Math Colloquium: Fri at 2:30pm-3:30pm, CH 443

Title: Cell Structures

Abstract: Alexsandroff in 1926 began approximating compact metric spaces by polyhedra in order to establish the foundations of algebraic topology for metric compacta and more general spaces. Compact metric spaces are obtained as inverse limits of inverse sequences of finite polyhedra. A sequence of mappings between factor spaces which commute with bonding mappings of two such inverse spectra defines a mapping between the the inverse limit spaces. Unfortunately the converse is not true in general. It took more than 60 years to fully develop Alexsandroff's idea. The resulting machinery is daunting.

Twenty five years ago Debski realized that the trouble with Alexsandroff's approach was that it requires too much information to be encoded in the inverse systems.

Debski-T in a paper to appear in Coll. Math.  define cell structures as inverse sequences of graphs (i.e. discrete sets with a reflexive and symmetric relation) with mild convergence conditions. They also define cell maps between cell structures. In that paper they show that these suffice to define all complete metric spaces and mappings between such spaces.

In this talk I will show by  how to define the real line and mappings of the line to itself using such discrete approximations. I will then explain how to extend the ideas in that example to obtain all topologically complete spaces and continuous maps between such spaces.

Conclusion: Most reasonable spaces and continuous mappings are determined by families of discrete approximations.
Math Colloquium: Fri at 2:30pm-3:30pm, CH 443

Title: On Weyl's embedding problem in Riemannian manifold

Abstract: Weyl's embedding problem is a classic isometric embedding problem in differential geometry. It concerns how to isometrically embed (S^2,g) with positive Gauss curvature into R^3. Recent study of quasi-local mass in General Relativity motivates us to study Weyl's embedding problem in general Riemannian manifold. In this talk, we focus on the mean curvature estimate for the embedding, which is the crucial part in view of continuity method. Together with work by Li-Wang, we are able to obtain an isometric embedding in certain Riemannian manifold. Moreover, we are able to recover the classic Weyl's embedding theorem in space form under the condition that g\in C^2 with D^2g Dini continuous, which is new even in Euclidean space.
Math Colloquium: Fri at 2:30pm-3:30pm, CH 443

Title: Semigroups of branched mapping classes

Recent work on classification problems arising in one-dimensional complex analytic dynamics suggest an underlying theory of mapping class semigroups. Let S2 denote the two-sphere, and fix a finite set P ⊂ S2. The set BrCov(S2, P) of orientation-preserving branched covering maps of pairs f: (S2, P) → (S2, P) of degree at least two and whose branch values lie in P is closed under composition and under pre- and post-composition by orientation-preserving homeomorphisms h: (S2, P) → (S2, P) fixing P set wise. Composition descends to a well-defined map on homotopy classes relative to P, yielding a countable semigroup BrMod(S2, P). In addition to the semigroup structure, BrMod(S2, P) is naturally equipped with two commuting actions of the mapping class group Mod(S2, P), induced by pre- and post-composition. This richer biset structure, and a related circle of constructions, turn out to be extraordinarily useful in this context. They lead to: algebraic invariants of elements of BrCov(S2, P) and an analog of the Baer-Dehn-Nielsen theorem; analogs of classical Hurwitz classes; conjugacy invariants; canonical decompositions and forms; an analog of Thurston's trichotomous classification of mapping classes; and induced dynamics on Teichmüller spaces.

This talk is based on algebraic and dynamical perspectives growing out of work of L. Bartholdi and V. Nekrashevych, and is based on ongoing conversations with S. Koch, D. Margalit, and N. Selinger.
Math Colloquium: Fri at 2:30pm-3:30pm, CH 443

A brief on short descriptions

Abstract: Given a binary string, can one find a short description for it?  The well-known answer is "no, Kolmogorov complexity is not computable.''  Faced with this barrier, one might instead seek a short list of candidates which includes a laconic description.  In fact, efficiently computable short lists do exist, and I will discuss the extent to which one can obtain them.  This talk will include a gentle introduction to Kolmogorov complexity followed by a discussion connecting list approximations to classical combinatorics and randomness extraction.  The program will roughly follow a recent SIGACT News survey coauthored with Marius Zimand.
Math Colloquium: Fri at 2:30pm-3:30pm, CH 443

Title: The Corona Theorem and its Applications

Abstract: The original proof of the Corona Theorem is due to L. Carleson and many different versions and generalizations of the Corona Theorem have been investigated. I will discuss one such generalization recently obtained with my graduate student. I will also explain how the technique widely used to solve the Corona problems can be applied to get results about operator equivalence (unitary equivalence and similarity) and about the degree bound in Hilbert's Nullstellensatz.
Title: When does a function belong to the union of Lebesgue spaces?

Abstract:  We survey some basic facts about Lebesgue spaces.  We show that the union of Lebesgue spaces is intimately related to the Hardy-Littlewood maximal function and the theory of weighted Lebesgue spaces -- Lebesgue spaces with a change of measure.  We give several simple characterizations of when a function belongs the union Lebesgue spaces. This presentation will be based on a joint work with Greg Knese and John McCarthy.  
Title: Finite group actions, orbifolds, and equivalence of group actions

Abstract: A handlebody orbifold consists of finitely many quotients of the 3-ball by spherical groups (Zn, Dn, A4, S4, and A5) connected by 1-handle orbifolds respecting singular axes and their orders, and such that topologically the outcome is an orientable handlebody. We will first examine the handlebody of genus two, V2, and the handlebody orbifolds V2/G, where G is a finite group. We will then discuss equivalence of group actions and see that, up to equivalence, there are 13 actions on V2. This will lead into my work, where we consider cyclic p-squared actions, where p is prime, on a handlebody of genus g.
Title: Sobolev Steepest Descent for Differential Equations

Abstract:  Solving differential equations using steepest descent methods based on the Euclidean norm has long been established as ineffective, although pre-conditioning techniques  may alleviate this problem to some extent. However, steepest descent in spaces with a better choice of norm can be quite efficient.    Beginning with an example accessible to undergraduates, we will outline Sobolev descent on a few elementary examples and demonstrate at least one interesting open problem in the area.  This talk should be accessible to undergraduates, graduate students and faculty not necessarily experts in numerical differential equations.