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4:00pm - RH 340N - Geometry and Topology Zhaoting Wei - (Texas A&M University-Commerce) Superconnections in geometry It is well-known that on a non-projective complex manifold, a |
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4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Jérôme Gilles - (SDSU) Empirical wavelets: principle, theory and applications Data driven techniques have been at the center of attention for several years. If supervised techniques have proven their efficacy in many fields, their main drawback is the need of extensive annotated datasets for their training. For certain applications, the availability of such huge datasets is not possible. On the other hand, time/spatial-frequency analysis has been used for decades to characterize signals and images. Data-driven time-frequency analysis techniques have been investigated this last decade. Among them, empirical wavelets have been proven to extract accurate information allowing further analysis. In this talk, we will review the concept of empirical wavelets and define a general mathematical framework. Finally, I will present some applications in signal/image processing. |
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4:00pm to 5:20pm - RH 340 R - Logic Set Theory Julian Eshkol - (UC Irvine) Combinatorics of Very Large Cardinals At and above the level of measurability, large cardinal notions are typically characterized by the existence of certain elementary embeddings of the universe into an inner model. We may contrast this with smaller large cardinal notions, whose characterizations tend to be strictly combinatorial. In this series of talks, we survey results from Magidor's thesis, in which he shows that the large notion of supercompactness can also be viewed combinatorially, and in this light supercompactness is seen to be a natural strengthening of ineffability. We will also survey modern results which show how these strong combinatorial principles can be forced to hold at small successor cardinals. |
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1:00pm to 2:00pm - RH 306 - Algebra Christopher O'Neill - (SDSU) Classifying numerical semigroups using polyhedral geometry A numerical semigroup is a subset of the natural numbers that is closed under addition. There is a family of polyhedral cones $C_m$, called Kunz cones, for which each numerical semigroup with smallest positive element $m$ corresponds to an integer point in $C_m$. Recent work has demonstrated that if two numerical semigroups correspond to points in the same face of $C_m$, they share many important properties, such as the number of minimal generators and the Betti numbers of their defining toric ideals. In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure.
In this talk, we survey what is known about the face structure of Kunz cones, and how studying Kunz cones can inform the classification of numerical semigroups. No familiarity with numerical semigroups or polyhedral geometry will be assumed for this talk. |
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3:00pm to 3:50pm - RH 306 - Analysis Dmitri Zaitsev - (Trinity College Dublin (TCD), Ireland) Global regularity in the d-bar-Neumann problem and finite type conditions The celebrated work of Catlin on global regularity of the $\bar\partial$-Neumann operator for pseudoconvex domains of finite type links local algebraic- and analytic geometric invariants through potential theory with estimates for $\bar\partial$-equation. Yet despite their importance, there seems to be a major lack of understanding of Catlin's techniques,resulting in a notable absence of an alternative proof, exposition or simplification. The goal of my talk will be to present an alternative proof based on a new notion of a ``tower multi-type''. The finiteness of the tower multi-type is an intrinsic geometric condition that is more general than the finiteness of the regular type, which in turn is more general than the finite type. Under that condition, we obtain a generalized stratification of the boundary into countably many level sets of the tower multi-type, each covered locally by strongly pseudoconvex submanifolds of the boundary. The existence of such stratification implies Catlin's potential-theoretic ``Property (P)'', which, in turn, is known to imply global regularity via compactness estimate. Notable applications of global regularity include Condition R by Bell and Ligocka and its applications to boundary smoothness of proper holomorphic maps generalizing a celebrated theorem by Fefferman. |
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2:00pm - 510R Rowland Hall - Combinatorics and Probability Luis Rademacher - (UC Davis) Expansion of random 0/1 polytopes and the Mihail and Vazirani conjecture A 0/1 polytope is the convex hull of a set of 0/1 d-dimensional vectors. A conjecture of Milena Mihail and Umesh Vazirani says that the graph of vertices and edges of every 0/1 polytope is highly connected. Specifically, it states that the edge expansion of the graph of every 0/1 polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a 0/1 polytope in R^d is greater than 1 over some polynomial function of d. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random 0/1 polytope in R^d is at least 1/12d with high probability. This is joint work with Brett Leroux. |
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3:00pm to 4:00pm - RH 306 - Number Theory Alex Barrios - (University of St. Thomas) Lower bounds for the modified Szpiro ratio Let $a,b,$ and $c$ be relatively prime positive integers such that $a+b=c$. How does c compare to $\operatorname{rad}(abc)$, where rad(n) denotes the product of the distinct prime factors of $n$? According to the explicit $abc$ conjecture, it is always the case that $c$ is less than the square of $\operatorname{rad}(abc)$. This simple statement is incredibly powerful, and as a consequence, one gets a (marginal) proof of Fermat's Last Theorem for exponent $n$ greater than $5$.
In this talk, we introduce Masser and Oesterlé's $abc$ conjecture and discuss some of its consequences, as well as some of the numerical evidence for the conjecture. We will then introduce elliptic curves and see that the $abc$ conjecture has an equivalent formulation in this setting, namely, the modified Szpiro conjecture. We conclude the talk by discussing a recent result that establishes the existence of sharp lower bounds for the modified Szpiro ratio of an elliptic curve that depends only on its torsion structure.
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4:00pm to 4:50pm - RH 306 - Colloquium Gunther Uhlmann - (University of Washington) Journey to the Center of the Earth We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It also has several applications in optics and medical imaging among others. The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will survey some of the known results about this problem. No previous knowledge of differential geometry will be assumed. |
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3:00pm to 4:00pm - RH 510R - Applied and Computational Mathematics Cheng Wang - (University of Massachusetts Dartmouth) Epitaxial thin film growth model and its numerical simulation A nonlinear PDE model of thin film growth model, with or without slope selection, are presented in the talk. A global in time solution with Gevrey regularity is established for the one with slope selection. For the numerical simulation, an idea of convex-concave splitting of the corresponding physical energy is applied, which gives to an implicit treatment for the convex part and an explicit treatment for the concave part. That in turn leads to a numerical scheme with a non-increasing energy. A first order accurate linear splitting and a second order accurate linear iteration algorithm are also considered, with some numerical simulation results presented. |