**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

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

**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.

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

**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.

**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 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.

**February 26
**

**Brooke Ullery
**

Harvard University

**Title**: **Cayley-Bacharach
theorems and polynomials vanishing on points
in projective space**

**Abstract:**If Z is a set of points in real n-space, we can ask which polynomials of degree d in n variables vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey several classical results including the original Cayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions in modern algebraic geometry.

**February 28
**

**Achilles Beros
**

University of Hawaii

**Title**: Homogeneous
perfect binary trees

**Abstract:**Perfect binary trees typically have highly inhomogeneous branches; for example, if a perfect tree is computable, then for any set A there is a branch of higher degree than A. I will examine properties such that it is possible to construct perfect binary trees every branch of which have the property. Futhermore, I will show conditions on the frequency of branching that permit or preclude the construction of such a tree. I will also survey other recent research in computability theory and its applications.

**March 4
**

**Konstantin Slutsky
**

nuTonomy Asia

**Title: Orbit equivalences
of Borel flows**

**Abstract:**We provide an overview of the orbit equivalence theory of Borel flows. In general, an orbit equivalence of two group actions is a bijective map between phase spaces that maps orbits onto orbits. Such maps are often further required to posses regularity properties depending on the category of group actions that is being considered. For example, Borel dynamics deals with Borel orbit equivalences, ergodic theory considers measure-preserving maps, topological dynamics assumes continuity, etc.

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**

**Abstract:**

**A pattern in a permutation is a subsequence with a specific relative order. What can we say about a typical large random permutation that avoids a particular pattern? We use a variety of approaches. For certain classes we give a geometric description that relates these classes to other types of well-studied objects like random walks or random trees. Using the right geometric description we can find the the distribution of certain statistics like the number and location of fixed points. This is based on joint work with Christopher Hoffman and Douglas Rizzolo.**

**March 7
**

**Christopher
Porter
**

Ohio State University

**Title****:
Perspectives on the
transformation of random
sequences**** **

**Abstract:**

**The theory of algorithmic randomness provides a framework for studying the behavior of algorithmically random sequences under effective transformations, i.e. transformations that are Turing computable. In this talk, I will take as a starting point a theorem due to Demuth, which states that if we apply a total computable transformation to a sufficiently random sequence—here taken to be a Martin-Löf random sequence—then as long as the resulting sequence is not computable, one can recover an unbiased random sequence from the transformed sequence. I will then discuss the efficiency of the procedure that recovers unbiased randomness from this transformed sequence. More specifically, I will survey results on (1) the conditions for the existence of a computable bound on the number of input bits needed for this procedure to yield a given number of output bits (based on joint work with Laurent Bienvenu and subsequent work with Rupert Hölzl), and (2) finding the average-case extraction rate of this procedure (based on joint work with Douglas Cenzer).**

**March 12
**

**Dustin Mixon
**

Ohio State University

**Title****: Squeeze fit:
Label-aware dimensionality reduction by
semidefinite programming**** **

**Abstract:**Given labeled points in a high-dimensional vector space, we seek a low-dimensional subspace such that projecting onto this subspace maintains some prescribed distance between points of differing labels. Intended applications include compressive classification. This talk will introduce a semidefinite relaxation of this problem, along with various performance guarantees. (Joint work with Culver McWhirter (OSU) and Soledad Villar (NYU)).

**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****: ****In****tegrability
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

**Title**: Hypergeometric decomposition of symmetric K3 quartic pencils

**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.