Revision as of 23:30, 1 March 2019

Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Spring 2019

date	speaker	title	host(s)
Jan 25 Room 911	Beata Randrianantoanina (Miami University Ohio) WIMAW	Some nonlinear problems in the geometry of Banach spaces and their applications	Tullia Dymarz
Jan 30 Wednesday	Talk rescheduled to Feb 15
Jan 31 Thursday	Talk rescheduled to Feb 13
Feb 1	Talk cancelled due to weather
Feb 5 Tuesday, VV 911	Alexei Poltoratski (Texas A&M University)	Completeness of exponentials: Beurling-Malliavin and type problems	Denisov
Feb 6 Wednesday, room 911	Li-Cheng Tsai (Columbia University)	When particle systems meet PDEs	Anderson
Feb 8	Aaron Naber (Northwestern)	A structure theory for spaces with lower Ricci curvature bounds	Street
Feb 11 Monday	David Treumann (Boston College)	Twisting things in topology and symplectic topology by pth powers	Caldararu
Feb 13 Wednesday	Dean Baskin (Texas A&M)	Radiation fields for wave equations	Street
Feb 15	Lillian Pierce (Duke University)	Short character sums	Boston and Street
Feb 22	Angelica Cueto (Ohio State)	Lines on cubic surfaces in the tropics	Erman and Corey
March 4	Vladimir Sverak (Minnesota)	Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models"	Kim
March 8	Jason McCullough (Iowa State)	On the degrees and complexity of algebraic varieties	Erman
March 15	Maksym Radziwill (Caltech)	TBA	Marshall
March 29	Jennifer Park (OSU)	TBA	Marshall
April 5	Ju-Lee Kim (MIT)	TBA	Gurevich
April 12	Eviatar Procaccia (TAMU)	TBA	Gurevich
April 19	Jo Nelson (Rice University)	TBA	Jean-Luc
April 22 Monday	Justin Hsu (Madison)	TBA	Lempp
April 26	Kavita Ramanan (Brown University)	TBA	WIMAW
May 3	Tomasz Przebinda (Oklahoma)	TBA	Gurevich

Abstracts

Beata Randrianantoanina (Miami University Ohio)

Title: Some nonlinear problems in the geometry of Banach spaces and their applications.

Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.

Lillian Pierce (Duke University)

Title: Short character sums

Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.

Angelica Cueto (The Ohio State University)

Title: Lines on cubic surfaces in the tropics

Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement any smooth surface of degree three in P^3 contains exactly 27 lines is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.

In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.

David Treumann (Boston College)

Title: Twisting things in topology and symplectic topology by pth powers

Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.

Dean Baskin (Texas A&M)

Title: Radiation fields for wave equations

Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.

Jianfeng Lu (Duke University)

Title: Density fitting: Analysis, algorithm and applications

Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.

Alexei Poltoratski (Texas A&M)

Title: Completeness of exponentials: Beurling-Malliavin and type problems

Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin problem was solved in the early 1960s and I will present its classical solution along with modern generalizations and applications. I will then discuss history and recent progress in the type problem, which stood open for more than 70 years.

Li-Cheng Tsai (Columbia University)

Title: When particle systems meet PDEs

Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.

Aaron Naber (Northwestern)

Title: A structure theory for spaces with lower Ricci curvature bounds.

Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.

Vladimir Sverak

Title: PDE aspects of the Navier-Stokes equations and simpler models

Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.

Jason McCullough

Title: On the degrees and complexity of algebraic varieties

Abstract: Given a system of polynomial equations in several variables, there are several natural questions regarding its associated solution set (algebraic variety): What is its dimension? Is it smooth or are there singularities? How is it embedded in affine/projective space? Free resolutions encode answers to all of these questions and are computable with modern computer algebra programs. This begs the question: can one bound the computational complexity of a variety in terms of readily available data? I will discuss two recently solved conjectures of Stillman and Eisenbud-Goto, how they relate to each other, and what they say about the complexity of algebraic varieties.