## Measure-Theoretic Dvoretzky Theorem and Applications to Data Science

- Probability,Uploaded Videos ( 1443 Views )SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">at this link (here).</a>

## Christine Heitsch : The Combinatorics of RNA Branching

- Mathematical Biology ( 301 Views )Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. For example, the branching of an RNA secondary structure is an important molecular characteristic yet difficult to predict correctly, especially for sequences on the scale of viral genomes. However, results from enumerative, probabilistic, analytic, and geometric combinatorics yield insights into RNA structure formation, and suggest new directions in viral capsid assembly.

## Sarah Schott : Computational Complexity

- Graduate/Faculty Seminar ( 272 Views )What does it mean for a problem to be in P, or NP? What is NP completeness? These are questions, among others, that I hope to answer in my talk on computational complexity. Computational complexity is a branch of theoretical computer science dealing with analysis of algorithms. I hope to make it as accessible as possible, with no prior knowledge of algorithms and running times.

## Paul Aspinwall : The Ubiquity of the ADE Classification

- Graduate/Faculty Seminar ( 262 Views )Many classes of mathematical objects turn out to be classified in the same way --- two infinite series and 3 "exceptional" objects. These include symmetries of 3-dimensional solids, rigid singularities, certain types of Lie algebras, positive definite even intersection forms, etc. Discovering why such classes should have the same classification has led to many beautiful ideas and observations. I will give a review of some of the basic ideas (assuming very little in the way of prerequisites) and I may have time to say why string theory has been important in this context.

## Arthur Szlam : A Total Variation-based Graph Clustering Algorithm for Cheeger Ratio Cuts

- Undergraduate Seminars ( 259 Views )I will discuss a continuous relaxation of the Cheeger cut problem on a weighted graph, and show how the relaxation is actually equivalent to the original problem. Then I will introduce an algorithm which experimentally is very efficient at approximating the solution to this problem on some clustering benchmarks. I will also give a heuristic variant of the algorithm which is faster but often gives just as accurate clustering results. This is joint work with Xavier Bresson, inspired by recent papers of Buhler and Hein, and Goldstein and Osher, and by an older paper of Strang.

## David Shea Vela-Vick : The equivalence of transverse link invariants in knot Floer homology

- Presentations ( 257 Views )The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces. Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined. The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams. The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo. We show that these two previously defined invariant agree. Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.

## Dmitry Vagner : Higher Dimensional Algebra in Topology

- Graduate/Faculty Seminar ( 237 Views )In his letter, "Pursuing Stacks," Grothendieck advocated to Quillen for the use of "higher" categories to encode the higher homotopy of spaces. In particular, Grothendieck dreamt of realizing homotopy n-types as n-groupoids. This powerful idea both opened the field of higher dimensional algebra but also informed a paradigm in which the distinction between topology and algebra is blurred. Since then, work by Baez and Dolan among others further surveyed the landscape of higher categories and their relationship to topology. In this talk, we will explore this story, beginning with some definitions and examples of higher categories. We will then proceed to explain "the periodic table of higher categories" and the four central hypotheses of higher category theory. In particular, these give purely algebraic characterizations of homotopy types, manifolds, and generalized knots; and account for the general phenomena of stabilization in topology. No prerequisites beyond basic ideas in algebraic topology will be expected.

## Jim Nolen : Bumps in the road: stability and fluctuations for traveling waves in an inhomogeneous medium

- Undergraduate Seminars ( 233 Views )If a partial differential equation has coefficients that vary with respect to the independent (spatial) variables, how do the fluctuations in the coefficients effect the solution? In particular, if these fluctuations have a statistical structure, can anything thing be said about the statistical behavior of the solutions? I'll consider these questions in the context of a scalar reaction diffusion equation. Without the variable coefficients, the equation admits stable traveling wave solutions. It turns out that the stability of wave-like solutions persists in a heterogeneous environment, and this fact can be used to derive a central limit theorem for the wave when the environment has a certain statistical structure.

I'll try to explain two interesting mathematical issues: First, how can one prove stability of the wave-like solution in this general setting, since spectral techniques don't seem applicable? Second, how can one use the structure of the problem to say something about how randomness in the environment effects the solution?

## Yanir Rubinstein : Einstein metrics on Kahler manifolds

- Geometry and Topology ( 229 Views )The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric. Kahler-Einstein (KE) metrics are a natural generalization of such metrics, and the search for them has a long and rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2. In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk I will try to give an introduction to Kahler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. One key ingredient is a new C^{2,\alpha} a priori estimate and continuity method for the complex Monge-Ampere equation. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a simple normal crossing divisor.

## Wolfgang Gaim : Semiclassical approximations to quantum mechanical equilibrium distributions

- Presentations ( 225 Views )In his 1932 paper, Eugene Wigner introduced the now famous Wigner function in order to compute quantum corrections to classical equilibrium distributions. We show how to extend this program and compute semiclassical approximations to quantum mechanical equilibrium distributions for slow, semiclassical degrees of freedom coupled to fast, quantum mechanical degrees of freedom. The main examples are molecules and electrons in crystalline solids. Where we will focus on the thermodynamics of the Hofstadter model as an application of the general results. The semiclassical formulas contain, in addition to quantum corrections similar to those of Wigner, also modifications of the classical Hamiltonian system used in the approximation: The classical energy and the Liouville measure on classical phase space turn out to have non-trivial-expansions in the semiclassical parameter. This talk is based on joint work with Stefan Teufel.

## Richard Hain : The Lie Algebra of the Mapping Class Group, Part 1

- Geometry and Topology ( 220 Views )In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.

## James Bremer : Improved methods for discretizing integral operators

- Presentations ( 217 Views )Integral equation methods are frequently used in the numerical solution of elliptic boundary value problems. After giving a brief overview of the advantages and disadvantages of such methods vis-a-vis more direct techniques like finite element methods, I will discuss two problems which arise in integral equation methods. In both cases, I take a contrarian position. The first is the discretization of integral operators on singular domains (e.g., surfaces with edges and curves with corners). The consensus opinion holds that integral equations given on such domains are exceedingly difficult to discretize and that sophisticated analysis, often specific to a particular boundary value problem, is required. I will explain that, in fact, the efficient solution of a broad class of such problems can be effected using an elementary approach. Exterior scattering problems given on planar domains with tens of thousands of corner points can be solved to 12 digit accuracy on my two year old desktop computer in a matter of hours. The second problem I will discuss is the evaluation of the singular integrals which arise form the discretization of weakly singular integral operators given on surfaces. Exponentially convergent algorithms for evaluating these integrals have been described in the literature and it is widely regarded as a "solved" problem. I will explain why this is not so and describe an approach which yields only algebraic convergence, but nonetheless performs better in practice than standard exponentially convergent methods.

## Li-Cheng Tsai : Interacting particle systems with moving boundaries

- Probability ( 215 Views )In this talk I will go over two examples of one-dimensional interacting particle systems: Aldous' up-the-river problem, and a modified Diffusion Limited Growth. I will explain how these systems connect to certain PDE problems with boundaries. For the up-the-river problem this connection helps to solve AldousÂ’ conjecture regarding an optimal strategy. For the modified DLA, this connection helps to characterize the scaling exponent and scaling limit of the boundary at the critical density. This talk is based on joint work with Amir Dembo and Wenpin Tang.

## Joseph Spivey! : Mapping Class Groups and Moduli Spaces

- Graduate/Faculty Seminar ( 212 Views )There are many different ways to make a compact 2-manifold of genus g into a Riemann surface. In fact, there is an entire space of dimension 3g-3 (when g>1) of possible holomorphic structures. This space is called the moduli space of Riemann surfaces of genus g. We will give a definition of moduli spaces and briefly talk about their construction, starting with the "easy" examples of g=0 and g=1. We will also talk about mapping class groups, which play an important part in the construction of moduli spaces.

## Kyoung Jin Lee : A scary, yet interesting, scenario to the fibrillating heart

- Nonlinear and Complex Systems ( 210 Views )Alternans, a beat-to-beat temporal alternation in the sequence of heart beats, is a known precursor of the development of cardiac fibrillation, leading to sudden cardiac death. The equally important precursor of cardiac arrhythmias is the rotating spiral wave of electro-mechanical activity, or reentry, on the heart tissue. In this talk, I will show that these two seemingly different phenomena can have a remarkable relationship: In well controlled in-vitro tissue cultures, isotropic populations of rat ventricular myocytes sustaining a temporal rhythm of alternans can support period-2 oscillatory re-entries, and vice versa. These re-entries bear `line defects' across which the phase of local excitation slips rather abruptly by $2\pi$, when a full period-2 cycle of alternans completes in $4\pi$. In other words, the cells belonging to the line defects are period-1 oscillatory whereas all the others in the bulk medium are period-2 oscillatory. We also find that a slowly rotating line defect results in a quasi-periodic like oscillation in the bulk medium. Some key features of these phenomena can be well reproduced in computer simulations of a nonlinear reaction-diffusion model.

## Hubert Bray : What do Black Holes and Soap Bubbles have in common?

- Graduate/Faculty Seminar ( 209 Views )We will begin with the idea of General Relativity, which Einstein called his "happiest thought," and then proceed with a qualitative and quantitative discussion of the curvature of space-time. We will describe the central role of differential geometry in the subject and the important role that mathematicians have played proving the conjectures of the physicists, as well as making a few conjectures of our own. Finally, we will describe the geometry of black holes and their relationship to soap bubbles.

## Yifeng Liu : Relative trace formulas and restriction problems for unitary groups

- Algebraic Geometry ( 208 Views )In this talk, I will introduce some new relative trace formulas toward the global Gan-Gross-Prasad conjecture for unitary groups, which generalize the trace formulas of Jacquet-Rallis and Flicker. In particular, I will state the corresponding conjecture of relative fundamental lemmas. A relation between the well-studied Jacquet-Rallis case the equal-rank case will also be discussed.

## Phillip Andreae : Spectral geometry and topology; Euler characteristic and analytic torsion

- Graduate/Faculty Seminar ( 207 Views )What do eigenvalues have to do with geometry and topology? The first part of the talk will provide a few answers to that very broad question, including a discussion of the Euler characteristic from a spectral theory perspective. The second part of the talk will be a brief introduction to my research in analytic torsion, a topological invariant defined in terms of eigenvalues. In particular I'll explain some similarities and differences between analytic torsion and Euler characteristic.