Greg Herschlag : Fluid Flow Through Permeable Channels
- Graduate/Faculty Seminar ( 110 Views )This talk will serve as a brief introduction to Newtonian fluids and focus on flow through channels. It will begin with a brief introduction to the Navier-Stokes and Stokes equations and continue with a derivation of fluid flow through an impermeable channel leading to classical Poiseuille flow. Next, I will present some results for flow through permeable channels. Finally, I will discuss a new result that has been found in collaboration with Jian-Guo Liu, Anita Layton and myself, in which we have determined an analytic solution for Stokes flow in an infinite permeable pipe.
Phil Andreae : An Introduction to Morse Theory
- Graduate/Faculty Seminar ( 103 Views )Morse theory relates the topology of a manifold M to the critical point set of a generic real-valued function on M. Most of this talk will be a non-rigorous review of the basic ideas of Morse theory and some simple applications. There will be lots of pictures of tori and other fun manifolds! In the last part of the talk, Id like to discuss briefly Wittens novel approach to Morse theory from the 1980s, which involves studying the asymptotics of a perturbed Laplace operator. I hope this will be interesting and new to most of the audience, even those who have seen some classical Morse theory. It should also illustrate the important role that analysis can play in topological problems.
Wenjing Liao : The MUSIC algorithm for line spectral estimation: stability and super-resolution
- Graduate/Faculty Seminar ( 128 Views )The problem of spectral estimation, namely recovering the frequency contents of a signal arises in various fields of science and engineering, including speech recognition, array imaging and remote sensing. In this talk, I will introduce the MUltiple SIgnal Classification (MUSIC) algorithm for line spectral estimation and provide a stability analysis of the MUSIC algorithm. Numerical comparison of MUSIC with other algorithms, such as greedy algorithms and L1 minimization, shows that MUSIC combines the advantages of strong stability and low computational complexity for the detection of well-separated frequencies on a continuum. Moreover, MUSIC truly shines when the separation of frequencies drops to one Rayleigh length and below while all other methods fail. This is a joint work with Albert Fannjiang at UC Davis. The talk involves basic linear algebra and Fourier analysis and it will be accessible to all.
Mark Stern : Frommers guide to vector bundles
- Graduate/Faculty Seminar ( 143 Views )I will give an introduction to the analysis, geometry, and topology of vector bundles for a general (i.e. nongeometric) audience. I will range from how vector bundles arise in Math 103 to how we can use partial differential equation techniques to extract interesting physical, algebraic, and topological information from them.
Ma Luo (Rome) : Galois theory for multiple zeta values and multiple modular values
- Graduate/Faculty Seminar ( 103 Views )Periods are numbers that can be expressed as integrals of algebraic differential forms over domains defined by polynomial inequalities with rational coefficients. They form a subring of complex numbers, which contains multiple zeta values and multiple modular values. Although some periods are transcendental, one can work out a Galois theory for them using their defining algebraic data, which is how the classical Galois theory for algebraic numbers were developed. I will discuss Francis Brown's results on multiple zeta values and more recent work on multiple modular values.
Fernando Schwartz : On the topology of black holes
- Graduate/Faculty Seminar ( 146 Views )3+1 dimensional black holes have spherical topology, but in higher dimensions this is no longer true. In this talk I will explain the preceding statement and show a construction, in terms of Riemannian geometry, of outermost apparent horizons with nonspherical topology.
Robert Bryant : The geometry of periodic equi-areal sequences
- Graduate/Faculty Seminar ( 126 Views )A sequence of functions $f = (f_i)$ ($-\infty < i < \infty$) on a surface $S$ is said to be \emph{equi-areal} (or sometimes, \emph{equi-Poisson}) if it satisfies the relations $$ df_{i-1}\wedge df_i = df_i\wedge df_{i+1}\ (\not=0) $$ for all $i$. In other words, the successive pairs $(f_i,f_{i+1})$ are local coordinates on $S$ that induce the same area form on $S$, independent of $i$. One says that $f$ is \emph{$n$-periodic} if $f_i = f_{i+n}$ for all $i$. The $n$-periodic equi-areal sequences for low values of $n$ turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras. In this talk, I will explain what is known about the classification (up to a natural notion of equivalence) of such sequences and their surprising relationships with differential geometry, cluster algebras, and the theory of overdetermined differential equations. I wont assume that the audience knows much differential geometry, just basic multi-variable calculus, and the emphasis will be on describing the interesting results rather than on technical details.
Humberto Diaz : A Tour of Heights & Rational Points
- Graduate/Faculty Seminar ( 104 Views )A very important (and difficult) problem for number theorists is to determine all the rational solutions to a polynomial equation defined over the rational numbers. The oldest nontrivial case, which dates back to Pythagoras, is that of finding all the rational points on a unit circle. In this talk, we will consider the case of elliptic curves, where the rational points have the structure of an Abelian group G under a curiously defined addition law. We will develop some preliminaries and introduce the classical height machinery, a powerful tool which helps us understand the complexity of the points of G. We will look at some important results about the height and about G and see what still remains very elusive.
Shishi Luo : How I learned to stop worrying and love mathematical biology
- Graduate/Faculty Seminar ( 136 Views )Biology has given mathematicians many new problems to work on in the last half century and the role of mathematics in biology research is only increasing. Through a series of examples, ranging from coat pattern formation to the evolution of RNA viruses, I will illustrate the insight that a mathematical treatment can give to problems in biology and will also discuss the difficulties involved in doing mathematical biology.
Anita Layton : Unraveling Kidney Physiology, Pathophysiology and Therapeutics: A Modeling Approach
- Graduate/Faculty Seminar ( 114 Views )The kidney not only filters metabolic wastes and toxins from the body, but it also regulates the body's water balance, electrolyte balance, and acid-base balance, blood pressure, and blood flow. Despite intense research, aspects of kidney functions remain incompletely understood. I will discuss how our group use mathematical modeling techniques to address a host of previously unanswered questions in renal physiology and pathophysiology: Why is the mammalian kidney so susceptible to hypoxia, despite receiving ~25% of the cardiac output? What are the mechanisms underlying the development of acute kidney injury in a patient who has undergone cardiac surgery performed on cardiopulmonary bypass? What is the effect of inhibiting sodium-glucose transport, a novel treatment for reducing renal glucose update in diabetes, on renal NaCl transport and oxygen consumption?
Brian Fitzpatrick : A Gentle Introduction to Categories and Categorification
- Graduate/Faculty Seminar ( 114 Views )Category theory is a language used to describe mathematical structures. Its focus is on the relationship between mathematical objects rather than the objects themselves. Categorification is then an attempt to find category-theoretic analogues of classical set-theoretic results. This talk will present the basic notions needed to develop a theory of categorification and will show how many familiar mathematical results are, in fact, "shadows" of a hidden categorical structure.
Chris O'Neill : An Introduction to Ehrhart Theory and Lattice Point Enumeration
- Graduate/Faculty Seminar ( 110 Views )A polytope is a subset of R^d which is the convex hull of a finite set of vertices. Given a polytope P, we can consider integer dilations of P, and ask how many integer points are contained in each dilation, as a function of the dilation factor. A theorem by Eugene Ehrhart tells us that, under the right conditions, this counting function is a polynomial, with some very interesting and unexpected properties. To demonstrate the usefulness of these results, we will give alternative proofs to some well known results from far outside the realm of geometry, including some basic facts about the chromatic polynomial of a graph. This talk will contain a little geometry, a little analysis, a little algebra, and a little combinatorics, and will be accessible to anyone who enjoys at least one of these topics.
Dave Rose : Graphical calculus and quantum knot invariants
- Graduate/Faculty Seminar ( 116 Views )At first glance, knot theory and representation theory seem to be unrelated fields of mathematics. In fact, this is not the case: in the early 90's, Reshetikhin and Turaev proved that knot invariants (and 3-manifold invariants) can be derived via the representation theory of quantum groups. The key link (no pun intended) between these areas is the observation that both the category of tangles and the category of representations share many similar structural features. In this talk we will explore these ideas, and if time permits, their categorified counterparts. If things like categories scare you, fear not; as the title suggests, all categories (and constructions on them) we encounter will have pictorial descriptions. In fact, no knowledge of category theory or representation theory is assumed. At the same time, if you have indeed taken Math 253, then this talk will provide context for the material in that course.
Joseph Spivey : A How-To Guide to Building Your Very Own Moduli Spaces (they make such great gifts)
- Graduate/Faculty Seminar ( 204 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).
Heekyoug Hahn : On tensor third $L$-functions of automorphic representations of $\GL_n(\A_F)$
- Graduate/Faculty Seminar ( 100 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.
William LeFew : Time-Reversal In Random Media: Current and Future Applications
- Graduate/Faculty Seminar ( 150 Views )This talk will discuss the basics of time-reversal theory in the context of wave propagation in random media. It will outline several of the interesting applications in the field including detection and encryption.
Jayce Getz : Trace formulae
- Graduate/Faculty Seminar ( 110 Views )All right, brain. You don't like me and I don't like you, but let's just do this and I can get back to killing you with trace formulae. -Homer Simpson (misquoted) We will discuss trace formulae starting with the Poisson summation formula and working towards the case of compact locally symmetric spaces. No background is assumed. Oh, and I'll bring beverages (both the big kid and little kid kind).
Rick Durrett : Random graphs as models of social networks.
- Graduate/Faculty Seminar ( 112 Views )We will describe the configuration model and discuss what happens when the people's opinions and the connections in the network coevolve. Despite the combined efforts of James Gleeson, Peter Mucha, Bill Shi, David Sivakoff, Josh Socolar, Chris Varghese and myself, we cannot prove any rigorous results so the talk should be accessible to almost anyone.
Michael Nicholas : An 3rd order accurate method in 3D period electromagnetic scattering
- Graduate/Faculty Seminar ( 150 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.
Shishi Luo : Getting a job after your PhD (and you thought graduating was hard...)
- Graduate/Faculty Seminar ( 118 Views )According to the NSF Survey of Earned Doctorates, of the Mathematics PhDs earned in 2011, 37% had an offer for postdoctoral research, 33% had definite employment in either academic, government, or industry positions, and 28% were seeking employment/postdoc. This means that although 100% of you are doing research right now, most likely only a third of you will continue to do research in a university setting. That means (a) getting a postdoc will be competitive and/or (b) you need to familiarize yourself with non-postdoc opportunities. To help you in this process, we've assembled a panel of local experts who have recently been through the job application process: Christine Berkesch (research faculty), Emily Braley (teaching faculty), Liz Munch (postdoc and non-academic), as well as senior faculty who can talk about qualities that are sought in both university (Mike Reed) and government research (Bill Pardon) settings. Come and learn what you can do now and in the future to make your job search more successful!