Spring 2019
Tuesdays 4:10 p.m. in Carver 0001 - Tea and cookies starting at 3:45 p.m. in Carver 404
January 14
Brian Collier
January 15
Ruoyu Wu
January 17
Ryan Goh
January 22
Carl Wang-Erickson
January 23
Rohit Nagpal
January 24
Radoslav Fulek
January 28
Noah Schweber
February 1
Gabriel Conant
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
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
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
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
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
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.
February 18
Florent Baudier
Texas A&M University
Title: Faithful embeddability
of graphs into Banach spaces and applications
-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.
February 19
Diane Holcomb
KTH Royal Institute of Technology
Title: On local point process
limits of random matrices
February 25
David Lipshutz
Technion - Israel Institute of Technology
Title: Sensitivity analysis of
reflected diffusions
February 26
Brooke Ullery
Harvard University
Title: Cayley-Bacharach theorems and polynomials vanishing on points in projective space
February 28
Achilles Beros
University of Hawaii
Title: Homogeneous
perfect binary trees
March 4
Konstantin Slutsky
nuTonomy Asia
Title: Orbit equivalences
of Borel flows
Since its origin in 1959 in the work of H. Dye, the concept of orbit equivalence has been studied quite extensively. While traditionally larger emphasis is given to actions of discrete groups, in this talk we concentrate on free actions of Rn-flows while taking the viewpoint of Borel dynamics.
For a free Rn-action, an orbit can be identified with an “affine” copy of the Euclidean space, which allows us to transfer any translation invariant structure from Rn onto each orbit. The two structures of utmost importance will be that of Lebesgue measure and the standard Euclidean topology. One may than consider orbit equivalence maps that furthermore preserve these structures on orbits. Resulting orbit equivalences are called Lebesgue orbit equivalence (LOE) and time-change equivalence respectively.
Properties of LOE maps correspond most closely to those of orbit equivalence maps between their discrete counterparts — free Zn actions. We illustrate this by discussing the analog for Rn-flows of Dougherty-Jackson-Kechris classification of hyperfinite equivalence relations.
Orbit equivalences of flows often arise as extensions of maps between cross sections — Borel sets that intersect each orbit in a non-empty countable set. Furthermore, strong geometric restrictions on crosssections are often necessary. Following this path, we explain why one-dimensional R-flows posses cross sections with only two distinct distances between adjacent points, and show how this implies classification of R-flows up to LOE.
If time permits, we conclude the talk with an overview of time-change equivalence, emphasizing the difference between Borel dynamics and ergodic theory.
March 5
Erik Slivken
University Paris VII
Title: Large random
pattern-avoiding
permutations
March 7
Christopher
Porter
Ohio State University
Title:
Perspectives on the
transformation of random
sequences
March 12
Dustin Mixon
Ohio State University
Title: Squeeze fit:
Label-aware dimensionality reduction by
semidefinite programming
April 9
Mark Lewis
Cornell University
Title: Constrained Optimization for Scheduling in Multi-Class Queueing
Abstract:
We consider the problem of scheduling
a single-server when there are
multiple parallel stations to serve. A
classic result in scheduling says to
create a station dependent index
consisting of the product of the
holding cost (per customer, per unit
time) times the rate at which the
service can be completed at that
station. The scheduler then
prioritizes work in the order of the
indices from highest to lowest.
Preferences are captured by the
various holding costs. A more natural
method for modeling preferences is to
assign constraints to the highest
priority customers (guaranteeing a
fixed quality of service level) and to
provide best effort service for the
other classes. We consider this
formulation, present conditions for
optimality and show how to construct
an optimal control. We initially focus
on the two station model, then explain
where the results can be extended.
Applications to patient flow in health
care are discussed.
April 16
Vladimir Chernyak
Wayne State University
Title: Integrability in Time-Dependent Quantum Problems: Factorization, Moduli Spaces, Spectral Curves, and Representations of Quantum Groups
Abstract: Quantum evolution with time-dependent Hamiltonians, $i\hbar \dot{\Psi}(t) = {H}(t) \Psi(t)$, as of today draws considerable attention, both in experimental and theoretical research. The simplest model with H(t) = A + Bt, A and B being N x N real hermitian matrices, has a known exact solution in special functions only for N=2, and is known as the multi-level Landau Zener problem (MLZ). However, for a certain class of N-dimensional problems that satisfy phenomenologically determined ``integrability'' conditions, the scattering matrix can be represented in a factorized form, with the elementary scattering events being represented in terms of known 2 x 2 scattering matrices. We reveal the reason that stands behind the aforementioned factorization: Each integrable MLZ problem can be embedded into a system of M linear first-order differential equations with respect to M-dimensional time, that satisfy consistency constraints. Classification of exactly solvable MLZ problems leads to moduli spaces that can be mapped into moduli spaces of smooth complex curves. We will speculate on possible relevance to number theory, and in particular modular forms. Finally we consider a more general, still integrable, so-called BCS model that describes N_ s identical quantum spins, with H(t) = H_C t^{-1} + H_G. It can be efficiently treated by making use of symmetry with respect to Artin's Braid Group B_{N_s}, due to effects of monodromy in the extended space. The compact Quantum Group SU_{q} (2) appears naturally as an auxiliary tool for describing relevant representation of the Braid group.May 3 (New Room and Time: Carver 268
at 3:10 p.m.)
Adriana Salerno
Bates College
Abstract: In this talk, we will show the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives. This is joint work with Charles F Doran, Tyler L Kelly, Steven Sperber, John Voight, and Ursula Whitcher.