# MSCS Seminar Calendar

Monday November 12, 2018

**Mathematical Computer Science Seminar**

Generalized Tur\'an problems for graphs and hypergraphs

Ruth Luo (UIUC)

3:00 PM in 427 SEO

We will talk about a generalization of the Tur\'an problem for hypergraphs: given a graph $F$, what is the maximum number of hyperedges an $r$-uniform $n$-vertex Berge $F$-free hypergraph can have? In particular, we will discuss tools used to reduce the hypergraph problem to problems for graphs. Finally, I will present some recent results for graphs without long Berge cycles. This is joint work with (different subsets of) Zoltan Furedi and Alexandr Kostochka.

**Geometry, Topology and Dynamics Seminar**

On Invariant Random Subgroups of Block-Diagonal Limits of Symmetric Groups

Constantine Medynets (U.S. Naval Academy)

3:00 PM in 636 SEO

We classify the ergodic invariant random subgroups of block-diagonal limits of symmetric groups in the cases when the groups are simple and the associated dimension groups have finite dimensional state spaces. These block-diagonal limits arise as the transformation groups (full groups) of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. Given a simple full group G admitting only a finite number of ergodic measures on the path-space X of the associated Bratteli diagram, we prove that every non-Dirac ergodic invariant random subgroup of G arises as the stabilizer distribution of the diagonal action on X^n for some n≥1. As a corollary, we establish that every group character χ of G has the form χ(g)=Prob(g∈K), where K is a conjugation-invariant random subgroup of G.

**Analysis and Applied Mathematics Seminar**

Multiscale/Multiphysics Coupling Framework for Bioprosthetic Heart Valve Damage

Yue Yu (Lehigh University)

4:00 PM in 636 SEO

Bioprosthetic heart valves (BHVs) are the most popular artificial replacements for diseased valves that mimic the structure of native valves. However, the life span of BHVs remains limited to 10-15 years, and the mechanisms that underlie BHVs failure remain poorly understood. Therefore, developing a unifying mathematical framework which captures material damage phenomena in the fluid-structure interaction environment would be extremely valuable for studying BHVs failure. Specifically, in this framework the computational domain is composed of three subregions: the fluid (blood) , the fracture structure (damaged BHVs) modeled by the recently developed nonlocal (peridynamics) theory, and the undamaged thin structure (undamaged BHVs). These three subregions are numerically coupled to each other with proper interface boundary conditions.
In this talk, I will introduce two sub-problems and the corresponding numerical methods we have developed for this multiscale/multiphysics framework. In the first problem the coupling strategy for fluid and thin structure is investigated. This problem presents unique challenge due to the large deformation of BHV leaflets, which causes dramatic changes in the fluid subdomain geometry and difficulties on the traditional conforming coupling methods. To overcome the challenge, the immersogemetric method was developed where the fluid and thin structure are discretized separately and coupled through penalty forces. To ensure the capability of the developed method in modeling BHVs, we have verified and validated this method. In the second problem, we proposed a Neumann-type interface boundary condition for the nonlocal model. In the nonlocal models the Neumann-type boundary conditions should be defined in a nonlocal way, namely, on a region with non-zero volume outside the surface, while in fluid—structure interfaces the hydrodynamic loadings from the fluid side are typically provided on a sharp co-dimension one surface. Therefore, we have shown that our new nonlocal Neumann-type boundary condition provides an approximation of physical boundary conditions on a sharp surface, with an optimal asymptotic convergence rate to the local counter part. Based on this new boundary condition, we have developed a fluid—peridynamics coupling framework without overlapping regions.

Tuesday November 13, 2018

Wednesday November 14, 2018

**Mathematics Education Colloquium**

Can Math Education Research Improve the Teaching of Abstract Algebra?

Aida Alibek (UIC)

12:00 PM in 612 SEO

Please join us for a discussion of the Fukawa-Connelly, Johnson & Keller paper from the Notices of AMS. No prior reading is required.
Everyone is welcome to join! Especially those involved in teaching Abstract Algebra for Undergraduate students!

**Statistics Seminar**

Factorizations and estimates of Dirichlet heat kernels for non-local operators with critical killings

Renming Song (UIUC)

4:00 PM in 636 SEO

In this talk I will discuss heat kernel estimates for critical perturbations
of non-local operators. To be more precise, let $X$ be the reflected
$\alpha$-stable process in the closure of a smooth open set $D$, and
$X^D$ the process killed upon exiting $D$. We consider potentials of the
form $\kappa(x)=C\delta_D(x)^{-\alpha}$ with positive $C$ and the
corresponding Feynman-Kac semigroups. Such potentials do not belong
to the Kato class. We obtain sharp two-sided estimates for the heat
kernel of the perturbed semigroups. The interior estimates of the
heat kernels have the usual $\alpha$-stable form, while the boundary
decay is of the form $\delta_D(x)^p$ with non-negative
$p\in [\alpha-1, \alpha)$ depending on the precise value of the
constant $C$. Our result recovers the heat kernel estimates of both
the censored and the killed stable process in $D$. Analogous
estimates are obtained for the heat kernel of the Feynman-Kac
semigroup of the $\alpha$-stable process in
${\mathbf R}^d\setminus \{0\}$ through the potential $C|x|^{-\alpha}$.
All estimates are derived from a more general result described as follows:
Let $X$ be a Hunt process on a locally compact separable metric space in
a strong duality with $\widehat{X}$. Assume that transition densities of
$X$ and $\widehat{X}$ are comparable to the function $\widetilde{q}(t,x,y)$
defined in terms of the volume of balls and a certain scaling function.
For an open set $D$ consider the killed process $X^D$, and a critical
smooth measure on $D$ with the corresponding positive additive functional
$(A_t)$. We show that the heat kernel of the the Feynman-Kac semigroup
of $X^D$ through the multiplicative functional $\exp(-A_t)$ admits the
factorization of the form
${\mathbf P}_x(\zeta >t)\widehat{\mathbf P}_y(\widehat{\zeta}>t)\widetilde{q}(t,x,y)$.
This is joint work with Soobin Cho, Panki Kim and Zoran Vondracek.

Thursday November 15, 2018

Friday November 16, 2018

Monday November 19, 2018

**Commutative Algebra Seminar**

GKZ-systems and mixed Hodge modules

Uli Walther (Purdue University)

11:00 AM in 427 SEO

I will define GKZ-systems, and talk a little about their properties from the
algebraic, analytic, and combinatorial point of view. Then I will
discuss a theorem of Gelfand et al, and a sharpening by Mathias Schulze and
myself, on the question which GKZ-systems arise as (D-module-)direct
image of a natural D-module on a torus. In such cases, the GKZ-system can
inherit a mixed Hodge module structure. I will then explain work with
Thomas Reichelt that computes the weight filtration of this MHM structure on a
class of GKZ-systems that comes up naturally in mirror symmetry. This
complements work of Reichelt and Christian Sevenheck who computed the Hodge
filtration, and supersedes computations of Batyrev who determined the
weight filtration in a generic point. Very few of such explicitly
computed structures seem to be known.

**Geometry, Topology and Dynamics Seminar**

Positivity in the asymptotic regime

Yanir Rubinstein (University of Maryland)

3:00 PM in 636 SEO

A general theme in geometry is the classification of algebraic/differential geometric structures which satisfy a positivity property. In this talk I will propose an ``asymptotic" version of this theme based on joint work with Cheltsov, Martinez-Garcia, and Zhang. On the algebraic side, we introduce the class of asymptotically log Fano varieties and state a classification theorem in dimension 2, generalizing the classical efforts of the Italian school. The novelty here is the use of tools of convex optimization. On the differential side, I will give a conjectural picture for existence of singular Kahler-Einstein , explain progress towards this conjecture, and, time permitting, relations to singular Kahler-Ricci solitons.

**Analysis and Applied Mathematics Seminar**

On a dissipative Gross-Pitaevskii-type model for exciton-polariton condensates

Ryan Obermeyer (University of Illinois at Chicago)

4:00 PM in 636 SEO

We study a generalized dissipative Gross-Pitaevskii-type model
arising in the description of exciton-polariton condensates. We derive rigorous
existence and uniqueness results for this model posed on the one dimensional
torus and derive various a-priori bounds on its solution. Then, we analyze
in detail the long time behavior of spatially homogenous solutions and their
respective steady states. In addition, we will present numerical simulations in
the case of more general initial data. We also study the corresponding adiabatic
regime which results in a single damped-driven Gross-Pitaveskii equation and
compare its dynamics to the one of the full coupled system.
Joint work with C. Sparber, P. Antonelli, P. Markowich, and J. Sierra

Tuesday November 20, 2018

**Graduate Groups and Dynamics Seminar**

Spectral gap for groups of algebraic matrices

Wouter van Limbeek (UIC)

3:00 PM in 1227 SEO

Previously we have established a spectral gap for weakly diophantine groups (first part of the work of Benoist-De Saxce). Today we'll cover the second part: We show that groups with algebraic entries are weakly diophantine.

**Logic Seminar**

An upper bound for uB-sealing

Grigor Sargsyan (Rutgers University)

3:30 PM in 427 SEO

Woodin showed that, assuming the existence of a supercompact cardinal and a class of Woodin cardinals, after collapsing a supercompact cardinal to be countable,
the theory of L(Gamma_{uB}) is sealed. Here, Gamma_{uB} is the collection of the universally Baire sets of reals. We say that the theory of L(Gamma_{uB}) is sealed if for any V-generic
g and a V[g]-generic h, there is an elementary embedding j: L(Gamma_uB)^{V[g]}-> L(Gamma_uB)^{V[g*h]}. It has been conjectured by the speaker that sealing has a weak large cardinal strength,
and its weakness is the reason why the core model induction becomes so much more complicated after passing the threshold given by sealing. In a very recent work, the speaker and Trang
showed that sealing is indeed weak, weaker than a Woodin cardinal that is itself a limit of Woodin cardinals. After stating the relevant theorems we will outline why exactly the core model induction becomes
rather difficult after this threshold.

Monday November 26, 2018

**Special Colloquium**

Bernoulli shifts and entropy theory

Brandon Seward (Courant Institute )

3:00 PM in 636 SEO

In ergodic theory, one often studies measure-preserving actions of countable groups on probability spaces.
Bernoulli shifts are a class of such actions that are particularly simple to define, but despite several decades
of study some elementary questions about them still remain open,
such as how they are classified up to isomorphism. Progress in understanding Bernoulli shifts
has historically gone hand-in-hand with the development of a tool known as entropy.
In this talk, I will review classical concepts and results, which apply in the case where the
acting group is amenable, and then I will discuss recent developments that are beginning to illuminate
the case of non-amenable groups.

There will be tea in SEO 300 starting at 4:15.

Wednesday November 28, 2018

Friday November 30, 2018

Monday December 3, 2018

**Mathematical Computer Science Seminar**

Algorithms for #BIS-hard problems on expander graphs

Matthew Jenssen (Oxford)

3:00 PM in 427 SEO

We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) d-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite d-regular tree. Joint work with Peter Keevash and Will Perkins.

**Analysis and Applied Mathematics Seminar**

Interfacial dynamics of dissolving objects in fluid flow

Christopher Rycroft (Harvard University)

4:00 PM in 636 SEO

An advection–diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow will be presented. By taking advantage of conformal invariance of the model, a numerical method will be introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of several dissolving objects will be shown, all of which show collapse to a single point in finite time. The simulations reveal a surprising connection between the position of the collapse point and the initial Laurent coefficients, which was subsequently derived analytically.

Wednesday December 5, 2018

Tuesday January 15, 2019

Wednesday January 23, 2019

Monday January 28, 2019

Monday February 11, 2019

Wednesday February 13, 2019

Wednesday February 20, 2019

Friday March 1, 2019

Monday March 4, 2019

Monday March 11, 2019

Wednesday March 13, 2019

Friday March 15, 2019

Monday March 18, 2019

Wednesday March 27, 2019

Wednesday April 10, 2019

Friday April 12, 2019

Wednesday April 17, 2019

Monday April 22, 2019

Wednesday April 24, 2019

Wednesday May 8, 2019