. The asymptotics of q-orthogonal turned out to have new features and lead to a q-analogue of the Airy function.
Refreshments will be served at 2:30p.m. in N620 Ross.
ABSTRACT: MCMC algorithms are a very popular method of approximately sampling from complicated probability distributions. A wide variety of MCMC schemes are available, and it is tempting to have the computer automatically "adapt" the algorithm while it runs, to improve and tune on the fly. However, natural-seeming adaptive schemes often fail to preserve the stationary distribution, thus destroying the fundamental ergodicity properties necessary for MCMC algorithms to be valid. In this talk, we review adaptive MCMC, and present simple conditions which ensure ergodicity (proved using intuitive coupling constructions, jointly with G.O. Roberts). The ideas are illustrated using a very simple example: an adaptive Metropolis algorithm on a six-point state space, animated by the java applet at probability.ca/jeff/java/adapt.html. This talk is intended for a general audience.
Refreshments will be served in N620 Ross Building at 3:00p.m.
ABSTRACT: Advanced-Retarded Functional Differential Equations arise in a surprisingly wide range of applications, most recently receiving significant attention because travelling wave solutions to lattice differential equations are defined by FDE boundary value problems on an unbounded domain. The presence of advances as well as delays makes the analysis of these problems tricky. Analytical studies would be greatly aided by the existence of good numerics. We propose two methods, one a collocation boundary value problem method, and the other an embedding of the FDE in a partial differential difference equation. We discuss the problem of truncation to a finite computational interval, and definition of suitable boundary functions in some detail. Travelling wave solutions of a spatially discrete Nagumo equation illustrate the issues that arise.
Refreshments will be served at 3:00p.m. in N620 Ross.
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: 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: 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: 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: 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: Experimental observations indicate that high molecular weight polymers in very dilute solution undergo a phase transition from an expanded open coil state to a compact state as the solvent quality or temperature is decreased. This transition is known as the collapse transition. The existence of a collapse transition has not been proved for the standard models of linear and ring polymers, that is the self-avoiding walk and self-avoiding polygon models, however, numerical evidence strongly supports its existence. The numerical evidence is also consistent with the conclusion that the the collapse transition of linear and ring polymers occurs at the same critical temperature.
In this talk, the known rigorous and numerical results related to the collapse phase transition of self-avoiding polygons and walks will be reviewed. Recent rigorous results will also be discussed. In particular, it will be shown that self-avoiding polygons in $\mathbb{Z}^2$, the square lattice, collapse at the same critical point as a wider class of lattice subgraphs, namely closed trails with a fixed number of vertices of degree 4. This is proved by establishing combinatorial bounds on closed trails in terms of self-avoiding polygons. A similar approach can be used to relate open trails with a fixed number of vertices of degree 4 and self-avoiding walks.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: We discuss how to establish differential equations for general orthogonal polynomials and compare this with Sturn-Liouville systems. We also discuss the connection with the Coulomb gas problem concerning the equilibrium position of N charged particles in an external field. Finally the role of discriminants is discussed and we show how to use them to give closed form expression for the energy of the system at equilibrium. q-analogues may be mentioned.
Refreshements will be served at 3:30p.m. in N620 Ross.
ABSTRACT: The theory of quasigroups and loops is a fairly young discipline which takes its roots from geometry, algebra and combinatorics. In geometry, it arose from the analysis of web structures; in algebra, from non-associative products; and in combinatorics, from Latin squares.
Today it has applications in many different parts of mathematics and physics (algebraic nets, differential geometry, designs theory (Steiner's systems), coding and encoding, cryptography, graph theory, ...)
Any associative quasigroup is a group and vice versa. Therefore the identities different from associativity are of the interest. We shall say a few words about the law of mediality ab . cd = ac . bd and relevant identities.
Refreshments will be served at 2:30p.m. in N620 Ross.
ABSTRACT: In this work, we study the stability for some linear neutral partial functional differential equations where the generator is not densely defined but satisfies the Hille-Yosida condition. In the nonhomogeneous case, we study the problem of the existence of periodic solutions.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: One of the basic tools in the study of nonlinear differential equations is normal form theory. It is particularly useful for analyzing the dynamical behaviour of a system such as instability and bifurcations. In solving physical and engineering problems it usually requires to compute the explicit expression of a normal form for a given system, which is not easy and very time consuming. Recently, further reduction of conventional normal forms (CNF) has received considerable considerations. Computing the simplest normal form (SNF) of differential equations is much more difficult than that of CNFs. Although computer algebra systems have been introduced in computing SNFs, novel methodologies are needed for improving computational efficiency. This talk will give a brief review of the recent development on the computation of SNFs, and presents several methods we have developed for computing the SNFs. Possible applications of SNFs will also be discussed.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: General classification problems and finite metric spaces In the seventies, a general program was proposed in Descriptive Set Theory to the effect of understanding simply definable (in particular Borel) mathematical structures. The motivation was to try to draw a dividing line between well-behaved and pathological structures, the existence of those being hopefully due only to the too wide generality of the notion of arbitrary set or structure.
I will talk about dividing Borel equivalence relations on Polish (separable, completely metrizable) spaces into ‘simply classifiable' and ‘wild' ones.
Some familiar examples of ‘simply classifiable' equivalence relations: Homeomorphism of compact orientable surfaces: two surfaces are homeomorphic if and only if they have the same number of ‘handles'. Similarity of $n\times n$ complex matrices: two matrices are similar if and only if they have the same Jordan canonical form. Isomorphism of countable torsion abelian groups: two groups are isomorphic if and only if they have the same ‘Ulm-invariants'.
It is more difficult to prove that an equivalence relation is not simply classifiable, and this problem will be the subject of my talk.
ABSTRACT: We consider Gibbs samplers for uniform distributions on regions. These samplers generalise the "slice sampler" Markov chains, which sample uniformly on the region underneath the graph of a density function. We examine the mathematical properties of these algorithms. In particular, we prove quantitative bounds on the convergence times of slice samplers, as well as very general qualitative convergence rates. We also show that certain convergence properties depend crucially on the smoothness of the boundary of the region itself. Finally, we describe the "polar slice sampler" modified algorithm.
This is joint work with G.O. Roberts (Lancaster University).
Refreshments will be served in N620 Ross at 3:00p.m.
ABSTRACT: We study the existence of positive solutions of some non-local boundary value problems, known as three point BVPs. We use the classical fixed point index for compact maps and some recent results of K.Q.Lan to establish the existence of at least one or at least two positive solutions.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: Let I[a,b] be the class of all subintervals, open or closed at either end, of an interval [a,b], and let mu be a real-valued additive and upper continuous function on I[a,b] which has bounded p-variation, 1\leq p<2. In the class of all real-valued (point) functions on [a,b] of bounded q-variation with 1/p+1/q>1, consider Kolmogorov-integral equations with respect to mu. This leads to a set up different from the customary constructions both in the classical and stochastic calculi. In this talk, we discuss several new important aspects of this set up.
Refreshments will be served in N620 Ross at 2:30p.m.
ABSTRACT: Direct numerical simulations(DNS) of multi-particle systems are conducted. In the present simulation, so-called immersed boundary method is used and the boundary conditions on each disperse phase are treated using the grids much smaller than the size of each disperse phase. Periodic box is used for the simulation to extract the averaged quantities. The dependence of drag coefficient on void fraction is investigated. The results show good agreement with the existing theory and experiment. The averaged flow fields around each spherical particle are reconstructed using the DNS data. They are expressed by the spherical harmonics to reduce the size of information.
Using these expressions, Sub-Grid Scale modeling is performed. Following the similar approach of the SGS model for the turbulence, we examine the several types of SGS models for dispersed flows. The results show that the present nonlinear model gives the better correlation with DNS results than Smagorinsky or Scale-Similarity type models.
Refreshments will be served in N620 Ross at 2:30p.m.
ABSTRACT: A class of algorithms for both structured and unstructured mesh adaption will be presented based on coordinate transformations. Various forms of the mesh equation that determines the coordinate transformation will be given for the one and higher dimensional cases. Practical issues of the implementation of the algorithms, including spatial balancing, scaling invariance, mesh movement of boundary points, selection of the monitor function, and spatial smoothing, will be addressed. Numerical results will also be shown to demonstrate the ability of the method.
Refreshments will be served in N620 Ross Building at 2:30p.m.
ABSTRACT: A wavelet or multi-wavelet generates a convenient orthonormal system of functions, starting with a few seed functions. In the systems displayed in this talk, a new procedure is used to find the functions, and the reward is expansions that behave pleasantly under some familiar groups of symmetries of 2- or 3- or 4-space.
Refreshments will be served in N620 Ross Building at 2:30p.m.
Professor Mikhail A. Shubin, Northeastern University, will speak on "Spectra of Magnetic Schr\"odinger Operators" at 2:30p.m. in N638 Ross.
ABSTRACT: It is well known that the condition $V(x)\to\infty$ as $x\to\nifty$ implies that the Schr\"odinger operator $H=-\Delta+V(x)$ in ${\bf R}^n$ has a discrete spectrum (K. Friedrichs, 1934). In physical language, this means that if a classical particle cannot escape to infinity (being forced to remain in a potential well), then the corresponding quantum particle is also localized. Similar results about magnetic Schr\"odinger operators will be explained in the talk. They are formulated in terms of effective potentials which are constructed from both electric and magnetic fields. The most advanced of these results use the Wiener capacity and in the case of vanishing magnetic field coincide with the necessary and sufficient conditions given by A.M. Molchanov in 1953.
Refreshments will be served in N620 Ross Building at 2:00 p.m.
Professor and Dr James Elder, Centre for Vision Research, York University, will speak on "Are Images One-Dimensional?" at 4:00p.m. in N638 Ross.
ABSTRACT: Abstract: In this talk I will develop arguments for the
representation of images by one-dimensional contours that account for less
than 10% of pixels in a typical natural image. I will begin by considering
the problem of edge detection and local blur estimation, and the
fundamental problem of scale which arises in reliably identifying and
characterizing discontinuities in a noisy and blurred discrete signal. Our
solution employs a model of an edge involving 2 intensity parameters and 1
blur parameter. To evaluate our solution, we invent a method for
inverting the edge representation to reconstruct an approximation of the
original image. The perceptual fidelity of the reconstruction attests to
the accuracy and completeness of the representation.
In order to model higher levels of human visual processing, and to
produce useful computer applications, we must also consider the problem of
perceptual grouping, i.e. how to chunk the representation into more global
primitives. I will report results of a recent study that identifies the
relative inferential power of several cues to the perceptual grouping of
contours in natural images.
Finally, I will show how these algorithms for edge detection,
reconstruction and grouping can be combined to produce novel systems for
editing and compressing images.
Refreshments will be served in N620 Ross Building at 3:30p.m.
ABSTRACT: How old is AIDS? In this talk I will review some of the modern history of the disease and then describe one approach, phylogenetic analysis, to the question. The idea is to examine the genetic variation of this very rapidly evolving virus and extrapolate to the time when the sequences associated with the current epidemic might have shared a common ancestor.
Refreshments will be served in N620 Ross Building at 3:00p.m.
Professor Henri Darmon, McGill University, will speak on "Analytic Solutions of Diophantine Equations" at 2:00p.m. in N638 Ross.
ABSTRACT: The solution of Diophantine equations is one of the main
pre-occupations of number theory. In the 19th century, mathematicians such
as Kronecker, developed a rich and elegant theory, designated under the
heading of "complex multiplication", allowing the solution of certain
Diophantine equations, known as elliptic curves, through special values of
elliptic modular functions. This theory has been much developed and
generalized in the 20th century by Shimura and Taniyama, among others. It
has led to the work of Gross-Zagier and Kolyvagin, which yields profound
insights into the arithmetic of elliptic curves.
I will mention briefly at the end of my lecture a tentative first step
toward a generalized theory of complex multiplication whose development
would seem to constitute a worthwhile goal for 21st century number
theory.
Refreshments will be served in N620 Ross at 3:00p.m.
ABSTRACT: After describing the chemostat, a laboratory apparatus used for the continuous culture of micro-organisms, and thought of by some ecologists as a simple lake in a laboratory, I will discuss various mathematical models, their analysis, and the implications of the predictions of these models with regards to managing the environment.
Refreshments will be served in the Departmental Common Room, N620 Ross, at 3:30p.m.
ABSTRACT: In his famous book on unsolved problems, S. Ulam asked when an approximate homomorphism into a metric group can be approximated by a true homomorphism. Results related to maps into Banach spaces and unitary groups will be briefly reviewed. The main attention will be given to a recently discovered new stability case: measurable maps into a countable product of discrete groups where the distance is defined via a submeasure on the index set.
Refreshments will be served in N620 Ross Building at 2:00p.m.
ABSTRACT: odulation spaces are function spaces that quantify the joint localization in both time and frequency of functions or distributions. We will discuss the importance and nature of time-frequency localization. We will show how operators mapping the Hilbert space $L^2(R^d)$ into itself can be realized by time-frequency superpositions--this is the classical "pseudo-differential operator calculus." Each such pseudo-differential operator is determined by a symbol that is defined on $R^{2d}$. The time-frequency characteristics of the symbol translate into spectral properties of the operator. We present some sufficient conditions on the time and frequency decay of a symbol that imply that the corresponding pseudo-differential operator is trace-class or bounded.
Refreshments will be served in N620 Ross Building at 3:30p.m.
ABSTRACT: The first part of the talk will discuss the necessity of the classical hypotheses for existence of compact global attractors. The second part will concentrate on diffusively balanced conservation laws.
Refreshments will be served the Department Common Room, N620R, at 3:30p.m.
ABSTRACT: This lecture is addressed to the classification problem for two-dimensional nonassociative algebras, possibly without a unit. Whereas previous investigations of this problem were confined either to algebraically closed fields of characteristic not 2 or 3 or, more often, to the base field of real numbers, the approach to be presented here is intrinsic and works over arbitrary base fields. Its main ingredients are principal Albert isotopes and the well known (and easy) classification of unital nonassociative algebras of dimension 2.
Refreshments will be served in N620 Ross Building at 3:30 p.m.
ABSTRACT: While quite diverse categories arise in mathematical practice, nonetheless group theory, lattice theory, linear algebra, etc. have a few distinctive features in common. The ubiquity of the contrast between an object and its many presentations and the Noetherian control of reduction modulo reflexive data are among the features well worth making explicit as a guide to using algebra.
Refreshments will be served in the Common Room, N620R, at 3:30p.m.
ABSTRACT: Each Discrete Cosine Transform (DCT) uses N real basis vectors whose components are cosines. These basis vectors are orthogonal and the transform is much used in image processing (drawbacks will be pointed out). The cosine series can be quickly computed by the fast Fourier transform. But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are in applications.
We prove orthogonality in a different way. Each DCT comes from the eigenvectors of a symmetric second-difference matrix. By varying the boundary conditions, we get the established transforms DCT-1 through DCT-4 (and also four more orthogonal bases of cosines). The boundary condition determines the centering (at a meshpoint or a midpoint) and decides on the entries cos [j or j+0.5] [k or k+0.5] pi/N .
Then we discuss bases from filter banks and wavelets. The key is to create a banded block Toeplitz matrix whose inverse is also banded. The algebra shows how the approximation properties of the wavelet basis are determined by the polynomials that can be reproduced exactly by wavelets.
In signal processing, so much depends on the choice of a good basis.
Refreshments will be served in N620 Ross at 2:30p.m.
ABSTRACT: We address the inverse problem of the identification of a passive three-dimensional impenetrable object in a shallow water environment. The latter is assumed to have flat reflecting (sound-soft top and sound-hard bottom) boundaries and therefore acts as a guide for acoustic waves. These waves are employed to interrogate the object and the scattered acoustic wavefield is measured (actually simulated herein) on the surface of a (virtual) vertically-oriented cylinder (of finite or infinite radius, corresponding to near-or far-field measurements) fully enclosing the object. To avoid the so-called inverse crime when simulating measurements, the corresponding direct scattering problem is resolved using a variant of a method due to Bruno and Reitich. The inverse problem is resolved using a variant of the canonical domain method.
Refreshments will be served in N624 Ross Building at 2:30p.m.
Refreshments will be served at 2:30p.m. in the Departmental Common Room, N620 Ross.
ABSTRACT: The Langlands programme is a series of far-reaching conjectures that relate analytic objects to fundamental arithmetic information. The trace formula is a powerful tool that has the potential to resolve some of these conjectures. We shall discuss the trace formula in general terms, and the problem of putting it into a form that can be applied to the conjectures.
Refreshments will be served in the Departmental Common Room, N620 Ross, at 2:30p.m.
ABSTRACT: Models of competition between normal and cancer cells are proposed with a chemotherapy agent acting as a predator on both. Analyses include three possible chemotherapy inputs; constant finite, constant continuous and periodic. Criteria for persistence or extinction of the cancer are derived and the existence of small amplitude periodicity is also investigated.
Refreshments will be served at 2:30p.m. in the Departmental Common Room, N620 Ross.
ABSTRACT: In recent years probabilistic methods have been used in the study of several seemingly unrelated problems concerning finite (and certain infinite)groups. These include conjectures concerning simple groups, permutation groups, free groups, and the modular group. I will survey some of these results, giving some hints of proofs. If time permits I will also discuss some open problems.
Refreshments will be served at 2:30p.m. in the Departmental Common Room--N620 Ross.
ABSTRACT: The Heisenberg group is the prototype of nilpotent Lie groups. I discuss the representation theory for this gorup and for its convolution algebra of bounded regular Borel measures. There are interesting connections to the "Berezin-Toeplitz quantization" and Weyl pseudo-differential operators.
Refreshments will be served at 2:30p.m. in the Departmental Common Room--N620 Ross.
ABSTRACT: We present a selection of recent results and questions related to topological groups. Among covered topics are: (i) convergent sequences in compact groups; (ii) closed subspaces which generate compact gorups algebraically or topologically; (iii) topology of independent subsets of compact groups; (iv) dimension theory of topological groups; (v) generating classes of topological groups via limit laws; (vi) categorical embeddings into topological groups.
The lecture requires only elementary knowledge of topology and group theory.
Refreshments will be served at 2:30p.m. in N620 Ross.
ABSTRACT: We will discuss a common "root" of the following three problems:
(a) existence of periodic solutions of non-autonomous dynamical systems whose principal parts respect certain group symmetries (Mawhen, Krasnoselskii's results and related topics);
(b) classification of represetnations of finite p-groups up to certain nonlinear equivalences (Atiyah-Tall Theorem and related topics);
(c) lower estimate of the number of critical orbits of smooth invariant functionals (Lusternik-Schnirelman Theory and related topics).
To study the above "root" we will present our recent Brouwer degree results for equivariant maps.
Applications to problems (a)-(c) as well as open questions will be also discussed.
Refreshments will be served at 3:00p.m. in the Departmental Common Room (N620 Ross)
PLEASE TAKE NOTE OF THE TIME. THIS IS HALF AN HOUR EARLIER THAN OTHER COLLOQUIA.
ABSTRACT: A simplified geometric model for a solid state is a discrete point set admitting three linearly independent translations (i.e. a 3-periodic Delone set). Every such point set D is "strictly ordered", that is to say: (1) For every radius rho htere are (up to rigid motions) only finitely many subsets od D fitting into a ball of radius rho.
There are many other structures satisfying (1) but failing to be 3-periodic. Some of them seem to yield good models for some quasicrystals. For others it is an open question, whether or not they occur in physical reality. Despite (1) most of these structures are "non-local" in the sense of Penrose (modified by van Ophuysen).
Refreshments will be served at 3:30p.m. in the Departmental Common Room--N620 Ross.
ABSTRACT: Investigation of the dynamics of majority nets (graphs of finite degrees with vertices loaded by a + or - charge, that reacharge simultaneously by local majority rule) reveals some surprising properties. These properties could be metaphors for the mysterious physical phenomena of transition from classical to quantum condition, and of phase transition.
ABSTRACT: Classical Galois theory was extended to schemes and successfully applied to abstract algebraic geometry by A. Grothendieck many years ago. The purpose of this talk is to describe (briefly) how one can further extend Galois theory to general categories, and to describe the theory of central extensions of groups and other algebraic structures as a new "non-Grothendieck" Galois theory.
ABSTRACT: Finance and risk management are nowadays a rich source of challenging mathematical problems. After a review of the subject, this talk addresses two recent results in this area. The problem of Value at Risk can be formulated as a problem of gaussian integration on the exterior of a quadric. This is an old problem first considered by Ruben in the 60s. Risk management makes one look at it in a new light and motivates the development of new elegant applications of harmonic analysis techniques. The problem of calibration risk is also a subject of high industrial interest. In this case, homogeneization techniques for stochastic differential equations developed by Papanicolao in the 70s are helpful and lead to new and more stable calibration techniques.
ABSTRACT: John McKay noticed in 1978 that 196884 nearly equals 196883: the former is the first nontrivial coefficient of the j-function, which generates all functions invariant under the modular group PSL(2,Z); the latter is the dimension of the first nontrivial irreducible representation of the Monster, which is the largest of th sporadic finite simple groups. Soon after that Conway and Norton proposed a number of conjectures, called Monstrous Moonshine, generalizing and exploring McKay's observation. Though most of these conjectures are now proven, these connections between finite groups and modular functions remain very mysterious, and connections with other algebraic structures are being made. In this elementary talk I will describe the Conway-Norton conjectures, their proof by Borcherds and Frenkel-Lepowsky-Meurman, and may own contribution to the subject.
ABSTRACT: In 1880 Poincare wrote an essay for a prize competition on differential equations. He followed this essay with three supplements that document his path to the discovery of automorphic functions. They have never been published, and they shed an interesting light on his way of working, and the importance non-Euclidean geometry had for him. In the 1920s Jacques Hadamard wrote a monograph for a Russian audience on Poincare's work, which was published in 1951 in Russian; it seems that the French original is lost. This fascinating commentary is now being translated by Abe Shenitzer, and it forms the subject of the secon dpart of the talk.