Seminars and Colloquia by Series

Group theory, geometry and dynamics of surface homeomorphisms

Series
Job Candidate Talk
Time
Thursday, January 7, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dan MargalitTufts University
Attached to every homeomorphism of a surface is a real number called its dilatation. For a generic (i.e. pseudo-Anosov) homeomorphism, the dilatation is an algebraic integer that records various properties of the map. For instance, it determines the entropy (dynamics), the growth rate of lengths of geodesics under iteration (geometry), the growth rate of intersection numbers under iteration (topology), and the length of the corresponding loop in moduli space (complex analysis). The set of possible dilatations is quite mysterious. In this talk I will explain the discovery, joint with Benson Farb and Chris Leininger, of two universality phenomena. The first can be described as "algebraic complexity implies dynamical complexity", and the second as "geometric complexity implies dynamical complexity".

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Series
Analysis Seminar
Time
Tuesday, December 8, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Xuan DuongMacquarie University
In this talk,we study weighted L^p-norm inequalities for general spectralmultipliersfor self-adjoint positive definite operators on L^2(X), where X is a space of homogeneous type. We show that the sharp weighted Hormander-type spectral multiplier theorems follow from the appropriate estimatesof the L^2 norm of the kernel of spectral multipliers and the Gaussian boundsfor the corresponding heat kernels. These results are applicable to spectral multipliersfor group invariant Laplace operators acting on Lie groups of polynomialgrowth and elliptic operators on compact manifolds. This is joint work with Adam Sikora and Lixin Yan.

Cancelled -- Asymptotics for implied volatility for local and stochastic volatility models

Series
Mathematical Finance/Financial Engineering Seminar
Time
Tuesday, December 8, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Peter LaurenceCourant Institute of Mathematical Science, New York University
We focus on time inhomogeneous local volatility models, the cornerstone of projection methods of higher dimensional models, and show how to use the heat kernel expansion to obtain new and, in some sense optimal, expansions of the implied volatility in the time to maturity variable. This is joint work with Jim Gatheral, Elton Hsu, Cheng Ouyang and Tai-Ho Wang.

Testing independence of regression errors with residuals as data

Series
Job Candidate Talk
Time
Tuesday, December 8, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Xia HuaMassachusetts Institute of Technology
In a regression model, say Y_i=f(X_i)+\epsilon_i, where (X_i,Y_i) are observed and f is an unknown regression function, the errors \epsilon_i may satisfy what we call the "weak'' assumption that they are orthogonal with mean 0 and the same variance, and often the further strong'' assumption that they are i.i.d. N(0,\sigma^2) for some \sigma\geq 0. In this talk, I will focus on the polynomial regression model, namely f(x) = \sum_{i=0}^n a_i x^i for unknown parameters a_i, under the strong assumption on the errors. When a_i's are estimated via least squares (equivalent to maximum likelihood) by \hat a_i, we then get the {\it residuals} \hat epsilon_j := Y_j-\sum_{i=0}^n\hat a_iX_j^i. We would like to test the hypothesis that the nth order polynomial regression holds with \epsilon_j i.i.d. N(0,\sigma^2) while the alternative can simply be the negation or be more specific, e.g., polynomial regression with order higher than n. I will talk about two possible tests, first the rather well known turning point test, and second a possibly new "convexity point test.'' Here the errors \epsilon_j are unobserved but for large enough n, if the model holds, \hat a_i will be close enough to the true a_i so that \hat epsilon_j will have approximately the properties of \epsilon_j. The turning point test would be applicable either by this approximation or in case one can evaluate the distribution of the turning point statistic for residuals. The "convexity point test'' for which the test statistic is actually the same whether applied to the errors \epsilon_j or the residuals \hat epsilon_j avoids the approximation required in applying the turning point test to residuals. On the other hand the null distribution of the convexity point statistic depends on the assumption of i.i.d. normal (not only continuously distributed) errors.

The Dehn function of SL(n,Z)

Series
Job Candidate Talk
Time
Monday, December 7, 2009 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Robert YoungIHES/Courant
The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an openquestion; one particularly interesting case is SL(n,Z). Thurston conjectured that SL(n,Z) has a quadratic Dehn function when n>=4. This differs from the behavior for n=2 (when the Dehn function is linear) and for n=3 (when it is exponential). I have proved Thurston's conjecture when n>=5, and in this talk, I will give an introduction to the Dehn function, discuss some of the background of the problem and, time permitting, give a sketch of the proof.

Southeast Geometry Seminar

Series
Other Talks
Time
Monday, December 7, 2009 - 08:00 for 8 hours (full day)
Location
University of Alabama, Birmingham
Speaker
Southeast Geometry SeminarUniversity of Alabama, Birmingham

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions:

  • The University of Alabama at Birmingham
  • The Georgia Institute of Technology
  • Emory University
  • The University of Tennessee Knoxville

The following five speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology.

  • Natasa Sesum (U Penn)
  • Alexandru Ionescu (U Wisconsin)
  • Sergiu Klainerman (Princeton U)
  • Alex Freire (U Tennessee Knoxville)
  • Christian Hainzl (UAB)

A poster session will be hosted. There will also be an evening public lecture by plenary speaker Sergiu Klainerman entitled The Mathematical Magic of Black Holes.

Undergraduate Research Seminar

Series
Other Talks
Time
Friday, December 4, 2009 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 168
Speaker
Michelle Delcourt and Leo ChenSchool of Mathematics, Georgia Tech

Leo Chen: The Shape and Stability of a Flexible Sheet in a von Karman Vortex Street

Michelle Delcourt: Dessin and Manturov bracket shuffles
In this talk we will explore the connections between knot theory and combinatorics. Links are related to Grothendieck's dessins d'enfants. Cartographic one-vertex dessins can be represented by chord diagrams. The diagrams can be recorded as "words" using a finite alphabet (k-bracket parenthesis system). Many combinatorial objects are related to these Manturov bracket structures.

Color-Critical Graphs on Surfaces

Series
Graph Theory Seminar
Time
Thursday, December 3, 2009 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Carl YergerMath, GT
A fundamental question in topological graph theory is as follows: Given a surface S and an integer t > 0, which graphs drawn in S are t-colorable? We say that a graph is (t+1)-critical if it is not t-colorable, but every proper subgraph is. In 1993, Carsten Thomassen showed that there are only finitely many six-critical graphs on a fixed surface with Euler genus g. In this talk, I will describe a new short proof of this fact. In addition, I will describe some structural lemmas that were useful to the proof and describe a list-coloring extension that is helpful to ongoing work that there are finitely many six-list-critical graphs on a fixed surface. This is a joint project with Ken-ichi Kawarabayashi of the National Institute of Informatics, Tokyo.

An Extension of the Cordoba-Fefferman Theorem on the Equivalence Between the Boundedness Maximal and Multiplier Operators

Series
Analysis Seminar
Time
Wednesday, December 2, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alexander StokolosGeorgia Southern University
I will speak about an extension of Cordoba-Fefferman Theorem on the equivalence between boundedness properties of certain classes of maximal and multiplier operators. This extension utilizes the recent work of Mike Bateman on directional maximal operators as well as my work with Paul Hagelstein on geometric maximal operators associated to homothecy invariant bases of convex sets satisfying Tauberian conditions.

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