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Series: Analysis Seminar

Sandro Levi and I have investigated variational strengthenings of uniform continuity and uniform convergence of nets or sequences of functions with respect to a family of subsets of the domain. Out of our theory comes an answer to this basic question: what is the weakest topology stronger than the topology of pointwise convergence in which continuity is preserved under taking limits? We argue that the classical theory constitues a misunderstanding of what is fundamentally a variational phenomenon.

Monday, January 11, 2010 - 13:00 ,
Location: Skiles 255 ,
Peter Blomgren ,
San Diego State University ,
Organizer: Sung Ha Kang

We describe two computational frameworks for the assessment of contractileresponses of enzymatically dissociated adult and neonatal cardiac myocytes.The proposed methodologies are variants of mathematically sound andcomputationally robust algorithms very well established in the imageprocessing community. The physiologic applications of the methodologies areevaluated by assessing the contraction in enzymatically dissociated adultand neonatal rat cardiocytes. Our results demonstrate the effectiveness ofthe proposed approaches in characterizing the true 'shortening' in thecontraction process of the cardiocytes. The proposed method not onlyprovides a more comprehensive assessment of the myocyte contraction process,but can potentially eliminate historical concerns and sources of errorscaused by myocyte rotation or translation during contraction. Furthermore,the versatility of the image processing techniques makes the methodssuitable for determining myocyte shortening in cells that usually bend ormove during contraction. The proposed method can be utilized to evaluatechanges in contractile behavior resulting from drug intervention, diseasemodeling, transgeneity, or other common applications to mammaliancardiocytes.This is research is in collaboration with Carlos Bazan, David Torres, andPaul Paolini.

Series: Geometry Topology Seminar

Series: Job Candidate Talk

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".

Series: Analysis Seminar

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.

Tuesday, December 8, 2009 - 15:00 ,
Location: Skiles 269 ,
Peter Laurence ,
Courant Institute of Mathematical Science, New York University ,
Organizer: Christian Houdre

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.

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

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.

Series: Other Talks

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.

Series: Other Talks

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.