Difference between revisions of "Colloquia"
(→Spring 2019) 
(Jo Nelson title and abstract.) 

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 [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)   [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)  
−  [[#  +  [[#Jo Nelson (Rice) Contact Invariants and Reeb Dynamics ]] 
 JeanLuc   JeanLuc  
    
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===Eviatar Procaccia===  ===Eviatar Procaccia===  
+  
Title: Can one hear the shape of a random walk?  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.  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)  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 nonintegrability. 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 Jholomorphic 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.  
== Past Colloquia ==  == Past Colloquia == 
Revision as of 11:04, 3 April 2019
Contents
 1 Mathematics Colloquium
 1.1 Spring 2019
 1.2 Abstracts
 1.2.1 Beata Randrianantoanina (Miami University Ohio)
 1.2.2 Lillian Pierce (Duke University)
 1.2.3 Angelica Cueto (The Ohio State University)
 1.2.4 David Treumann (Boston College)
 1.2.5 Dean Baskin (Texas A&M)
 1.2.6 Jianfeng Lu (Duke University)
 1.2.7 Alexei Poltoratski (Texas A&M)
 1.2.8 LiCheng Tsai (Columbia University)
 1.2.9 Aaron Naber (Northwestern)
 1.2.10 Vladimir Sverak (Minnesota)
 1.2.11 Jason McCullough (Iowa State)
 1.2.12 Maksym Radziwill (Caltech)
 1.2.13 Jennifer Park (OSU)
 1.2.14 Eviatar Procaccia
 1.2.15 Jo Nelson (Rice)
 1.3 Past Colloquia
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: BeurlingMalliavin and type problems  Denisov  
Feb 6 Wednesday, room 911  LiCheng 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 Monday  Vladimir Sverak (Minnesota)  Wasow lecture "PDE aspects of the NavierStokes equations and simpler models"  Kim  
March 8  Jason McCullough (Iowa State)  On the degrees and complexity of algebraic varieties  Erman  
March 15  

Marshall  
March 29  Jennifer Park (OSU)  Rational points on varieties  Marshall  
April 5  JuLee Kim (MIT)  TBA  Gurevich  
April 12  Eviatar Procaccia (TAMU)  Can one hear the shape of a random walk?  Gurevich  
April 19  Jo Nelson (Rice University)  Contact Invariants and Reeb Dynamics  JeanLuc  
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 socalled character sum. For example, both understanding the Riemann zeta function or Dirichlet Lfunctions 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 wellknow 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 "Ffield." An Ffield on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an Ffield should remind you of a Bfield, 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 DeligneLusztig theory, and I found something like a cluster structure on the DeligneLusztig 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 equalcharacteristic version of the FarguesFontaine 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 lowrank 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 lowrank 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: BeurlingMalliavin 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^2space. The BerulingMalliavin 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.
LiCheng 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 realworld 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 NavierStokes equations and simpler models
Abstract: Does the NavierStokes 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 EisenbudGoto, 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 Lfunctions, 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 ChowlaElliott 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.
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 nonintegrability. 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 Jholomorphic 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.