# Colloquia/Spring2019

# Mathematics Colloquium

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

## Spring 2019

## 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 (Minnesota)

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 (Iowa State)

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.

### Maksym Radziwill (Caltech)

Title: Recent progress in multiplicative number theory

Abstract: Multiplicative number theory aims to understand the ways in which integers factorize, and the distribution of integers with special multiplicative properties (such as primes). It is a central area of analytic number theory with various connections to L-functions, harmonic analysis, combinatorics, probability etc. At the core of the subject lie difficult questions such as the Riemann Hypothesis, and they set a benchmark for its accomplishments. An outstanding challenge in this field is to understand the multiplicative properties of integers linked by additive conditions, for instance n and n+ 1. A central conjecture making this precise is the Chowla-Elliott conjecture on correlations of multiplicative functions evaluated at consecutive integers. Until recently this conjecture appeared completely out of reach and was thought to be at least as difficult as showing the existence of infinitely many twin primes. These are also the kind of questions that lie beyond the capability of the Riemann Hypothesis. However recently the landscape of multiplicative number theory has been changing and we are no longer so certain about the limitations of our (new) tools. I will discuss the recent progress on these questions.

### Jennifer Park (OSU)

Title: Rational points on varieties

Abstract: The question of finding rational solutions to systems of polynomial equations has been investigated at least since the days of Pythagoras, but it is still not completely resolved (and in fact, it has been proven that there will never be an algorithm that answers this question!) Nonetheless, we will discuss various techniques that could answer this question in certain cases, and we will outline some conjectures related to this problem as well.

### Ju-Lee Kim (MIT)

Title: Types and counting automorphic forms

Abstract: We review the theory of types in representations of p-adic groups and discuss some applications for quantifying automorphic forms.

### Eviatar Procaccia

Title: Can one hear the shape of a random walk?

Abstract: We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary . We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes. Joint work with Marek Biskup (UCLA)

### Jo Nelson (Rice)

Title: Contact Invariants and Reeb Dynamics

Abstract: Contact geometry is the study of certain geometric structures on odd dimensional smooth manifolds. A contactstructure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. I will explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant, contact homology, whose chain complex is generated by closed Reeb orbits. In particular, I will explain the pitfalls in defining contact homology and discuss my work, in part joint with Hutchings, which provides rigorous constructions and applications to dynamics via geometric methods. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.

### Justin Hsu (Madison)

Title: From Couplings to Probabilistic Relational Program Logics

Abstract: Many program properties are relational, comparing the behavior of a program (or even two different programs) on two different inputs. While researchers have developed various techniques for verifying such properties for standard, deterministic programs, relational properties for probabilistic programs have been more challenging. In this talk, I will survey recent developments targeting a range of probabilistic relational properties, with motivations from privacy, cryptography, and machine learning. The key idea is to meld relational program logics with an idea from probability theory, called probabilistic couplings. The logics allow a highly compositional and surprisingly general style of program analysis, supporting clean proofs for a broad array of probabilistic relational properties.

### Kavita Ramanan (Brown)

Title: Tales of Random Projections

Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.

### Tomasz Przebinda (Oklahoma)

Title: Resonances of the Laplace-Beltrami operator on a symmetric space of non-compact type and geometry of hyperplane arrangements in a complex sphere.

Abstract: The resonances, mentioned in the title are poles of a meromorphic extension of the resolvent of the Laplacian, when its domain is restricted from the Hilbert space of the square integrable functions on the symmetric space to the space of compactly supported smooth functions. The corresponding residues yield irreducible admissible spherical representations of the group of the isometries of the symmetric space.

The Helgason Fourier transform provides an expression for the resolvent in terms of an integral over the unit sphere in a real Euclidean space of dimension equal to the rank of the symmetric space. The problem of finding the desired meromorphic extension leads to the problem of deforming that sphere within its complexification while avoiding the hyperplanes defined by the singularities of the Harish-Chandra c-function. (In the classical analysis this is equivalent to the standard procedure of a deforming the contour of integration on the complex plane avoiding a finite number of points.)

The theory of resonances on a symmetric space of non-compact type is far from completion. Even the existence of the resonances is not known in general. We shall explain the underlying geometry in the case when the rank of the symmetric space is 3.

The talk is based on ongoing works with Joachim Hilgert (Universit\"at Paderborn, Germany) and Angela Pasquale (Universit\'e de Lorraine – Metz, France).