## Colloquia

Colloquia for the Department of Mathematics and Statistics are normally held in **University Hall 4010 on Fridays at 4:00 p.m.** Any departures from this are indicated below. Light refreshments are served before
the colloquia in 2040 University Hall. Driving directions, parking information and maps are available on the university website.

## 2015-2016 Colloquia

Below is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.

**Spring Semester**

**May 2, 2016, Monday, 4:00-5:00pm in UH 4170**

Pablo Roldan (ITAM Instituto Tecnológico Autónomo de México) *Instabilities in the Restricted Three Body Problem*

Abstract: The stability of the Solar System is a longstanding problem. Over the centuries, mathematicians have spent an inordinate amount of energy proving stronger and stronger stability theorems for dynamical systems related to the Solar System within the frame of the Newtonian N-body problem. A key result in this direction is Arnold's theorem, which proves the existence of a set of positive Lebesgue measure filled by invariant tori in planetary systems, provided that the masses of the planets are small. However, in the phase space, the gaps left by the invariant tori leave room for instability.

In fact, the numerical computations of Sussman, Wisdom and Laskar have showed that over the life span of the Sun, collisions and ejections of inner planets are probable. Our Solar System is now widely believed to be unstable, and the general conjecture about the N-body problem is quite the opposite of what it used to be: no topological stability whatsoever holds, in a very strong sense. (Herman called this "the oldest open problem in dynamical systems").

Currently, the above conjecture is largely out of reach. A more modest but still very challenging goal is a local version of the conjecture: If the masses of the planets are small enough, the wandering set (unstable orbits) accumulates on the set of circular, coplanar, Keplerian motions.

In this talk, I will show the existence of large instabilities in a simplified planetary system (the Sun-Jupiter-Asteroid three body problem) and I will describe the associated instability mechanism. This provides a step towards the proof of Herman's local conjecture. Such instabilities occur near mean-motion resonances. This is the first time that large instabilities are established in the three body problem.

I will also describe the methods that we use, which are quite diverse (analytic, geometric, topologic, and numeric), and are representative of my research. If time permits, I will review some other results that we have obtained showing instability of trajectories in different regimes, and propose some open problems.

**April 29, 2016**

Chunhua Shan (University of Alberta, Canada) *Nilpotent Singularities in Dynamical Systems and Hilbert's 16th Problem*

Abstract: Nilpotent singularities of high codimension can be the organizing center for a complex system. The study of nilpotent singularities is essential in the theory and applications of differential equations and dynamical systems. Particularly it is closely related to the bifurcation theory, geometric singular perturbation theory and finiteness part of Hilbert's 16th problem.

In this talk I will briefly introduce the recent progress of finiteness part of Hilbert's 16th problem and show that 4 families of graphics with nilpotent singularities have finite cyclicity. Then I will present some applications of bifurcation theory and geometric singular perturbation theory in mathematical epidemiology by studying nilpotent singularities. Relaxation oscillations observed in epidemiology will be investigated. Stochastic bifurcations will also be discussed.

**April 27, 2016, Wednesday, 4:00-5:00pm in UH 4170**

Sean Rostami (University of Wisconsin - Madison) *On the Canonical Representatives of a Finite Weyl Group*

Abstract: Let K be a field and G a split connected reductive affine algebraic K-group. Let T be a split maximal torus of G, W its finite Weyl group, and R its root system. After fixing a realization of R in G and choosing a simple system for R, one gets a system of representatives for W in G(K), called the Canonical Representatives. It is well-known that these representatives rarely form a subgroup, and it is necessary for some questions to understand and quantify this failure. Various new formulas are given which constitute progress in this direction. An application of such formulas to the simple supercuspidals of Gross-Reeder and Reeder-Yu is provided.

**April 25, 2016, Monday, 4:00-5:00pm in UH 4410**

Christsopher Ormerod (Caltech) *Reductions of discrete nonlinear wave equations and discrete isomonodromy*

Abstract: The group-invariant solutions of the Korteweg-de Vries (KdV) equation may be expressed in terms of elliptic functions and Painlevé transcendents. We present discrete analogues of these results. In particular, we show how reductions of partial difference equations known as the discrete KdV equation and discrete Schwarzian KdV equation give us discrete analogues of elliptic equations and discrete Painlevé equations. Using discrete isomonodromic deformations, we present reductions expressed in terms of higher discrete Painlevé equations than those previously obtained. More generally, we show how reductions of these partial difference equations give discrete Garnier systems. Applications to orthogonal polynomials, continuous isomonodromy and tropical geometry are specified.

**April 22, 2016**

Pengfei Zhang (University of Mississippi) *Dynamical Systems with Some Hyperbolicity*

Abstract: In this talk we will discuss dynamical systems with some hyperbolicity. In the first part of the talk, we will introduce the dynamical billiards, and investigate the properties of a general convex billiard. We also give a new construction of convex billiard, the asymmetric lemons, on which the dynamics are completely hyperbolic. In the second part of the talk, we discuss the dynamical systems with partial hyperbolicity. We obtain some interesting dichotomies. For example, we show that either the system is completely dissipative, or one can physically observe the transitivity of the system. In the third part of the talk, we describe several characterizations of a chaotic system. We will show that there exist some geometric obstructions for a system to be simple.

**April 18, 2016, Monday, 4:00-5:00pm in UH 4410**

Alimjon Eshmatov (Cornell University) *Dixmier Groups. Non-Commutative Poisson Structures on Calabi-Yau Algebras.*

Abstract: In the first part of my talk I will discuss our recent works which answer to old questions posed by T.Stafford about automorphism groups of differential operators on curves. We give a geometric presentation of these groups $\{G_n\}_{n\ge 0}$ using the Bass-Serre theory of groups acting on trees. This result generalizes well-known theorems of Dixmier and Makar-Limanov on automorphisms of the first Weyl algebra. Then we show that $G_n$ have natural infinite-dimensional (ind-) algebraic group structures. We prove that the conjugacy classes of non-abelian Borel subgroups of $G_n$ are in bijection with the partitions of $n$. This implies that the $G_n$ are pairwise non-isomorphic as abstract groups. This joint work with Y. Berest and F. Eshmatov.

In the second part of my talk, we introduce the notion of a derived Poisson structure on an associative (not necessarily commutative) algebra A. This structure is characterized by the property of being a "weakest" structure on A that induces natural Poisson structures on the derived moduli spaces of finite-dimensional representations of A. A derived Poisson structure gives rise to a graded Lie algebra structure on cyclic homology; it can thus be viewed as a higher homological extension of the notion of $H_0$-Poisson structure introduced by W. Crawley-Boevey (2011). We will give new examples and present recent results on Calabi-Yau algebras obtained in joint work with X. Chen, F. Eshmatov and S. Yang. If time permits, I will explain the relation between derived Poisson structures and the Chas-Sullivan brackets arising in string topology.

**April 8, 2016**

Naomi Tanabe (Dartmouth College) *Nonvanishing of central values of L-functions*

Abstract: Analyzing the special values of L-functions has been a significant target of research in Number Theory ever since the Riemann zeta function was introduced in the eighteenth century. In this talk, I will discuss various L-functions and study their special values. In particular, I will give a special attention to some nonvanishing properties of L-functions attached to Hilbert modular forms.

**Fall Semester**

**November 13, 2015**

Jeffrey Morton (University of Toledo) *Introduction to Topological Quantum Field Theory in 2 Dimensions*

Abstract: This talk gives an introductory view of topological quantum field theory (TQFT) from a category-theoretic point of view. I will explain what a TQFT is and introduce the necessary ideas from category theory, leading to an explanation of how a TQFT can be described as a kind of monoidal functor. Then I will describe how, for 2-dimensional spacetimes, this means that a TQFT is equivalent to a certain kind of bialgebra called a Frobenius algebra.

**October 16, 2015**

Logan Hoehn (Nipissing University) *A Complete Classification of Homogeneous Plane Continua*

Abstract: A space X is homogeneous if for every pair of points in X, there is a homeomorphism of X onto itself taking one point to the other. Kuratowski and Knaster asked in 1920 whether the circle is the only homogeneous compact connected space (continuum) in the plane. Explorations of this problem fueled a significant amount of research in continuum theory, and among other things, led to the discovery of two new homogeneous continua in the plane: the pseudo-arc and the circle of pseudo-arcs. I will describe our recent result which shows that there are no more undiscovered homogeneous compact connected spaces in the plane. Our result actually can be used to show that we have determined all compact homogeneous spaces in the plane.

This is joint work with Lex G. Oversteegen, University of Alabama at Birmingham.

**October 9, 2015**

Morley Davidson (Kent State University) *Calculating the maximum number of moves needed to solve Rubik's Cube*

Abstract: The problem of determining the Cube's diameter, ie. maximum number of moves required to solve an arbitrary scrambling, goes back to the puzzle's origins circa 1980. In the last few years the problem was finished off for the two most popular ways of counting moves, the "half-turn" and "quarter-turn" metrics, with the help of supercomputers at Google and the Ohio Supercomputer Center, respectively. In this talk we discuss the mathematical and algorithmic tricks that made these computations possible and affordable with present-day machinery.

**October 2, 2015 (UH 4480)**

Yichuan Zhao (Georgia State University) *Smoothed Jackknife Empirical Likelihood Inference for the Difference of ROC Curves*

Abstract: For the comparison of two diagnostic markers at a flexible specificity, people apply the difference of two correlated receiver operating characteristic (ROC) curves to identify the diagnostic test with stronger discriminant ability. In this paper, we employ jackknife empirical likelihood (JEL) method to construct confidence intervals for the difference of two correlated continuous-scale ROC curves. Using the jackknife pseudo-sample, we can avoid estimating several nuisance variables which have to be estimated in existing methods. We prove that the smoothed jackknife empirical log likelihood ratio is asymptotically chi-squared distribution. Furthermore, the simulation studies in terms of coverage probability and average length of confidence intervals show the good performance in small samples with a moderate computational cost. A real data set is used to illustrated our method.

**October 2, 2015**

Leonard Scott (The University of Virginia) *Finite and algebraic groups, some history and a recent application*

Abstract: The interrelationship of finite and algebraic group theory, together with modern computer calculations (let me mention Frank Luebeck, and my own student, Tim Sprowl), led to the demise in 2012 of a 1961 conjecture of G. E. Wall related to finite group permutation actions. Stated in abstract group theory terms, the conjecture asserted that the number of maximal subgroups of a given finite group G was less than the number of elements of G. It was known to be true for all solvable finite groups and almost all finite simple groups, yet there are infinitely many counterexamples. The part played by algebraic group theory brought together several quite substantial developments in the subject, part and parcel of a historical march and a bigger picture, involving not only group actions and subgroups, but also representation theory and cohomology. The role of a conjecture of Lusztig on algebraic group representations . true for very large prime field characteristics . stands out, as does work of Cline-Parshall-Scott-van der Kallen relating cohomology of these representations to the finite group case.

**Shoemaker Lecture Series September 23-25, 2015**

Amie Wilkinson (University of Chicago)

**Lecture 1: The Ergodic Hypothesis and Beyond - "The General Case"**

September 23, 4:00-5:00pm in GH 5300

Abstract: The celebrated Ergodic Theorems of George Birkhoff and von Neumann in the 1930's gave rise to a mathematical formulation of Boltzmann's Ergodic Hypothesis in thermodynamics. This reformulated hypothesis has been described by a variety of authors as the conjecture that ergodicity -- a form of randomness of orbit distributions -- should be "the general case" in conservative dynamics. I will discuss remarkable discoveries in the intervening century that show why such a hypothesis must be false in its most restrictive formulation but still survives in some contexts. In the end, I will begin to tackle the question, "When is ergodicity and other chaotic behavior the general case?"

**Lecture 2: Robust mechanisms for chaos, I: Geometry and the birth of stable ergodicity**

September 24, 4:00-5:00pm in UH 4010

Abstract: The first general, robust mechanism for ergodicity was developed by E. Hopf in the 1930's in the context of Riemannian geometry. Loosely put, Hopf showed that for a negatively curved, compact surface, the "typical" infinite geodesic fills the manifold in a very uniform way, a property called equidistribution. I will discuss Hopf's basic idea in both topological and measure-theoretic settings and how it has developed into a widely applicable mechanism for chaotic behavior in smooth dynamics.

**Lecture 3: Robust mechanisms for chaos, II: Stable ergodicity and partial hyperbolicity**

September 25, 10:00-11:00am in UH 4010

Abstract: Kolmogorov introduced in the 1950's a robust mechanism for non-ergodicity, which is now known as the KAM phenomenon (named for Kologorov, Arnold and Moser). A current, pressing problem in smooth dynamics is to understand the interplay between KAM and Hopf phenomena in specific classes of dynamical systems. I will describe a class of dynamical systems, called the partially hyperbolic systems, in which the two phenomena can in some sense be combined. I'll also explain recent results that give strong evidence for the truth of a modified ergodic hypothesis in this setting, known as the Pugh-Shub stable ergodicity conjecture.