Title:Just so stories a la carte around Geometry, Dynamics, and PDEs.
Speaker: Prof. Dmitri Burago, Penn State University
Date and Time: Tuesday, April 4, 2017, 2:20-3:20pm
Place: Rome 351
Abstract: The lecture consists of several mini-talks with just definitions, motivations, some ideas of proofs, and open problems. I will discuss some (hardly all) of the following topics, based on the input from the audience:
Short Bio: Dmitri Burago is a Distinguished Professor of Mathematics at Penn State University. His research interests include dynamical systems, algorithmic complexity, Finsler geometry, combinatorial group theory, and partial differential equations. In 1997, Burago was the recipient of an Alfred P. Sloan Research fellowship and, in 1995, he received Penn State's Faculty Scholar Medal for Outstanding Achievements. He has been a member of the St. Petersburg Mathematical Society since 1992. Before joining the Eberly College of Science faculty at Penn State in 1994, Burago was a faculty member at the University of Pennsylvania, a researcher at the St. Petersburg Institute for Informatics and Automation, and an assistant professor in the Department of Mathematics and Mechanics at St. Petersburg State University in Russia. He received doctoral and master's degrees from St. Petersburg State University in 1992 and 1986, respectively.
Title:Making nonelementary classes more elementary
Speaker: William Boney, Harvard University
Date and Time: Friday, February 24, 2017, 1:30-2:30pm
Place: MPA 305
Abstract: Classification theory seeks to organize classes of structures (such as algebraically closed fields, random graphs, dense linear orders) along dividing lines that separate classes into well-behaved on one side and chaotic on the other. Since its beginning, classification theory has discovered a plethora of dividing lines for classes axiomatizable in first-order logic and has been applied to solve problems in algebraic geometry, topological dynamics, and more.
However, when looking at examining nonelementary classes (such as rank 1 valued fields or pseudoexponentiation), the lack of compactness is a serious impediment to developing this theory. In the past decade, Grossberg and VanDieren have isolated the notion of tameness. Tameness can be seen as a fragment of compactness that is strong enough to allow the construction of classification theory, but weak enough to be enjoyed by many natural examples. We will discuss the motivation for classification theory in nonelementary classes and some recent results using tameness, focusing on illuminating examples. No logic background will be assumed.
Title:Model theory and Painlevé equations
Speaker: James Freitag, University of Illinois at Chicago
Date and Time: Thursday, February 23, 2017, 1:30-2:30pm
Place: Phillips 217
Abstract: The Painlevé equations are six families of nonlinear order two differential equations with complex parameters. Around the start of the last century, the equations were isolated for foundational reasons in analysis, but the equations have since arisen naturally in various mathematical contexts. In this talk, we will discuss how to use model theory, a part of mathematical logic, to answer several open questions about Painlevé equations. We will also describe several other applications of model theory and differential algebraic equations to number theory.
Title:q-analogues of factorization problems in the symmetric group
Speaker: Joel Lewis, University of Minnesota
Date and Time: Friday, February 17, 2017, 2:15-3:15pm
Place: Rome 351
Abstract: Given a nice piece of combinatorics for the symmetric group S_n, there is often a corresponding nice piece of combinatorics for the general linear group GL_n(F_q) over a finite field F_q, called a q-analogue. In this talk, we'll describe an example of this phenomenon coming from the enumeration of factorizations. In S_n, the number of ways to write an n-cycle as a product of n - 1 transpositions is Cayley's number n^(n - 2). In GL_n(F_q), the corresponding problem is to write a Singer cycle as a product of n reflections. We show that the number of such factorizations is (q^n - 1)^(n - 1), and give some extensions. Mysteriously, the second answer is closely related to the first as q approaches 1. Our proofs do not provide an explanation for this relationship; instead, they proceed by exploiting the (complex) representation theory of GL_n(F_q).
Title: Connectivity and structure in matroids
Speaker: Stefan van Zwam, Louisiana State University
Date and Time: Wednesday, February 15, 2017, 11:00am-12:00pm
Place: Rome 459
Abstract: A general theme in matroid structure theory is that highly connected matroids exhibit more structure than matroids with low-order separations. We will discuss several examples of this phenomenon, as well as an application to the theory of error-correcting codes.
Title: Gaussian measures on infinite dimensional spaces and applications
Speaker: Nathan Totz, University of Massachusetts Amherst
Date and Time: Monday, February 13, 2017, 1:45-2:45pm
Place: Rome 459
Abstract: We review the classical extension of the Gaussian probability measure from finite dimensional spaces to infinite dimensional spaces. Such Gaussian measures (along with their weighted relatives) play an important role as invariants of flows defined on infinite dimensional spaces. As an application of this idea, we employ Gaussian measures to address the question of the long time existence of a flow corresponding to a family of modified surface quasigeostrophic equations, regarded as a flow on a space of Fourier coefficients. We present recent results (joint with Andrea Nahmod, Natasha Pavlovic, and Gigliola Staffilani) showing that such flows are global in time on a subset of a rough Sobolev space of full measure.
Title: Shapes of polynomial Julia sets
Speaker: Kathryn Lindsey, University of Chicago
Date and Time: Friday, February 10, 2017, 2:15-3:15pm
Place: Rome 351
Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. W. Thurston asked “What are the possible shapes of polynomial Julia sets?” For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name? It turns out the answer to all of these is “yes.” I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes.
Title: Some almost sharp scattering results for the cubic nonlinear wave equation
Speaker: Benjamin Dodson, John Hopkins University
Date and Time: Tuesday, January 31, 2017, 3:00-4:00pm
Place: Rome 771
Abstract: In this talk we will discuss some scattering results for the cubic nonlinear wave equation. We will prove these results for radial data nearly lying in the critical Sobolev space. We prove this using hyperbolic coordinates.
Title: Examples of Linear Algebra over Division Algebras
Speaker: Salahoddin Shokranian, University of Brasilia
Date and Time: Friday, January 27, 2017, 1:00-2:00pm
Place: Rome 204
Abstract: Matrices over some division algebras are considered. In the case of finite fields, applications are in coding theory, and in the case of non‐commutative division algebras, Hermitian matrices over quaternions provide examples toward geometry of such matrices and analysis.
Short Bio: Salahoddin Shokranian has studied mathematics, graduate level, at the University of California, Berkeley where he did his Ph.D. in automorphic forms. He moved to Brasilia since then, working at the University of Brasilia where he is to be retired. He has been visitor to several research institutes such as the Institute for Advanced Study, Tata Institute of Fundamental Research, Max-Planck Institute for Mathematics and worked at the Universities of Purdue, Toronto and Yale. He has lectured at many universities and research centers. He likes to write books; on linear algebra, number theory, modular forms, cryptography and history of modern number theory. Some of them have already been republished or reprinted.