# Colloquium 2019

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday February 4, 2019 at 16:00, Wachman 617
Mathematical Aspects of Arbitrage

Ioannis Karatzas, Columbia University

We introduce models for financial markets and, in their context, the notions of portfolio rules and of arbitrage. The normative assumption of absence of arbitrage is central in the modern theories of mathematical economics and finance. We relate it to probabilistic concepts such as "fair game", "martingale", "coherence" in the sense of deFinetti, and "equivalent martingale measure".

We also survey recent work in the context of the Stochastic Portfolio Theory pioneered by E.R. Fernholz. This theory provides descriptive conditions under which opportunities for arbitrage, or outperformance, do exist; then constructs simple portfolios that implement them. We also explain how, even in the presence of such arbitrage, most of the standard mathematical theory of finance still functions, though in somewhat modified form.

• Monday February 18, 2019 at 16:00, Wachman 617
Counting closed geodesics: classical and non-classical behavior

Ilya Kapovich, CUNY

The problem of counting closed geodesics of bounded length, originally in the setting of negatively curved manifolds, goes back to the classic work of Margulis in 1960s about the dynamics of the geodesic flow. Since then Margulis' results have been generalized to many other contexts where some whiff of hyperbolicity is present. Thus a 2011 result of Eskin and Mirzakhani shows that for a closed hyperbolic surface S of genus $g\ge 2$, the number $N(L)$ of closed Teichmuller geodesics of length $\le L$ in the moduli space of $S$ grows as $e^{hL}/(hL)$ where $h=6g-6$. The number $N(L)$ is also equal to the number of conjugacy classes of pseudo-Anosov elements $\phi$ in the mapping class group $MCG(S)$ with $\log\lambda(\phi)\le L$, where $\lambda(\phi)>1$ is the "dilation" or "stretch factor" of $\phi$.

We consider an analogous problem in the $Out(F_r)$ setting, for the action of $Out(F_r)$ on a "cousin" of Teichmuller space, called the Culler-Vogtmann outer space $X_r$. In this context being a "fully irreducible" element of $Out(F_r)$ serves as a natural counterpart of being pseudo-Anosov. Every fully irreducible $\phi\in Out(F_r)$ acts on $X_r$ as a loxodromic isometry with translation length $\log\lambda(\phi)$, where again $\lambda(\phi)$ is the stretch factor of $\phi$. We estimate the number $N_r(L)$ of fully irreducible elements $\phi\in Out(F_r)$ with $\log\lambda(\phi)\le L$. We prove, for $r\ge 3$, that $N_r(L)$ grows doubly exponentially in $L$ as $L\to\infty$, in terms of both lower and upper bounds. These bounds reveal new behavior not present in classic hyperbolic dynamical systems. The talk is based on a joint paper with Catherine Pfaff.

• Monday March 11, 2019 at 16:00, Wachman 617
Molecular metaprogramming: Software and hardware to create enzyme like catalysts and atomically precise membranes using molecular Lego

Christian Schafmeister, Department of Chemistry, Temple University

My group has developed a radical new approach to creating large, complex molecules to carry out complex catalytic and molecular recognition functions that will work like enzymes and membrane channels but be more robust and “designable” (see inset figure). Our approach is to synthesize stereochemically pure cyclic building blocks (bis-amino acids) that we couple through pairs of amide bonds to create spiro-ladder oligomers with programmed shapes (molecular Lego). The shape of each molecular Lego structure is pre-organized and controlled by the sequence and stereochemistry of its component bis-amino acids. We are scaling up molecular Lego both in quantity and size to achieve molecular Lego structures that approach the size of small proteins whereupon they will unlock new capabilities. They will display complex three-dimensional structures and present pockets and complex surfaces (1,500 – 5,000 Daltons). We have developed a computer programming environment called Cando that enables the rational design of molecular Lego structures for catalytic and molecular recognition capabilities. I will describe our approach to molecular Lego and several applications of functionalized molecular Lego including catalysis to carry out C-H activation, hydrolyze nerve agents and stereochemically controlled poly-ester polymerization reactions. I will also describe our approach to developing atomically precise membranes to carry out separations with high flux and selectivity. I will also demonstrate how we are using our unique computational tools to design large, complex macromolecules and materials with catalytic and separation capabilities.

• Monday March 18, 2019 at 16:00, Wachman 617
Introduction to the asymmetric simple exclusion process (from a combinatorialist’s point of view)

Lauren Williams, Harvard University

The asymmetric simple exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice, subject to the condition that there is at most one particle per site. This model was introduced in 1970 by biologists (as a model for translation in protein synthesis) but has since been shown to display a rich mathematical structure. There are many variants of the model – e.g. the lattice could be a ring, or a line with open boundaries. One can also allow multiple species of particles with different “weights.” I will explain how one can give combinatorial formulas for the stationary distribution using various kinds of tableaux. I will also explain how the ASEP is related to interesting families of orthogonal polynomials, including Askey-Wilson polynomials, Koornwinder polynomials, and Macdonald polynomials. Based on joint work with Sylvie Corteel (Paris) and Olya Mandelshtam (Brown).

• Monday March 25, 2019 at 16:00, Wachman 617
Singular integrals and boundary problems on Riemannian manifolds

Marius Mitrea, University of Missouri

In this talk I will discuss, in a methodical manner, the process that lets us consider singular integral operators of boundary layer type in a given compact Riemannian manifold M, and then use these to solve boundary value problems in subdomains of M of a general nature, best described in the language of Geometric Measure Theory. The talk is intended for a general audience, and it only requires a basic background in analysis.

• Monday April 1, 2019 at 16:00, Wachman 617
Random matrices and stochastic geometry

Todd Kemp, UCSD

Random Matrix Theory has become one of the hottest fields in probability and applied mathematics. With deep connections to analysis, combinatorics, and even number theory and representation theory, in the age of big data it is also finding its place at the heart of data science.

The field has largely focused on two kinds of generalizations of Gaussian random matrices, either preserving entry-wise independence or preserving rotational invariance. From another point of view, however, the classical Gaussian matrix ensembles can be viewed as Brownian motion on Lie algebras, and this Lie structure goes a long way in explaining some of their known fine structure. This suggests a third, geometric generalization of these ensembles to study: Brownian motion on the corresponding matrix Lie groups.

In this lecture, I will discuss the state of the art in our understanding of the behavior of eigenvalues of Brownian motion on Lie groups, focusing on unitary groups and general linear groups. No specialized background knowledge is required. There will be lots of pictures.

• Monday April 8, 2019 at 16:00, Wachman 617
Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models

Marta Lewicka, University of Pittsburgh

We discuss some mathematical problems combining geometry and analysis, that arise from the description of elastic objects displaying heterogeneous incompatibilities of strains. These strains may be present in bulk or in thin structures, may be associated with growth, swelling, shrinkage, plasticity, etc. We will describe the effect of such incompatibilities on the singular limits' bidimensional models, in the variational description pertaining to the "non-Euclidean elasticity" and discuss the interaction of nonlinear PDEs, geometry and mechanics of materials in the prediction of patterns and shape formation.