Departmental Colloquium
Spring 2011
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 27 |
Chenyang Xu
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In 19 century, Hurwitz proved that for any compact Riemann surfaces of genus g>1, the order of its automorphism group is bounded by 84(g-1). We prove a natural generalization of this result to all dimensions. In fact, this generalization itself is a special case of a conjecture on volumes of algebraic varieties due to Alexeev-Koll\'ar. Our work verifies this conjecture. If time permits, I will talk about various other applications to justify that our results play important roles in the study of birational geometry and related fields. This is a joint work with Christopher Hacon and James McKernan.
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February 17 |
Martin Short
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Criminologists have often observed that crimes tend to form "epidemic" like spatio-temporal patterns, whereby criminal events seem to spawn further events in nearby space-time regions. This often leads to the formation of crime hotspots, which are localized regions of intense criminal activity. However, descriptive and/or predictive mathematical models of these hotspots have been lacking. In this talk, I will describe two possible models for this phenomenon. One starts from the bottom up, modeling some basic aspects of human criminal behavior and arriving at a set of PDEs reminiscent of others in the mathematical biology literature. I will discuss some of the results and implications of the analysis of these PDEs, especially in terms of police response to hotspot formation. The second model starts from the top down, taking a data-driven statistical approach to the problem. This method is closely related to others used in the modeling and prediction of earthquakes, and yields practical results that can help police to best utilize their limited resources.
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March 1 |
Brian Smithling
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Shimura varieties are arithmetically interesting algebraic varieties defined over number fields. A basic problem for them is to define and study "good" models of them over (localizations of) rings of integers, so that (for example) the equations defining them can be reduced mod p. Amongst the most accessible Shimura varieties are those that admit interpretations as moduli spaces of abelian varieties with additional structure; for certain of these, Rapoport and Zink have defined natural integral models and, moreover, reduced the local study of them to their local models, which are projective schemes defined purely in terms of linear algebra. I will present an overview of some aspects of the theory of local models, focusing in particular on cases in which the Rapoport-Zink models are not "good" ones.
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March 3 |
R. Michael Range
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It is quite well known, and surely not surprising, that–in analogy to multivariable real calculus–classical complex analysis, and in particular the Cauchy-Riemann equations,generalize to several complex variables. One of the many new phenomena of the higher dimensional theory involves the restriction of the Cauchy-Riemann equations to lower dimensional submanifolds of complex Euclidean space and related abstract versions. We shall discuss the basic concepts, review some little known historical details, and describe some intriguing examples and fundamental problems and results.
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March 10 |
Gunther Uhlmann
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We will give a general survey on travel time tomography. This consists in determining the internal properties of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology, oil exploration, medical imaging and ocean acoustics among several areas of applications. Travel time tomography is also related to the boundary rigidity problem and lens rigidity problem considered in differential geometry. This connection will also be discussed.
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March 31 |
Smadar Karni
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Non-conservative hyperbolic systems arise in a wide range of applications, which makes their theoretical study and numerical approximation very important. The mathematical theory of weak solutions has been generalized to the nonconservative setup using the notion of vanishing viscosity and viscous path. The difficulty lies in the fact that shock relations depend not only on the immediate states ahead/ behind the shock, but also on the viscous path that connects them. While advances have been made on the theoretical front, those advances have been slow to translate into successful numerical methods. In this talk, we shed light on some of the difficulties involved by considering the so-called path-conservative numerical scheme applied to an illuminating example from gas dynamics, where the choice of linear path happens to give the correct jump conditions. This is joint work with Remi Abgrall, University of Bordeaux, France
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April 1 |
Gopal Prasad
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In this talk I will describe recent joint work with Andrei Rapinchuk in which we have used number theoretic techniques to prove the existence of elements with interesting properties in any Zariski-dense subgroup of a real semi-simple Lie group. These elements have been used in several different contexts. In our recent work we have used them to decide when two "weakly commensurable" arithmetic subgroups are actually commensurable. We have used known results, and a well-known conjecture, in transcendental number theory to show that for the symmetric space X of a noncompact absolutely simple real Lie group G, if the quotients X/\Gamma_1 and X/\Gamma_2, where \Gamma_1 and \Gamma_2 are lattices in G and at least one of them is arithmetic, are either isolength, or they are compact and isospectral, then the subgroups \Gamma_1 and \Gamma_2 are weakly commensurable. Therefore, our results on weakly commensurable arithmetic groups have important consequences for the Riemannian Geometry of locally symmetric spaces. Our work has led us to investigate local-global principles for embedding of fields with involution into a central simple algebra with involution. The failure of such a local-global principle for certain central simple algebras of degree 4n, given with an orthogonal involution, implies interesting results, for example, for compact arithmetic hyperbolic spaces of dimension 4n-1.
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April 7 |
Pierre Colmez
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The p-adic Langlands program is still in infancy, but the case of GL_2(Q) is rather well understood by now, following Wiles's breakthrough which lead to the proof of Fermat's last theorem. In this lecture, I will try to explain what this program is good for and what we know with respect to its local component.
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April 14 |
Jaigyoung Choe
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A minimal surface is locally the surface with minimum area. Therefore the Euclidean coordinates $x_1,x_2,x_3$ are harmonic on a minimal surface in $\mathbb R^3$. And the Euclidean coordinates $x_i (i=1,2,3,4)$ satisfy $\Delta x_i+2x_i=0$ on a minimal surface in $\mathbb S^3(\subset\mathbb R^4)$. Then Yau conjectured that the first eigenvalue of the Laplacian on a compact embedded minimal surface in $\mathbb S^3$ should be just 2. In this talk I will first show how minimal surfaces are constructed in $\mathbb S^3$, and give an easy proof of Yau's conjecture for most of the minimal surfaces. (Joint work with M. Soret)
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April 21 |
Wilfried Schmid
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Understanding the unitary dual of a general reductive Lie group is the major open problem in the representation theory of such groups. After a discussion of the problem, I shall describe an application of Hodge theory towards its solution. This is joint work with Kari Vilonen.
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