# MSCS Seminar Calendar

Monday October 12, 2015

**Combinatorics Seminar**

Planted Partitions in Random Graphs

Sam Cole (UIC)

3:00 PM in SEO 427

In the

*planted partition problem*, $n = ks$ vertices of a random graph are partitioned into $k$ unknown ``clusters,'' each of size $s$. Edges between vertices in the same cluster and different clusters are included with constant probability $p$ and $q$, respectively (where $0 \le q < p \le 1$). The goal is to recover the unknown clusters from the randomly generated graph. This talk will give a brief survey of results for this problem and present a simple spectral algorithm that, with high probability, recovers the partition as long as the clusters sizes are at least $\Omega(\sqrt n)$.**Graduate Student Colloquium**

Counting lines on a cubic surface

Yajnaseni Dutta (Northwestern University)

4:00 PM in BH 209

A classical problem in algebraic geometry is to study the space of algebraic subspaces of an algebraic space. A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single algebraic equations of degree 3. One of the oldest and most beautiful statements in algebraic geometry is that on a smooth cubic surface there are exactly 27 lines. We begin with the classical approach, in a terms of cubic forms, and then proceed to the more modern approach, in which the cubic surface is considered as a blow up of the projective planes in 6 points.
Food will be provided.

**Analysis and Applied Mathematics Seminar**

Sharp interface limit in a phase field model of cell motility

Leonid Berlyand (Penn State)

4:00 PM in SEO 636

We study the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters such as strength of protrusion by actin filaments and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network.
The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion perturbed by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters. In the subcritical regime, the well-posedness of the problem is proved (M. Mizuhara et al., 2015). Our main focus is the supercritical regime where we established surprising features of the motion of the interface such as discontinuities of velocities and hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in the 2D model.
Because of properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. This is joint work with V. Rybalko and M. Potomkin.

Tuesday October 13, 2015

**Quantum Topology / Hopf Algebra Seminar**

Dold-Kan Theorem, Homotopy and Link Homology

Louis H Kauffman (UIC)

3:00 PM in SEO 612

This talk is a summary of joint work with Chris Gomes. We give a
self-contained statement of the Dold-Kan Theorem and discuss how it can be
applied to give a homotopy interpretation of Khovanov Homology.

**Logic Seminar**

Some new logical zero-one laws

Caroline Terry (UIC)

4:00 PM in SEO 427

Suppose $\mathcal{L}$ is a finite first-order language and for each integer $n$,
suppose $F(n)$ is a set of $\mathcal{L}$-structures with underlying set
$\{1,\ldots, n\}$. We say the family $F=\bigcup_{n\in \mathbb{N}}F(n)$ has
a zero-one law if for every first order sentence $\phi$,
the proportion of elements in $F(n)$ which satisfy $\phi$ goes to zero or one
as $n\rightarrow \infty$. In this talk we give a brief overview of the history
of this topic, then present some new examples of families with zero-one laws.
This is joint work with Dhruv Mubayi.

Wednesday October 14, 2015

**Graduate Geometry, Topology and Dynamics Seminar**

Entropy of Holomorphic Maps

Daniel Ingebretson

3:00 PM in SEO 612

If $ f $ is a smooth map of the Riemann sphere, its entropy is exactly the logarithm of its degree. This result is due to Gromov. We'll sketch an alternate proof of this theorem using some machinery from the theory of nonuniformly hyperbolic dynamical systems.

**Algebraic Geometry Seminar**

Loci of curves with subcanonical points in low genus

Nicola Tarasca (University of Utah)

4:00 PM in SEO 427

In this talk, I will discuss loci of curves with subcanonical points inside moduli spaces of curves. For instance, the locus of curves of genus 3 with a marked subcanonical point has two components: the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. I will show how to compute the classes of the closures of these codimension-two loci in the moduli space of stable curves of genus 3 with a marked point. Similarly, I will present the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. Finally, I will discuss the geometric consequences of these computations. This is joint work with Dawei Chen.

**Statistics Seminar**

Weighted Optimality Criteria and Design Search Algorithms

Jonathan Stallings (NCSU)

4:00 PM in SEO 636

Standard design criteria like the A-, E-, and D-criterion implicitly assume the experimenter is equally interested in all estimable functions. Because of this, efficient designs under these criteria spread information evenly across the estimation space. In some cases, optimal designs can be analytically derived under these criteria but researchers are beginning to rely on design search algorithms to find these designs. These computer-generated designs are often found under the D-criterion because of its fast computations with point- and coordinate-exchange algorithms. However, the D-criterion is a poor assessment of a design when the goals of an experiment imply differential interest among the estimable functions. To reflect relative importance, Stallings and Morgan (Biometrika, 2015, in press) introduced general weighted optimality criteria, which assign weights to variances so that greater weight implies greater interest. These criteria are natural extensions of standard design criteria so that design search algorithms can be easily modified to perform optimization with respect to this new class of criteria. This talk first reviews the theory of general weighted optimality criteria and shows how the weighted analogues of standard criteria behave. A straightforward modification of typical design search algorithms is then shown to perform weighted optimization. The algorithm is implemented in SAS PROC OPTEX to find efficient blocked treatment-versus-control designs; unblocked and blocked factorial experiments that are focused on main effect estimation; and factorial experiments under a baseline parameterization.

Thursday October 15, 2015

Friday October 16, 2015

**Departmental Colloquium**

From model theory to Diophantine geometry

David Marker (UIC)

3:00 PM in Lecture Center C4

In the 90s model theorists introduced the notion of
o-minimal geometry to find more general settings where results from
real algebraic geometry could be extended. In recent years these
ideas have found applications in Pila's work on the Andre-Oort
Conjecture. I will survey some of these ideas and Pila and Zannier's
application to give a new proof of the Manin-Mumford Conjecture.

Tea after 4 PM in the lounge on the 3rd floor of SEO

Monday October 19, 2015

**Geometry, Topology and Dynamics Seminar**

Counting non-simple closed curves on surfaces

Jenya Sapir (University of Illinois at Urbana-Champaign)

3:00 PM in SEO 636

We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has produced asymptotics for the growth of the number of simple closed curves and curves with one self-intersection (respectively) with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result.

**Analysis and Applied Mathematics Seminar**

Semi-Analytical Time Differencing Methods for Stiff Problems

Chang-Yeol Jung (Ulsan National Institute of Science and Technology)

4:00 PM in SEO 636

A semi-analytical method is developed based on conventional integrating factor (IF) and exponential time differencing (ETD) schemes for stiff problems.
The latter means that there exists a thin layer with a large variation in their solutions. The occurrence of this stiff layer is due to the multiplication of a very small parameter $\epsilon$
with the transient term of the equation. Via singular perturbation analysis, an analytic approximation of the stiff layer, which is called a corrector, is sought for and embedded into the IF and ETD methods. These new schemes are then used to approximate the non-stiff part of the solution. Since the stiff part is resolved analytically by the corrector, the new method outperforms the conventional ones in terms of accuracy. In this paper, we apply our new method for both problems of ordinary differential equations and some partial differential equations.

Tuesday October 20, 2015

**Dissertation Defense**

Dynamics of Equicontinuous Group Actions on Cantor Sets

Jessica Dyer (University of Illinois, Chicago)

1:00 PM in SEO 612

A Vietoris solenoid is the inverse limit of $n-to-1$ covering maps over the torus, which are all homogeneous.McCord introduced generalized weak solenoids and showed that they are homogeneous if the monodromy action is defined by a normal chain. Schori showed by construction that non-homogeneous weak solenoids exist. Rogers and Tollefson showed that there exists a weak solenoid that is not defined by a normal chain, but is homogeneous. They also constructed a non-homogeneous solenoid given by covering maps which are regular from level $i$ to $i-1$, but whose composition onto the base space is non-regular. Fokkink and Oversteegen gave a criterion in terms of defining group chains for a weak solenoid to be homogeneous, i.e. for the monodromy action to be regular. In this work, we investigate further the properties of the dynamics of group chains. We show that each group chain yields a minimal equicontinuous Cantor dynamical system. Conversely, we use a method of Kakutani-Rokhlin partitions to show that minimal equicontinuous Cantor dynamical systems can be represented by group chains. We then use their associated chains to classify minimal equicontinuous Cantor dynamical systems as regular, weakly regular, or irregular. We show that this classification is an invariant of the cardinality of the set of orbits of the Automorphism group. We consider the set $\mathfrak{G_{\phi}}$ of all group chains associated to a dynamical system, and show that the classification as regular, weakly regular, or irregular is an invariant of the number of equivalence classes of chains in $\mathfrak{G_{\phi}}$. We introduce a new invariant of a dynamical system called the

*discriminant group*, and show that its cardinality is related to the degree of non-homogeneity of the system. We give new proofs using group chains of the irregularity of the Schori and Rogers and Tollefson solenoids, and we introduce new examples of group chains which are weakly regular and have either finite or infinite discriminant group.**Logic Seminar**

Shelah's eventual categoricity conjecture in universal classes

Sebastien Vasey (CMU)

4:00 PM in SEO 427

Abstract elementary classes (AECs) are an axiomatic framework encompassing classes of models of an $L_{\lambda, \omega}$ sentence, as well as numerous algebraic examples. They were introduced by Saharon Shelah in the mid seventies. One of Shelah's goals was to study generalizations of Morley's categoricity theorem to the infinitary setup. Among several variations, Shelah conjectured the following eventual version: An AEC categorical in a high-enough cardinal is categorical on a tail of cardinals.
In this talk, we will prove the conjecture for universal classes. It is an interesting type of AEC introduced by Shelah in a milestone 1987 paper [Sh:300] (the work was done in 1985). They correspond approximately to classes of models of a universal $L_{\lambda, \omega}$ sentence. The proof of the conjecture proceeds by first observing that any universal class satisfies tameness: a locality property isolated by Grossberg and VanDieren which says that orbital types are determined by their small restrictions. Next, several structural properties are derived from categoricity: the class has amalgamation on a tail and in fact admits a well-behaved forking-like independence relation. Finally, a definition of a unidimensionality-like property (due to Shelah) is shown to follow from categoricity in a single cardinal and imply categoricity on a tail of cardinals. The argument generalizes to tame AECS which have primes over sets of the form $Ma$.

Wednesday October 21, 2015

**Algebraic Geometry Seminar**

Normal functions over locally symmetric varieties

Matthew Kerr (Washington University)

4:00 PM in SEO 427

An algebraic cycle homologous to zero on a variety leads to an extension of Hodge-theoretic data, and in a variational context to a family of extensions called a normal function. These may be viewed as "horizontal" sections of a bundle of complex tori, and are used to detect cycles modulo algebraic (or rational) equivalence. Conversely, the existence of normal functions can be used to predict that interesting cycles are present...or absent: a famous theorem of Green and Voisin states that for projective hypersurfaces of large enough degree, there are no normal functions (into the intermediate Jacobian bundle associated to these hypersurfaces) over any etale neighborhood of the coarse moduli space.
Inspired by recent work of Friedman-Laza on Hermitian variations of Hodge structure and Oort's conjecture on special (i.e. Shimura) subvarieties in the Torelli locus, R. Keast and I wondered about the existence of normal functions over etale neighborhoods of Shimura varieties. Here the function is supposed to take values in a family of intermediate Jacobians associated to a representation of a reductive group. In this talk I will explain our classification of the cases where a Green-Voisin analogue does *not* hold and where one therefore expects interesting cycles to occur, and give some evidence that these predictions might be "sharp".

Thursday October 22, 2015

Friday October 23, 2015

Monday October 26, 2015

**Geometry, Topology and Dynamics Seminar**

Hamiltonian monodromy: an overview and new perspectives

Konstantinos Efstathiou (University of Groningen)

3:00 PM in SEO 636

Torus bundles are one of the most prominent features of integrable
Hamiltonian systems. The monodromy of such torus bundles over circles is
called Hamiltonian monodromy. In this talk I will give an overview of
Hamiltonian monodromy and discuss the development of the subject. I will
also present the ``geometric monodromy theorem'' which associates
monodromy to the focus-focus singular points of the Hamiltonian system.
Then I will consider the case of $n$-DOF (degree of freedom) integrable
Hamiltonian systems with global $\mathbb T^{n-1}$ actions and discuss
recent results obtained for such systems together with N. Martynchuk. In
particular, we have shown that monodromy in such systems is associated
to points where the isotropy is an $\mathbb S^1$ subgroup of the
$\mathbb T^{n-1}$ action. In the special case of $2$-DOF systems with an
$\mathbb S^1$ action this result implies that monodromy is associated to
the fixed points of the action. Finally we give a general and easy to
apply formula for computing monodromy in $n$-DOF systems.

Wednesday October 28, 2015

Monday November 2, 2015

**Departmental Colloquium**

Adjacency and coherency preservers

Peter Semrl (University of Ljubljana, Slovenia)

3:00 PM in SEO 636

Two matrices are said to be adjacent if their difference is of
rank one. Fundamental theorems of geometry of matrices proved by L.-K.
Hua describe the general form of bijective maps on various spaces of
matrices preserving adjacency in both directions. We will present
several recent improvements of these results and some applications in
mathematical physics.

**Analysis and Applied Mathematics Seminar**

Analysis and computations of convection dominated flows in the presence of a boundary

Youngjoon Hong (UIC)

4:00 PM in SEO 636

In this talk, I will present convergence results of singularly perturbed problems in the sense of PDEs, which is related to the vanishing viscosity limit. I also provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.

Wednesday November 4, 2015

Friday November 6, 2015

Monday November 9, 2015

Tuesday November 10, 2015

Wednesday November 11, 2015

Thursday November 12, 2015

Friday November 13, 2015

Monday November 16, 2015

**Geometry, Topology and Dynamics Seminar**

A construction of limiting solutions of Hitchin's equations

Laura Fredrickson (University of Texas at Austin)

3:00 PM in SEO 636

I'll describe a construction of solutions of Hitchin's equations on a compact Riemann surface near the ``ends'' of the SL(n,C)-Hitchin moduli space. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's construction of SL(2,C) solutions of Hitchin's equations where the Higgs field is ``simple''. In the generalized construction, moduli spaces of irregular connections arise. This is ongoing work.

Tuesday November 17, 2015

**Logic Seminar**

Elementary amenable groups and the space of marked groups

Phillip Wesolek (Université catholique de Louvain)

4:00 PM in SEO 427

(Joint work with Jay Williams) The space of marked groups is a cantor space that parameterizes all countable groups. This space allows for tools from descriptive set theory to be applied to group-theoretic questions. The class of elementary amenable groups is the smallest class that contains the abelian groups and the finite groups and that is closed under group extension, taking subgroups, taking quotients, and taking directed unions. In this talk, we first give a characterization of elementary amenable marked groups in terms of well-founded trees; as a consequence, elementary amenability is equivalent to a chain condition. We then show the set of elementary amenable marked groups is coanalytic and non-Borel. This gives a new, non-constructive proof of a theorem of Grigorchuk: There are amenable non-elementary amenable groups. We conclude by discussing further questions and possible generalizations of the techniques.

Friday November 20, 2015

**Departmental Colloquium**

Graphs, vectors and groups

Noga Alon (Tel Aviv University)

3:00 PM in SEO 636

The study of Cayley sum-graphs of finite abelian groups is related to the investigation of pseudo-random graphs and to problems in Combinatorial Number Theory, Geometry and Information Theory. I will discuss this topic, describing the motivation
and focusing on several results that illustrate the interplay between Graph Theory, Geometry and Number Theory.

Tea on the 3rd floor at 4:15 PM

Wednesday December 2, 2015

Monday February 1, 2016

Monday February 8, 2016

Monday February 22, 2016

Friday February 26, 2016