## Miles M. Crosskey : Mathematics in Magic

- Graduate/Faculty Seminar ( 224 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.

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

- Graduate/Faculty Seminar ( 187 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!

## Shrawan Kumar : Topology of Lie groups

- Graduate/Faculty Seminar ( 186 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.

## Leonardo Mihalcea : What is Schubert calculus?

- Graduate/Faculty Seminar ( 166 Views )Do you ever wanted to know how many lines in 3−space intersect 4 given random lines ? (Answer: 2.) One way to prove this is to do explicit computations in the cohomology of the Grassmannian of lines in the projective space. But interestingly enough, one can also use Representation Theory, or symmetric functions (Schur polynomials), to answer this question. The aim of this talk is to present the basics of Schubert Calculus, as seen from the cohomological point of view. I will define Schubert varieties in Grassmannians, and discuss about how they intersect. The final goal is to show that 2 = 1+ 1 (and I may also use KnutsonĀ?s puzzles for another proof of this).

## 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.

## Ben Gaines : Fun with Fans: Quotient Construction of Toric Varieties

- Graduate/Faculty Seminar ( 152 Views )A toric variety is a special class of algebraic variety, which can be expressed by a picture that makes some of it's properties much easier to analyze. These varieties are an area of ongoing research in algebraic geometry, due in part to their applications in many different fields (such as mathematical physics). In this talk I will discuss how we can use a fan to construct a toric variety as a quotient. This talk will be accessible to all graduate students and will focus on examples, to illustrate the usefulness of this method.

## Benoit Charbonneau : Instantons and reduction of order via the Nahm transform

- Graduate/Faculty Seminar ( 152 Views )The instanton equations appear in gauge theory and generalize both the Maxwell equations and the harmonic equation. Their study has been and continues to be a very fertile ground for interactions between physicists and mathematicians. The object of this talk is a description of instanton solutions on S^1xR^3 due to Hurtubise and myself using the Nahm transform, a non-linear transformation that takes a system of PDE and produces a system of ODE or even a system of algebraic equations. This description allows us to answer existence questions for calorons.

## Kash Balachandran : The Kakeya Conjecture

- Graduate/Faculty Seminar ( 148 Views )In 1917, Soichi Kakeya posed the question: What is the smallest amount of area required to continuously rotate a unit line segment in the plane by a full rotation? Inpsired by this, what is the smallest measure of a set in $\mathbb{R}^n$ that contains a unit line segment in every direction? Such sets are called Kakeya sets, and can be shown to have arbitrarily small measure w.r.t. n-dimensional Lebesgue measure [and in fact, measure zero]. The Kakeya conjecture asserts that the Hausdorff and Minkowski dimension of these sets in $\mathbb{R}^n$ is $n$. In this talk, I will introduce at a very elementary level the machinery necessary to understand what the Kakeya conjecture is asking, and how the Kakeya conjecture has consequences for fields diverse as multidimensional Fourier summation, wave equations, Dirichlet series in analytic number theory, and random number generation. I'll also touch on how tools from various mathematical disciplines from additive combinatorics and algebraic geometry to multiscale analysis and heat flow can be used to obtain partial results to this problem. The talk will be geared towards a general audience.

## Luca F. Di Cerbo : Some facts and speculations about the full Yamabe problem in dimension four

- Graduate/Faculty Seminar ( 129 Views )In the first part of this talk I will review the basic properties of the Yamabe problem on compact manifolds and recall some well-known results of LeBrun concerning the computation of the Yamabe invariant in dimension four. In the second part I will suggest some possible generalizations for finite volume 4-manifolds. Hopefully some grad student will want to resume this work were I left it...

## Dave Rose : Cartans theorem on maximal tori

- Graduate/Faculty Seminar ( 128 Views )Cartan's theorem on maximal tori in compact Lie groups can be thought of as a generalization of the spectral theorem for unitary matrices. The goal of this talk will be to sketch the `topological' proof of this theorem, based on the Lefschetz fixed point theorem. Along the way, we'll encounter the flag variety, an interesting object whose geometry encodes the representation theory of the Lie group. Those who don't specialize in geometry or topology fear not, we will give examples showing that these concepts are very concrete objects familiar from linear algebra.

## Dino J. Lorenzini : Linear algebra: my lack, your luck?

- Graduate/Faculty Seminar ( 126 Views )Given a (n x n)-matrix M over a commutative integral domain R, one can try to associate to it a diagonal matrix called the Smith Normal Form of M. This can be done when R is the ring of integers, or the polynomial ring F[x] over a field F, and various applications of the existence of the Smith Normal Form are discussed in matrix theory. Which commutative integral domains R have the property that every matrix with coefficients in R admits a Smith Normal form? This is a very old question, as for instance Wedderburn in 1915 already discussed the case where R is the ring of holomorphic functions. I will review all necessary concepts, and discuss several easily stated open problems in this circle of ideas.

## Benoit Charbonneau : Gauge theory and modern problems in geometry

- Graduate/Faculty Seminar ( 121 Views )I will survey some modern questions in geometry that were solved or that could be solved using tools of gauge theory. This talk should be accessible to first year grad students, and of interest to anyone who is curious about what happens in the field of geometry.

## Jim Nolen : On the great effect of small noise

- Graduate/Faculty Seminar ( 119 Views )This talk will include some PDE, some probability, and some asymptotic analysis. The FKPP equation is a nonlinear partial differential equation that admits traveling wave solutions. It has been used as a simple model for many phenomena involving a stable state invading an unstable state (for example, an advantageous gene spreading through a population). Experiments and numerical simulations show that the speed at which the wave moves is much slower than what is predicted by the deterministic, continuum equation. One way to resolve this discrepancy is to account for the role of noise in the model by adding a stochastic term in the equation (i.e. a stochastic partial differential equation). Analysis of the SPDE has shown that very small noise in the equation introduces a very large correction to the speed associated with the deterministic model. I will explain the basics of the deterministic and stochastic equations, and I will explain some ideas about the asymptotic analysis of the stochastic waves. I hope to have time to explain some open problems related to this topic.

## Mark Iwen : Compressed Sensing for Manifold Data

- Graduate/Faculty Seminar ( 118 Views )We will discuss techniques for approximating a point in high-dimensional Euclidean space which is close to a known low-dimensional compact submanifold when only a compressed linear sketch of the point is available. More specifically, given a point, x, close to a known submanifold of R^D, we will consider linear measurement operators, M: R^D -> R^m, which have associated nonlinear inverses, A: R^m -> R^D, so that || x - A(Mx) || is small even when m << D. Both the design of good linear operators, M, and the design of stable nonlinear inverses, A, will be discussed. An algorithmic implementation of a particular nonlinear inverse will be presented, along with related stability bounds for the approximation of manifold data.

## Mauro Maggioni : A primer on wavelets and their applications

- Graduate/Faculty Seminar ( 117 Views )Wavelets are widely used in signal processing (e.g. analysis of sounds and music) and imaging, for tasks such as denoising and compression (ever wondered how jpeg works?). In harmonic analysis they have been used to understand and solve problems involving integral operators motivated by PDEs. In numerical PDEs they lead to fast algorithms for solving certain types of integral equations and PDEs. I will give a gentle introduction to wavelets and some of their motivating applications, accompanied by live demos. If time allows, I will discuss shortcomings and how they have been addressed in more recent developments and generalizations.

## Hangjun Xu : Constant Mean Curvature Surfaces in Asymptotically Flat Manifolds

- Graduate/Faculty Seminar ( 114 Views )The study of surfaces with constant mean curvature (CMC) goes back to 1841 when Delaunay classified all CMC surfaces of revolution. There has been consistent work on finding CMC hypersurfaces in various ambient manifolds. In this talk, we will discuss some nice properties of CMC surfaces, and then the existence of CMC surfaces in the Schwarzschild, and in general, asymptotically flat manifolds.

## Heekyoug Hahn : On tensor third $L$-functions of automorphic representations of $\GL_n(\A_F)$

- Graduate/Faculty Seminar ( 109 Views )Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic representation of $\GL_n(\A_F)$ where $F$ is a global field.