Department of Mathematics and Statistics
Texas Tech University
The
Pure Mathematics Colloquium: Current Advances in Mathematics is dedicated to different topics mostly in pure mathematics, and not limited to any specific areas. Therefore, it will be of interest of all faculty whose research may include algebra, number theory, topology, logic, geometry, and analysis. The goal of the Colloquium is to further promote research in pure mathematics in our department and to develop and maintain communications with outside experts about current advances in mathematics. We will invite mathematicians from our department and from around the world to deliver online lectures on recent progress of significance.
Website.
We give a survey on our new method to study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of mass-conserving solutions to the Coagulation-Fragmentation equation. These solve partly some long standing open problems in the field. Based on joint works with Hiroyoshi Mitake (University of Tokyo) and Truong-Son Van (CMU).
In 1979 Kaufman constructed a surprising counterexample to the classical Sard theorem: there is a surjective map $f:[0,1]^{n+1}\to [0,1]^n$ of class $C^1$ such that rank $df\leq 1$ everywhere. A natural question is whether there is a topologically non-trivial version of this example: a mapping $f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)$ such that rank $df < n$ everywhere and $f$ is not homotopic to a constant map. Clearly such a map has to be surjective and it cannot be of class $C^2$ (because of Sard's theorem). I will discuss the following result: If $n=2,3$ and $f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)$ is not homotopic to a constant map, then there is an open set $\Omega\subset \mathbb{S}^{n+1}$ such that rank $df = n$ on $\Omega$, while for any $n\geq 4$, there is a map $f\in C^1(\mathbb{S}^{n+1}, \mathbb{S}^n)$ that is not homotopic to a constant map and such that rank $df < n$ everywhere. The result in the case $n\geq 4$ answers a question of Larry Guth. I will also discuss an application of the result to a solution of a recent conjecture of Jacek Galeski. In particular I will show that there is a $C^1$ mappings in $\mathbb{R}^5$ with the derivative of rank at most $3$ that cannot be uniformly approximated by $C^2$ mappings with the derivative of rank at most 3. The methods use analysis, algebraic topology and geometric measure theory. The talk will be accessible to graduate students. The presentation is based on my two joint papers. One with P. Goldstein and one with P. Goldstein and P. Pankka.Given an ultra-metric space $X$ with a reference measure and a distance distribution function on $[0,\infty)$, we construct a symmetric Markov semigroup whose generator $L$ can be regarded as a non-local Laplace operator on such a space. Besides, it generates a jump process on $X$ that is analogous to symmetric stable Levy processes in the Euclidean space. We obtain explicit formulas and estimates for the heat kernel and the Green function of $L$. In a particular case when $X$ is the space of $p$-adic numbers (or its power), this construction recovers the Taibleson Laplacian. We apply this theory also to study of the Vladimirov Laplacian. Even in this well-established setting several of the results are new.“Mysterious Duality" was suggested by mathematical physicists Iqbal, Neitzke, and Vafa in 2001. They noticed that (already mysterious for a mathematician) toroidal compactifications of M-theory lead to the same series of combinatorial objects as the del Pezzo surfaces (equally mysterious for all but algebraic geometers) do, along with numerous mysterious coincidences: both toroidal compactifications and del Pezzo surfaces give rise to the exceptional series $E_k$; a collection of various M- and D-branes corresponds to a set of divisors on a del Pezzo surface; the brane tension is related to the ?area? of the corresponding divisor, etc. The mystery is that it is not at all clear where this duality comes from. In the talk, I will present another series of mathematical objects: certain versions of multiple loop spaces of the four-sphere $S^4$, which are, on the one hand, directly connected to M-theory and its compactifications, and, on the other hand, possess the same combinatorics as the del Pezzo surfaces. This solves the physics-mathematics mystery and transfers it to the geometry-topology plane: what is the relation between the del Pezzo surfaces and loop spaces of the four-sphere? This is a preliminary report on a work with Hisham Sati.We will give an overview of how physics and homological algebra
have met in the setting of gauge theory, with an emphasis on how the new
subject of derived geometry provides a clarifying framework. The talk's
concrete aim is to explain the Higgs mechanism as a case study. Our
approach will be low-tech and will emphasize the motivations; anyone
familiar with notions like vector bundle and cochain complex should be able
to follow. (This talk is based on a paper with Chris Elliott.)