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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.
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.
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.
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.
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).
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.
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.
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.
John Bush, MIT
Yves Couder and coworkers in Paris discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be exclusive to the microscopic, quantum realm. These walking droplets propel themselves by virtue of a resonant interaction with their own wavefield, and so represent the first macroscopic realization of a pilot-wave system of the form proposed for microscopic quantum dynamics by Louis de Broglie in the 1920s. New experimental and theoretical results allow us to rationalize the emergence of quantum-like behavior in this hydrodynamic pilot-wave system in a number of settings, and explore its potential and limitations as a quantum analog.
Lisa Fauci, Tulane University
Respiratory cilia that transport mucus in the lungs, spermatozoa that collectively move through the female reproductive tract, paddling appendages that propel a crawfish, and fish swimming in a school are all examples of oscillators that exert force on a surrounding fluid. Do the synchronous or phase-shifted periodic motions that we observe arise due to hydrodynamic coupling? We will discuss experiments and models of the self-organized pattern of beating flagella and cilia — from minimal models of colloidal particles driven by optical traps to more detailed models that include dynamics of the molecular motors driving the motion. We will also examine the role of fluid inertia on the dynamics of synchronization of such systems.
David Harbater, University of Pennsylvania
Local-global principles have long played an important role in number theory and in the study of curves over finite fields, beginning with the Hasse-Minkowski theorem on quadratic forms. After reviewing the classical situation, this talk will discuss local-global principles that have recently been found to hold in the context of certain "higher dimensional" fields, using new methods.
Ofer Zeitouni, Weizmann Institute and Courant Institute, NYU
We discuss the spectrum of high dimensional non-Hermitian matrices under small noisy perturbations. That spectrum can be extremely unstable, as the maximal nilpotent matrix $J_N$ with $J_N(i,j)=1$ iff $j=i+1$ demonstrates. Numerical analysts studied worst case perturbations, using the notion of pseudo-spectrum. Our focus is on finding the locus of most eigenvalues (limits of density of states), as well as studying stray eigenvalues ("outliers"). I will describe the background, show some fun and intriguing simulations, and present some theorems. No background will be assumed. The talk is based on joint work with Anirban Basak and Elliot Paquette.
Howard Nuer, University of Illinois at Chicago
The question of determining if a given algebraic variety is rational is a notoriously difficult problem in algebraic geometry, and attempts to solve rationality problems have often produced powerful new techniques. A well-known open rationality problem is the determination of a criterion for when a cubic hypersurface of five-dimensional projective space is rational. After discussing the history of this problem, I will introduce the two conjectural rationality criteria that have been put forth and then discuss a package of tools I have developed with my collaborators to bring these two conjectures together. Our theory of Relative Bridgeland Stability has a number of other beautiful consequences such as a new proof of the integral Hodge Conjecture for Cubic Fourfolds and the construction of full-dimensional families of projective HyperKahler manifolds.
Pavel Safronov, University of Zurich
Skein modules are certain vector spaces associated to 3-manifolds built from embedded links which may be viewed as a generalization of the Jones polynomial of links in the 3-sphere. In this talk I will explain their connection to quantum groups, Floer theory and supersymmetric gauge theories. I will also outline a recent proof of a conjecture of Witten on finite-dimensionality of skein modules for closed 3-manifolds. This is joint work with Sam Gunningham and David Jordan.
Jaclyn Lang, University of Paris 13
Although studying numbers seems to have little to do with shapes, geometry has become an indispensable tool in number theory during the last 70 years. Deligne's proof of the Weil Conjectures, Wiles's proof of Fermat's Last Theorem, and Faltings's proof of the Mordell Conjecture all require machinery from Grothendieck's algebraic geometry. It is less frequent to find instances where tools from number theory have been used to deduce theorems in geometry. In this talk, we will introduce one tool from each of these subjects -- Galois representations in number theory and cohomology in geometry -- and explain how arithmetic can be used as a tool to prove some important conjectures in geometry. More precisely, we will discuss ongoing joint work with Laure Flapan in which we prove the Hodge and Tate Conjectures for self-products of 16 K3-surfaces using arithmetic techniques.
Eric Ramos, University of Oregon
Perhaps one of the most well-known theorems in graph theory is the celebrated Graph Minor Theorem of Robertson and Seymour. This theorem states that in any infinite collection of finite graphs, there must be a pair of graphs for which one is obtained from the other by a sequence of edge contractions and deletions. In this talk, I will present work of Nick Proudfoot, Dane Miyata, and myself which proves a categorified version of the graph minor theorem. As an application, we show how configuration spaces of graphs must display some strongly uniform properties. We then show how this result can be seen as a vast generalization of a variety of classical theorems in graph configuration spaces. This talk will assume minimal background knowledge, and will display few technical details.
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