## 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.

## 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.

## Joseph Spivey : A How-To Guide to Building Your Very Own Moduli Spaces (they make such great gifts)

- Graduate/Faculty Seminar ( 236 Views )I'll be talking about how to construct the moduli space for genus g Riemann surfaces with r boundary components. I'll draw lots of pictures and focus a lot of attention on genus 1 Riemann surfaces with 1 boundary component. As an application, I'll probably talk about H^1(SL2(Z)) with coefficients in various representations--and the correspondence to modular forms (briefly, and without a whole lot of proofs).

## George Lam : The Positive Mass Theorem in General Relativity

- Graduate/Faculty Seminar ( 235 Views )The Positive Mass Theorem in general relativity states that a spacelike hypersurface of a spacetime satisfying the dominant energy condition must have nonnegative total mass. In the special case in which the hypersurface is totally geodesic, local energy density coincides with scalar curvature, and the above theorem becomes a purely geometric statement about complete, asymptotically flat Riemannian manifolds. I will try to present the necessary background for one to understand the statement of the theorem. I will also discuss attempts to better understand the relationship between scalar curvature and total mass. Note that this talk is especially geared towards early graduate students and people specializing in other fields, and thus I will assume no previous knowledge of smooth manifolds, Riemannian geometry or general relativity.

## Zhennan Zhou : Semi-classical Schrodinger equation in the electromagnetic field: approximations and numerics

- Graduate/Faculty Seminar ( 225 Views )I will discuss the semi-classical Schrodinger equation with vector potentials, and its challenges in analysis and in numerical simulations. The time splitting spectral method method will be introduced to solve the equation directly, which is believed to have the optimal mesh strategy. Afterwards. a series of wave packet based approximation approaches will be introduced, like the Gaussian beam method, Hagedorn wave packets method and the Gaussian wave packet transformation method.

## Miles M. Crosskey : Mathematics in Magic

- Graduate/Faculty Seminar ( 221 Views )Many simple card tricks rely on mathematical principles and logic. I will be talking about some of these tricks, and the interesting ideas behind them. Hopefully I will have time to show you two or three tricks, and the proof to how they work. I will be using work from Mathematical Magic by Diaconis and Graham. The exciting thing about these tricks is they do not rely upon sleight of hand, and come out looking stunning nonetheless.

## 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.

## 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.

## 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.

## Shahed Sharif : Class field theory and cyclotomic fields

- Graduate/Faculty Seminar ( 202 Views )We'll undertake a gentle introduction to class field theory by investigating cyclotomic fields, including a proof of quadratic reciprocity. The results we'll discuss complement Les Saper's Grad Faculty seminar talk, though by no means is the latter a prerequisite. As a special treat, I will reveal a completely new, elementary proof of Fermat's Last Theorem.

## Jianfeng Lu : Surface hopping: Mystery and opportunities for mathematicians

- Graduate/Faculty Seminar ( 198 Views )Surface hopping is a very popular approach in theoretical chemistry for mixed quantum-classical dynamics. Yes, the above sentence looks scary. Let us start over again ... We will examine from a mathematical point of view how stochastic trajectories can be used to approximate solutions to a Schrodinger equation (which is different from what Feynman did). Besides some applications in chemistry, this is a nice topic since it combines ideas from asymptotic analysis, applied probability, and applied harmonic analysis. The only background assumed in this talk is "separation of variables" (and of course some PDEs where separation of variables is applied to).

## Yuriy Mileyko : Enter Skeleton: a brief overview of skeletal structures

- Graduate/Faculty Seminar ( 197 Views )Skeletal structures, such as medial axis and curve skeleton, are a particular class of shape descriptors. They have numerous applications in shape recognition, shape retrieval, animation, morphing, registration, and virtual navigation. This talk will give a brief overview of the medial axis and the curve skeleton. The focus will be on the properties of the two objects crucial to applications. We shall show that the rigorous mathematical definition of the medial axis has allowed for an extensive and successful study of such properties. The curve skeleton, on the other hand, is typically defined by the set of properties it has to possess. As a result, numerous methods for computing the curve skeleton have been proposed, each providing mostly experimental verification of the required properties. If time permits, I will mention my work on defining shape skeleta via persistent homology, thus providing a powerful platform for investigating their properties.

## Tom Witelski : Perturbation analysis for impulsive differential equations: How asymptotics can resolve the ambiguities of distribution theory

- Graduate/Faculty Seminar ( 194 Views )Models for dynamical systems that include short-time or abrupt forcing can be written as impulsive differential equations. Applications include mechanical systems with impacts and models for electro-chemical spiking signals in neurons. We consider a model for spiking in neurons given by a nonlinear ordinary differential equation that includes a Dirac delta function. Ambiguities in how to interpret such equations can be resolved via perturbation methods and asymptotic analysis of delta sequences.

## Kevin Kordek : Geography of Mapping Class Groups and Moduli Spaces

- Graduate/Faculty Seminar ( 191 Views )Mapping class groups are topological objects which can be used to describe the continuous symmetries of a surface. On the other hand, every compact orientable surface has a moduli space, a complex variety whose points parametrize all of its inequivalent complex structures. These concepts turn out to be closely related. In this talk, we'll cover the basics of both mapping class groups and moduli of Riemann surfaces, as well as explore their relationship.

## Timothy Lucas : Numerical Solutions of an Immunology Model

- Graduate/Faculty Seminar ( 188 Views )The immune system in vertebrates is composed of individual cells called lymphocytes which work together to combat antigens such as bacteria and viruses. Upon detecting foreign molecules these immune cells secrete soluble factors that attract other immune cells to the site of the infection. The soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the cells is inherently stochastic, but biased toward the gradient of the soluble factors. I will discuss a numerical method for solving the reaction-diffusion stochastic system based on a first order splitting scheme. This method makes use of known first order schemes for solving the diffusion, the reaction and the stochastic differential equations separately.

## Robert Bryant : Curves, Surfaces, and Webs: An Episode in 19th Century Geometry

- Graduate/Faculty Seminar ( 187 Views )An old question about surfaces in 3-space is: When can a surface be written as a sum of two curves? For example, the elliptic paraboloid z = x^2 + y^2 can be thought of as the sum of the two space curves (x,0,x^2) and (0,y,y^2). However, a little thought shows that most surfaces in space should not be expressible parametrically as X(s) + Y(t) where X and Y are space curves. Surfaces for which this can be done are called `surfaces of translation'. This raises the question of determining whether or not this is possible for a given surface and in how many ways. This simple question leads to some surprisingly deep mathematics, involving complex analysis and overdetermined systems of PDE, and to other questions that are still open today. I will explain some of these developments (and what they have to do with my own work). There will even be a few pictures.

## Robert Bryant : The Concept of Holonomy

- Graduate/Faculty Seminar ( 186 Views )In the 19th century, people began to study mechanical systems in which motion in a configuration space was constrained by 'no slip' conditions, such as, for example, a wheel or a ball rolling on a plane without slipping. It was immediately noticed that there were many cases in which these 'rolling' constraints did not prevent one from being able to join any two points in a configuration space by an admissible path, and these situations were called 'non-holonomic'. The notion of 'holonomy' arose as a way to quantify and study these 'non-holonomic' systems, and it has turned out to be very fruitful, with many applications in differential geometry and mathematical physics as well as in practical mechanics problems (such as figuring out how to use robot hands to manipulate 3-dimensional objects). In this talk, I'll introduce the ideas that led to the development of the concept of holonomy, show how some simple examples are computed, and describe how even very simple systems, such as a convex surface rolling over another surface without slipping or twisting, can lead to some surprising and exceptional geometry. No expertise in differential geometry will be assumed; if you are comfortable with vector calculus, you can enjoy the talk.

## Bianca Santoro : Nice person speaks of ... ?

- Graduate/Faculty Seminar ( 185 Views )THIS JUST IN - An Abstract: I plan to speak about the good old Calabi Conjecture, and its beautiful solution by Yau, that gave gim the Fields Medal. I will start with some background material, and see how far we can get into the proof!

## Aubrey HB : Persistent Homology

- Graduate/Faculty Seminar ( 184 Views )Persistent Homology is an emerging field of Computational Topology that is developing tools to discover the underlying structure in high-dimensional data sets. I will discuss the origins and main concepts involved in Persistent Homology in an accessible way, with illustrations and comprehensive examples. If time allows, I will also describe some current, as well as, future applications of Persistent Homology.

## Bill Allard : The Boundary Finder

- Graduate/Faculty Seminar ( 184 Views )(This abstract is in TeX source code. Sorry!) Fix a small positive number $h$. Let $$G=h\mathbb{Z}^2=\{(ih,jh):i,j\in\mathbb{Z}\};$$ thus $G$ is a rectangular grid of points in $\mathbb{R}^2$. Let $\Omega$ be an bounded open subset of $\mathbb{R}^2$ with $C^1$ boundary and let $E=\{x\in G:x\in\Omega\}$. {\bf Question One.} Given $E$ can one determine the length of $\partial\Omega$ to within $O(h)$? The answer to this question is ``yes'', provided $\Omega$ satisfies a certain natural ``thickness'' condition; without this additional assumption the answer may be ``no''. {\bf Question Two.} Is there a fast algorithm for determining the length of $\partial\Omega$. The answer to this question also ``yes''. In this talk I will describe the proof that the answer to Question One is ``yes'' and I will describe the fast algorithm whose existence is implied in the answer to Question Two. If time permits, I will describe some applications.

## Shrawan Kumar : Topology of Lie groups

- Graduate/Faculty Seminar ( 183 Views )I will give an overview of some of the classical results on the topology of Lie groups, including Hopf's theorem which fully determines the cohomology algebra over the real numbers of any Lie group. We will also discuss how the deRham cohomology of a compact Lie group can be represented by bi-invariant forms. In addition, we will discuss first and the second homotopy groups of Lie groups.

## Michael Jenista : Dynamical Systems and the Conley Index

- Graduate/Faculty Seminar ( 179 Views )An introductory lecture to the Conley Index theory. We consider the flow case and introduce the key object of study: an index pair of an isolated invariant set. Index pairs are robust under perturbations and their homotopy type is invariant, making them an ideal tool for problems with error terms or even data-generated systems. The relevant tools are algebraic topology and some knowledge of continuous flows.

## Leslie Saper : Quadratic Reciprocity from Euler to Langlands

- Graduate/Faculty Seminar ( 177 Views )The law of quadratic reciprocity was conjectured by Euler and first proved in full generality by Gauss. I will not prove quadratic reciprocity but I will discuss it in the context of the general reciprocity law due to Emil Artin. I will then explain how Langlands's program is a nonabelian generalization of this. If there is time, I will indicate how my work fits into this program.

## Michael Reed : The Ear for Mathematicians

- Graduate/Faculty Seminar ( 176 Views )The ear from the outside in. Eardrum, middle ear, cochlea, 8th nerve, brainstem, cortex. What happens anyway when you listen to Mozart or Van Halen? How do pressure waves become electrical signals? What happens next? Is there deep mathematics in the auditory system? And what are those carteliginous things doing flapping in the breeze on the side of your head? Who says an abstract has to have declarative sentences? Will some of these questions be answered? Come and see!

## Abraham Smith : DEs to EDS: How to solve PDEs without being clever

- Graduate/Faculty Seminar ( 172 Views )This talk is directed to anyone who cares about anything, at all levels. In particular, it will be a soft introduction to exterior differential systems (EDS). EDS is often associated with differential geometry, but it is really just a language for understanding the solution space of differential equations. The EDS viewpoint is temporarily mind-bending, but its concise and clean description of integrability, from conservation laws to geometric invariants, justifies the initial cramps.