## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## Hubert Bray : An Overview of General Relativity

- Graduate/Faculty Seminar ( 167 Views )After brief introductions to special relativity and the foundations of differential geometry, we will discuss the big ideas behind Einstein's theory of general relativity. Einstein's theory replaces Newtonian physics not only as the best description of gravity according to experiments, but also as a philosophically pleasing and very geometric idea, which Einstein called his "happiest thought." We will also discuss the predictions made by general relativity, including the big bang and black holes, both of which are strongly supported by observations. We will discuss these ideas from a geometric perspective, and discuss some of the open problems and future directions that are currently being studied.

## Michael Nicholas : An 3rd order accurate method in 3D period electromagnetic scattering

- Graduate/Faculty Seminar ( 165 Views )Periodic electromagnetic scattering problems are interesting and challenging for various reasons. I will outline these problems and discuss my research in how to deal with singularities that arise. My methods include some analysis, some asymptotics, some numerics, a bunch of pictures I ripped off the web, and - as long as there are no follow up questions - a little bit of geometry.

## Mark Stern : Gauge theory : the geometry and physics of the ambiguity of acceleration

- Graduate/Faculty Seminar ( 163 Views )I will discuss the rich mathematical structures which arise when one asks how to define acceleration in the absence of a preferred coordinate system. I will introduce the Yang-Mills equations, which specialize to give electromagnetism and much of the physics of the standard model. I'll discuss aspects of the geometry, topology, and analysis of the Yang-Mills equations and how too much symmetry can actually make an analysis problem more difficult.

## Francis Motta : A Dynamical Systems Perspective on Complex Hadamard Matrices

- Graduate/Faculty Seminar ( 157 Views )Simply put, complex Hadamard matrices are scaled unitary matrices with entries drawn from unit complex numbers. They appear as an essential ingredient in quantum information theory and their real members have deep connections to finite geometry and number theory. For us, in this talk, they will be the fixed points of both discrete and continuous dynamical systems. We begin by introducing complex Hadamards and some essential preliminaries. We then discuss a discrete-time dynamical system which can be used to generate complex Hadamards as well as closely related objects known as mutually unbiased bases. Finally, we construct a continuous system whose fixed points are complex Hadamards and exploit classical results from dynamical systems theory to study local structure in spaces of complex Hadamards.

## Henry Adams : Evasion Paths in Mobile Sensor Networks

- Graduate/Faculty Seminar ( 155 Views )Suppose ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data (i.e. Cech complexes) as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends on more than just the fibrewise homotopy type of the region covered by sensors. In the setting of planar sensors that also measure weak rotation information, we provide necessary and sufficient conditions for the existence of an evasion path, and we pose an open question concerning Cech and alpha complexes. Joint with Gunnar Carlsson.

## Wai (Jenny) Law : Approximately Counting Perfect and General Matchings in Bipartite andGeneral Graphs

- Graduate/Faculty Seminar ( 154 Views )Approximating the permanent of a matrix with nonnegative entries is a well studied problem. The most successful approach to date uses Markov chains, and Jerrum, Sinclair, and Vigoda developed such a method that runs in polynomial time O(n^7 (log n)^4). We present a very different approach using self-reducible acceptance/rejection, and show that for a class of dense problems, our method has an O(n^4 log n) expected running time. Also, we extend our approach to approximate the number of perfect matchings in non-bipartite graphs and general matchings in general graphs.

## Mauro Maggioni : Random walks on data sets in high dimensions, and a new hot system of coordinates

- Graduate/Faculty Seminar ( 153 Views )I will motivate the need to analyze data sets in high dimensions, their geometrical properties and the properties of functions on them with several examples. I will focus on techniques based on random walks on data sets, and introduce a new nonlinear system of coordinates based on heat kernels, similar in spirit to the GPS system, for parametrizing data sets. If time allows, I will also discuss simple but surprisingly successful applications of the heat kernel to fit functions on data, that performs at the state-of-art or better as a classifier on a variety of benchmark data sets.

## Harold Layton : Irregular Flow Oscillations in the Nephrons of Spontaneously Hypertensive Rats

- Graduate/Faculty Seminar ( 151 Views )The nephron is the functional unit of the kidney. The flow rate in each nephron is regulated, in part, by tubuloglomerular feedback, a negative feedback loop. In some parameter regimes, this feedback system can exhibit oscillations that approximate limit-cycle oscillations. However, nephron flow in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the associated complex power spectra in SHR. A bifurcation analysis of the TGF model equation was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. Four potential sources of spectral complexity in SHR were identified: (1) bifurcations that produce qualitative changes in solution type, leading to multiple spectrum peaks and their respective harmonic peaks; (2) continuous lability in delay parameters, leading to broadening of peaks and their harmonics; (3) episodic lability in delay parameters, leading to multiple peaks and their harmonics; and (4) coupling of small numbers of nephrons, leading to broadening of peaks, multiple peaks, and their harmonics. We conclude that the complex power spectra in SHR may be explained by the inherent complexity of TGF dynamics, which may include solution bifurcations, variation in TGF parameters, and coupling between small numbers of neighboring nephrons.