Department of Mathematics Colloquium

 
 

The ISU Department of Mathematics Colloquium is organized by

Pablo Raúl Stinga (stinga@iastate.edu)




Spring 2019

Tuesdays 4:10 p.m. in Carver 0001 - Tea and cookies starting at 3:45 p.m. in Carver 404

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February 18

Florent Baudier

Texas A&M University

Title: Faithful embeddability of graphs into Banach spaces and applications

Abstract: Faithful embeddability of graphs and metric spaces into Banach spaces is pivotal to research areas as diverse as:
-the design of approximation algorithms in theoretical computer science (sparsest cut problem, multi-commodity flows, approximate nearest neighbor search, sketching…),
-topology (Novikov conjecture),
-noncommutative geometry (coarse Baum-Connes conjecture),
-geometric group theory (Von Neumann’s amenability, Gromov’s program).
This non-exhaustive list can be stretched at will since metric spaces, with a wide variety of features, arise in nearly all areas of mathematics.
In this talk, I will focus on bi-Lipschitz and coarse embeddings of graphs (finite and infinite) into Banach spaces with some desirable geometric properties. I will discuss fundamental geometric problems of either local or asymptotic nature, in particular purely metric characterizations of “linear” properties of Banach spaces in the spirit of the Ribe program. One of the main goal of the talk is to present some fundamental ideas and techniques, as well as to convey the geometric intuition behind them.




Upcoming



February 19

Diane Holcomb

KTH Royal Institute of Technology

Title: On local point process limits of random matrices

Abstract: The study of random matrices goes back to the works of Wishart (1920’s) and Wigner (1930’s). At the time they introduced a special class of models that had explicitly computable joint density functions for their eigenvalues. These joint densities turn out to be specific cases of a more general physical model called Coulomb gas models which describe particles interacting through some Hamiltonian. In this talk we will discuss a specific class of these Coulomb gas models called beta-ensembles which are a partial generalization of the random matrix eigenvalue process. I will begin by introducing a random matrix model and the generalization of its eigenvalue process to a beta-ensemble. We will then discuss how one can study the interactions of individual eigenvalues as the number of them grows to infinity. I will introduce the Sine-beta process, one of the limit processes that appears when the eigenvalues are rescaled to see these “local” interactions, and discuss several results and techniques that may be used for studying this process.




February 21

Alex Kruckman

Indiana University Bloomington

Title: Interpolative fusions

Abstract: Model theory is the subfield of logic which studies ordinary mathematical objects (e.g. fields, graphs, linear orders) from the point of view of definability in first-order logic. In this talk, we will focus on the following situation: we have a mathematical object equipped with extra structure in multiple different ways. For example, a vector space with an inner product and an ordering, or a field with a distinguished subfield and a distinguished automorphism. Abstractly, we consider a first-order theory $T$ in a language $L$, and two theories $T_1$ and $T_2$ extending $T$, in languages $L_1$ and $L_2$ extending $L$. Then the “interpolative fusion” of $T_1$ and $T_2$ over $T$ is the the theory of structures which are simultaneously models of $T_1$ and $T_2$, in which the definable sets in $L_1$ and $L_2$ interact “as randomly as possible” subject to constraints imposed by $T$. (Don’t worry, there will be examples!) This setting unifies many existing examples in model theory and provides general tools for new applications. We study two basic questions: (1) When does the interpolative fusion exist, and how can we axiomatize it? (2) How can we understand properties of the interpolative fusion in terms of properties of the component theories $T, T_1$, and $T_2$? This is joint work with Minh Chieu Tran and Erik Walsberg.



February 25

David Lipshutz

Technion - Israel Institute of Technology

Title: Sensitivity analysis of reflected diffusions

Abstract: Reflected diffusions (RDs) constrained to remain in convex polyhedral domains arise in a variety of contexts, including as “heavy traffic” limits of queueing networks and in the study of rankbased interacting particle models. Sensitivity analysis of such an RD with respect to its defining parameters is of interest from both theoretical and applied perspectives. In this talk I will characterize pathwise derivatives of an RD in terms of solutions to a linear constrained stochastic differential equation whose coefficients, domain and directions of reflection depend on the state of the RD. I will demonstrate how pathwise derivatives are useful in Monte Carlo methods to estimate sensitivities of an RD, and also in characterizing sensitivities of the stationary distribution of an RD. The proofs of these results involve a careful analysis of sample path properties of RDs, as well as geometric properties of the convex polyhedral domain and the associated directions of reflection along its boundary.




March 12

Dustin Mixon

Ohio State University

Title:

Abstract:




April 9

Mark Lewis

Cornell University

Title:

Abstract:





Past


January 14

Brian Collier

University of Maryland



January 15

Ruoyu Wu

University of Michigan



January 17

Ryan Goh

Boston University



January 22

Carl Wang-Erickson

Imperial College London



January 23

Rohit Nagpal

University of Michigan




January 24

Radoslav Fulek

IST Austria




January 28

Noah Schweber

University of Wisconsin-Madison




February 1

Gabriel Conant

University of Notre Dame
Title: Arithmetic regularity with forbidden bipartite configurations

Abstract: Szemeredi’s regularity lemma is a fundamental result in graph theory, which says that sufficiently large finite graphs can be partitioned into a small number of pieces so that the edges between most pairs of pieces are randomly distributed. In other words, the regularity lemma processes finite graphs into ingredients that are either highly structured (namely, the partition of the graph) or highly random (namely, the edges between regular pairs). In 2005, Green introduced arithmetic regularity, which is a group-theoretic analogue of Szemeredi’s result. Green’s work involves decompositions of finite abelian groups into algebraically structured pieces, which behave randomly with respect to some chosen subset of the group. In this talk, I will consider a set $A$ in an arbitrary finite group $G$, such that the bipartite graph on $G\times G$ induced by $x\cdot y\in A$ omits some induced bipartite subgraph of size bounded by a fixed integer $k$. I will present strengthened versions arithmetic regularity in this setting, which yield algebraic structure theorems for sets $A$ as above. This work combines tools from model theory, additive combinatorics, and the structure of compact topological groups. Joint with A. Pillay and C. Terry.



February 4

Sun Kim

Illinois State University

Title: Two identities in Ramanujan’s Lost Notebook & Bressoud’s conjecture

Abstract: In this talk, we discuss two topics. The first topic is about two identities that Ramanujan recorded without proofs in his lost notebook. These two identities are intimately connected with the classical Circle and Divisor problems in number theory, respectively. They involve doubly infinite series of Bessel functions, and in each case, there are three possible interpretations for the double series. We proved the first identity under all three interpretations, and the second under two of them. Furthermore, we discuss many analogues and generalizations of them. This is joint work with Bruce C. Berndt and Alexandru Zaharescu.

The second topic is about Bressoud’s conjecture. In 1980, Bressoud obtained an analytic generalization of the Rogers-Ramanujan-Gordon identities. He then tried to establish a combinatorial interpretation of his identity, which specializes to many well-known Rogers-Ramanujan type identities. He proved that a certain partition identity follows from his identity in a very restrictive case and conjectured that the partition identity holds true in general. We discuss Bressoud’s conjecture for the general case and bijective proofs of it.



February 7

Jack Jeffries

University of Michigan

Title: From Zariski-Nagata to local fundamental groups

Abstract: Hilbert's Nullstellensatz gives a dictionary between algebra and geometry; e.g., solution sets to polynomial equations over the complex numbers (varieties) translate to (radical) ideals in polynomial rings. A classical theorem of Zariski-Nagata gives a deeper layer to this correspondence: polynomial functions that vanish to certain order along a variety correspond to a natural algebraic notion called symbolic powers.
In this talk, we will explain this theorem, and then pursue a couple of variations on this theme. First, we will consider how the failure of this theorem over ambient spaces with bends and corners allows to study the geometry of such spaces; in particular, we will give bounds on size of local fundamental groups. Second, we will consider what happens when we replace the complex numbers by the integers; we will show that "arithmetic differential geometry" (in the sense of Buium) allows us to obtain a Zariski-Nagata theorem in this setting. Only a passing familiarity with polynomials and complex numbers is assumed.
This is based on joint projects with Holger Brenner, Alessandro De Stefani, Eloísa Grifo, Luis Núñez-Betancourt, and Ilya Smirnov.




February 11

Chen-Yun Lin

University

Title: Data and Curse of Dimensionality; Spectral Geometry comes to rescue!

Abstract: High dimensional data is increasingly available in many fields. However, the analysis of such data suffers the so-called curse of dimensionality. One powerful approach to nonlinear dimensionality reduction is the diffusion-type maps. Its continuous counterpart is the embedding of a manifold using the eigenfunctions of the Laplace-Beltrami operator. Accordingly, one may ask, how many eigenfunctions are required in order to embed a given manifold. In this talk, I will give some background regarding the dimensionality reduction problem, spectral geometry, and show theoretical results for a generalized diffusion map. Specifically, I will show a closed Riemannian manifold can be embedded into a finite dimensional Euclidean space by maps constructed based on the connection Laplacian at a certain time. This time and the embedding dimension can be bounded in terms of the dimension and geometric bounds of the manifold. Furthermore, the map based on heat kernels can be made arbitrarily close to an isometry. In addition, I will give a ‘’real world” example pertaining to paleonthology, that demonstrates how heat kernels and diffusion maps can be used to quantify the similarity of shapes. The empirical results suggest that this framework is better than the metric commonly used in biological morphometrics.




February 12

Christopher Henderson

University of Chicago

Title: Front slowdown due to nonlocal interactions

Abstract: Reaction-diffusion equations arise as models of systems in which spreading and growing forces interact in nontrivial ways, often creating a front (i.e., a moving interface). In many applications it is natural to consider nonlocal interactions, but, mathematically, the resulting equations have a number of new features and technical difficulties; in particular, the comparison principle, which implies that initially ordered solutions remain ordered, no longer applies. I will survey classical results, present several examples of nonlocal reaction-diffusion equations, and then focus on a particular one, the cane toads equation, which is inspired by the invasive species in Australia. In all cases, the emphasis will be on the influence of long-range interactions due to nonlocal terms on the behavior of fronts. In particular, I will show that, surprisingly, the cane toads front propagates slower than the standard methods predict.



February 14

Maja Taskovic

University of Pennsylvania

Title: Exponential tails for the Boltzmann equation

Abstract: In kinetic theory, a large system of interacting particles is described by a particle probability distribution function. One of the first equations derived in such a way was the Boltzmann equation (derived by Maxwell in 1866 and by Boltzmann in 1872). The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable with a belief that such an assumption does not affect the equation significantly. However, in last 20 years it has been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting.
We study the behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments, which can be understood as a generalization of exponential moments. An interesting aspect of the result is that the singularity rate of the angular kernel affects the order of tails that can be propagated in time. This is based on joint works with Alonso, Gamba, and Pavlovic.





Fall 2018

Tuesdays 4:10 p.m. in Hoover 1213 - Tea and cookies starting at 3:45 p.m. in Carver 404


September 7 (Friday at 3:10pm - Carver 305 - Colloquium / Discrete Math seminar)

Shira Zerbib

University of Michigan

Title: Colorful coverings of polytopes -- the hidden topological truth behind different colorful phenomena

Abstract: The topological KKMS theorem, a powerful extension of Brouwer's Fixed-Point theorem, was proved by Shapely in 1973 in the context of game theory. We prove a colorful and polytopal generalization of the KKMS Theorem, and show that our theorem implies some seemingly unrelated results in discrete geometry and combinatorics involving colorful settings. For example, we apply our theorem to provide a new proof of the Colorful Caratheodory Theorem due to Barany. We further apply our theorem to obtain a new upper bound on the piercing numbers in colorful $d$-interval families, extending results of Tardos, Kaiser and Alon for the non-colored case. Finally, we apply our theorem to questions regarding fair division.
Joint with Florian Frick.


September 11

Maja Taskovic

University of Pennsylvania

Title: On the relativistic Landau equation

Abstract: In 1936, Landau derived a model for a dilute hot plasma where fast moving particles interact via Coulomb interactions. Instead of tracking every particle separately, which would lead to a large number of equations, the dynamics is being described by the particle density function. This model, also known as the Landau equation, does not include the effects of Einstein's theory of special relativity. However, when particle velocities are close to the speed of light, which happens frequently in a hot plasma, then relativistic effects become important. A model that captures these effects, the relativistic Landau equation, was derived by Budker and Beliaev in 1956.
We study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. The difficulty of the problem lies in the extreme complexity of the kernel of the collision operator. We present a new decomposition of such kernel. This is then used to prove the global Entropy dissipation estimate, the propagation of any polynomial moment for a weak solution, and the existence of a true weak solution for a large class of initial data.
This is joint work with Robert M. Strain.


September 18

Krishna B. Athreya

Iowa State University

Title: David Harold Blackwell

Abstract: David Harold Blackwell was a great mathematician, statistician that belonged to the African American Community from Central Illinois. He was born in 1919 and died in 2010. He got his PhD in mathematics from University of Illinois in 1941 under the great American mathematician J. L. Doob. David Blackwell worked in many areas in mathematics. These include Probability Theory, Game Theory, Information Theory and Bayesian Analysis. He was the first African American to be elected to full professorship at UC Berkeley. He was a member of the US Academy of Sciences, Fellow of the UK Royal Statistical Society, and received many other honors. In this talk we shall outline some of his major research contributions.


September 25

Caroline Terry

University of Chicago

Title: A stable arithmetic regularity lemma in finite-dimensional vector spaces over fields of primer order

Abstract: Arithmetic combinatorics studies the additive and multiplicative structure of subsets of groups, especially finite abelian groups such as cyclic groups of prime order, or finite dimensional vector spaces over finite fields. Insipired by Szemerèdi's regularity lemma, arithmetic regularity lemmas are tools used in arithmecit combinatorics to produce group theoretic analogues of results from graph theory. The arithmetic regularity lemma for $F_p^n$ (first proved by Green in 2005) states that given $A\subseteq F_p^n$, there exists $H\leq F_p^n$ of bounded index such that $A$ is Fourier-uniform with respect to almost all cosets of $H$. In general, the growth of the codimension of H is required to be of tower type depending on the degree of uniformity, and must also allow for a small number of non-uniform elements. The main result of this talk is that, under a natural model theoretic assumption called stability, the bad bounds and non-uniform elements are not necessary. Specifically, we present an arithmetic regularity lemma for $k$-stable sets $A\subseteq F_p^n$, where the bound on the codimension of the subspace is only polynomial in the degree of uniformity, and where there are no non-uniform elements. This result is a natural extension to the arithmetic setting of the work on stable graph regularity lemmas initiated by Malliaris and Shelah.
This is joint work with Julia Wolf.


October 2

Tuncay Aktosun

The University of Texas at Arlington

Title: Determining the shape of a human vocal tract from speech sounds

Abstract: The elementary units for human speech are called phonemes, and the utterance of each phoneme by a person is governed by a particular shape of that person's vocal tract. A mathematical description is presented for the shape of the vocal tract during the creation of each phoneme, which corresponds to the direct problem. Then, a corresponding inverse problem is analyzed; namely, the determination of the shape of the human vocal tract from the sound pressure measurements at the mouth associated with an uttered phoneme.
The talk is based on joint work with P. Sacks of Iowa State University.
 


October 9

Naiomi Cameron

Clark College

Title: Inversion generating functions for signed pattern avoiding permutations

Abstract: We consider the classical Mahonian statistics on the set $B_n(\Sigma)$ of signed permutations in the hyperoctahedral group $B_n$ which avoid all patterns in $\Sigma$, where $\Sigma$ is a set of patterns of length two.  In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\left|B_n(\Sigma)\right|$ is constant whenever $\left|\Sigma\right|=1$, whereas in most but not all instances where $\left|\Sigma\right|=2$, $\left|B_n(\Sigma)\right|=(n+1)!$.  We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\left|B_n(\Sigma)\right|=(n+1)!$.  In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on $B_n(21, \bar{2}\bar{1})$ and by giving the major index on $D_n(\Sigma)$ for $\Sigma =\{21, \bar{2}\bar{1}\}$ and $\Sigma=\{12,21\}$.  The main result of this paper is to give the inversion generating functions for $B_n(\Sigma)$ for almost all sets $\Sigma$ with $\left|\Sigma\right|\leq2.$


October 16

Vlad Vicol

Courant Institute

Title: Nonuniqueness of weak solutions to the Navier-Stokes equation

Abstract: We prove that distributional solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Hölder continuous weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.
This is a joint work with Tristan Buckmaster.



October 23

Eric Weber

Iowa State University

Title: The Kaczmarz algorithm: theory and applications

Abstract: The Kaczmarz algorithm is an iterative method for solving systems of linear equations that was introduced by Stefan Kaczmarz in 1937. The algorithm is now enjoying a resurgence in interest, as it has been found useful in data science applications. It also has remarkably deep connections to complex and harmonic analysis. We shall introduce the algorithm, demonstrate some of its features, and present some of its applications. Towards the end of the talk, we shall outline these deep connections to complex and harmonic analysis.


October 30

Brooke Ullery

Harvard University

Title: Measures of irrationality for algebraic varieties

Abstract: In algebraic geometry, a smooth curve is said to be rational if it is isomorphic to $P^1$, the projective line. More generally, the gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality even further to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. We will discuss these measures of irrationality and various methods of calculating and bounding them. We will mainly focus on the examples of hypersurfaces and, more generally, complete intersections in projective space. All of these terms will be defined and there will be lots of accessible examples!


November 2 (Friday at 3:10pm - Carver 305 - Colloquium / Discrete Mathematics seminar
                      - Tea and cookies will be afterwards, at 4:00pm in Carver 404)

Emily Sergel

University of Pennsylvania

Title: Parking functions, schedules, and the Delta Conjecture

Abstract: A parking function is a kind of labeled lattice path. The Shuffle Conjecture states that a certain enumeration of parking functions is closely related to an important symmetric function basis called the Macdonald polynomials. The Delta Conjecture is a generalization with similar ties to Macdonald polynomials. We discuss the combinatorics of parking functions and the notion of schedules as they pertain to these two settings.
Joint work with Jim Haglund.



November 6

Marta D'Elia

Sandia National Laboratories

Title: Nonlocal models in computational science and engineering: challenges and applications

Abstract: Nonlocal continuum theories such as peridynamics and nonlocal elasticity can capture strong nonlocal effects due to long-range forces at the mesoscale or microscale. For problems where these effects cannot be neglected, nonlocal models are more accurate than classical Partial Differential Equations (PDEs) that only consider interactions due to contact. However, the improved accuracy of nonlocal models comes at the price of a computations cost that is significantly higher than that of PDEs.

In this talk I will present nonlocal models and the Nonlocal Vector Calculus, a theory that allows one to treat nonlocal diffusion problems in almost the same way as PDEs. Furthermore, I will present current open challenges related to the numerical solution of nonlocal problems and show how we are currently addressing them. Specifically, I will describe an optimization-based local-nonlocal coupling strategy and briefly introduce a technique to improve the performance of Finite Element (FE) approximations.

The goal of local-nonlocal coupling methods is to combine the computational efficiency of PDEs with the accuracy of nonlocal models. These couplings are imperative when the size of the computational domain or the extent of the nonlocal interactions are such that the nonlocal solution becomes prohibitively expensive to compute, yet the nonlocal model is required to accurately resolve small scale features. Our approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. I will present consistency and convergence studies and, using three-dimensional geometries, I will also show that our approach can be successfully applied to challenging, realistic problems.

Finally, I will introduce a new concept of nonlocal neighborhood that helps improving the performance of FE methods and show how our approach allows for fast assembling in two- and three-dimensional computations.


November 13

Catherine Searle

Wichita State University

Title: Symmetries of spaces with lower curvature bounds

Abstract: The classification of manifolds of positive and non-negative sectional curvature is a long standing problem in Riemannian geometry. In particular, restricting our attention to closed, simply-connected manifolds, there are no topological obstructions that allow us to distinguish between positive and non-negative curvature, that is, we have no examples of manifolds that admit a metric of non-negative curvature that do no admit a metric of positive curvature. However, with the introduction of symmetries, we are able to distinguish between these two classes. In this context, I will discuss recent joint work with Christine Escher and work with Zheting Dong and Christine Escher on non-negatively curved manifolds with abelian symmetries. 


December 12

Shira Zerbib

University of Michigan


December 13

Shlomo Gelaki

Technion - Israel Institute of Technology