Given a space of ``states'', a dynamical system on this space is described by a semiflow, that is by a mapping taking an initial state at time zero to the state of the system at a subsequent time. A monotone dynamical system is one on an ordered metric space which has the property that ordered initial states lead to ordered subsequent states. The theory of monotone dynamical systems has found many effective applications in the study of population biology, chemical reactions, epidemic dynamics and neural network models of various differential equations.
This series of talks will give a detailed introduction to the main theory of monotone dynamical systems. By introducing the basic definitions and developing some tools, we will arrive at the main result, for the strongly order-preserving dynamical systems, of the generic quasiconvergence (i.e., the typical orbit converges to the set of all equilibrium points) and generic convergence (i.e., the typical orbit converges to an equilibrium point). Some other important results of their own interests such as the convergence criterion, the limit set dichotomy, the sequential limit trichotomy, the order interval trichotomy, the global asymptotic stability and connecting orbit theorems will also be proved.
This series of talks is accessible for general audience including graduate students. Background in differential equations or dynamical systems will be introduced. Knowledge in general topology is the only necessity of the audience.
In the talk I will survey some recent results on real functions that have been proved using modern set theoretic methods. I will also discuss some recent results concerning cardinal functions in analysis. These cardinal invariants are associated to different classes of real functions and are defined in the context of closing these classes into addition and multiplication operations.
The talk will contain the following results:
(1) In ZFC, there is a +-Ramsey filter which is not first countable;
(2) (Dow) Under p=c, there is a +-Ramsey filter which is not \alpha_3;
(3) If we add Cohen reals(any number) in a model of CH, then in the model there is a +-Ramsey filter which is \omega_1 generated but not \alpha_3.
The third uses Simon's construction of a compact Frechet whose square is not Frechet.
These partially answer Michael Hrusak's questions. Michael is not yet satisfied because the following question is still open: Open Question: There is in ZFC a +-Ramsey filter which is not \alpha_3?
(I already spoke on the topic at the U of T seminar some time ago)
We'll show that for a discrete semigroup S there exists a uniquely determined complete Boolena algebra B(S) - the algebra of clopen subsets of M(S). M(S) is the phase space of the universal minimal dynamical (topological) system for S and it is an extremally disconnected compact Hausdorff space. We shall deal with this connection of semigroups and complete Boolean algebras, focusing on structural properties of these algebras. We shall show that B(S) is either atomic or atomless; that B(S) is weakly homogenous provided S has a minimal left ideal; and that for countable semigroups B(S) is semi-Cohen (and for special cases even Cohen). We shall also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reflection G(S) of a group-like semigroup S can be constructed via universal minimal dynamical system for S and, moreover, B(S) and B(G(S)) are the same.
For those who attanded my previous talk on the topic at U of T, there will not be too much of a new material, but solutions to some of the unresolved problems of that time will be presented.
Theorems of characterization of normality are proved using extendability of continuous real-valued functions, upper and lower semicontinuous functions and sequences of locally finite families of open sets.
Extendability of continuous functions
A theorem that characterizes the extendability of a bounded real-valued continuous function defined on a subset of a topological space is proved. The Tietze-Urysohn Theorem becomes a corollary of this new result.
We prove some consistency results in set theory and topology.
We prove from the existence of a supercompact cardinal that there is a representable algebras with a non representable subalgebra.
We prove following results regarding embedding certain space into a regular pseudoradial spaces.
We also construct, in ZFC, an ultrafilter p on \omega with a special almost disjoint refinement and a strong Noetherian base.