Topic: Sharp Lieb-Thirring inequalities in high dimension and stability of matter April 14

Presenter: Ari Laptev, Royal Institute of Technology, Stockholm

Topic: Gauss-Bonnet integral and conformal compactification in dimension four April 14

Presenter: Paul Yang, Princeton University

Abstract: In this talk I will motivate the discussion by reviewing the Cohn-Vossen inequality for complete surfaces of finite

total curvature and the results of Huber and Finn relating the Gauss-Bonnet integral to an isoperimetric constant for the surface. I then introduce a fourth order invariant as a variant of the four dimensional Chern-Gauss-Bonnet integrand. Analysis of the this equation allows us to extend the results of Cohn-Vossen/Huber/Finn to four dimensions.

Topic: Semi-Classical Limits of Eigenvalues and Witten's Proof April 15

of the Morse Inequalities

Presenter: Jeff Schenker, Princeton University

Note: This is the final GS Seminar this semester. Pizza will be provided.

Abstract: A proof, due to Witten, of the Morse inequalities uses the ``semi-classical'' limit of certain Schr\"odinger type

operators. I will sketch Witten's proof and describe in more detail the proof of the existence of the ``semi-classical" limit.

Topic: Cheeger's Inequality for infinite graphs April 15

Presenter: Shahriar Mokhtari-Sharghi, Long Island University

Abstract: In this talk we prove analogs of Cheeger's inequality for infinite graphs endowed with some particular measure

for edges and vertices. Cheeger's inequality says that if M is an n-dimensional Reimannian manifold, then the first

non-zero eigenvalue of the Laplacian of M is at least the infimum of vol(M')/{min(vol(M_1,M_2))}, as M' ranges over all

(n-1)-dimensional submanifolds which divide M into two connected components M_1 and M_2.

We state the inequality in a discrete setting using the "natural" analogy between the discrete and continuous cases, and

prove it.

Topic: Kazhdan-Lusztig Polynomials for 321-hexagon-avoiding permutations April 15

Presenter: Sara Billey, MIT

Abstract: In 1990, Deodhar proposed a combinatorial framework for determining the Kazhdan-Lusztig polynomials

$P_{x,w}$ in the case where $W$ is any Coxeter group. We explicitly describe the combinatorics in the case where

$W=A_n$ (the symmetric group on $n+1$ letters) and the permutation $w$ is 321-hexagon-avoiding. Our formula can

be expressed in terms of a simple statistic on all subexpressions of of any fixed reduced word for $w$. As a

consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincar\'e polynomial of the intersection

cohomology of the Schubert variety corresponding to $w$ is $(1+q)^{l(w)}$ if and only if $w$ is 21-hexagon-avoiding.

We also give a sufficient condition for the Schubert variety $X_w$ to have a small resolution. The results extend easily to

those Weyl groups whose Coxeter graphs have no branch points ($B_n$, $F_4$, $G_2$). This work is joint with

Gregory Warrington.

Topic: Topological Methods for the Cohomology of Galois Groups April 15

Presenter: Dikran Karagueuzian, University of Wisconsin at Madison

Date: Thursday, April 15, 1999

Time: 4 p.m.

Room: Fine 314

Topic: Adaptive structured mesh refinement for possible singularities in April 16

3D fluid and plasma flows

Presenter: Christiane Marliani, NYU Courant Institute

Topic: Numerical Evidence for Blowup in a Class of Symmetric Flows April 16

Presenter: Richard B. Pelz, Rutgers University

Topic: Applications and Algorithms for Fourier Analysis on SL_2(p) April 19

Presenter: D.Rockmore, Dartmouth Univeristy

Topic: Exact Expression for the Effective Elastic Tensor of April 19

Disordered Composites

Presenter: Salvatore Torquato, Civil Engineering & Operations Research & Princeton Materials Institute

Abstract: The problem of determining exact expressions for the effective elastic tensor of macroscopically anisotropic,

two-phase composite media of arbitrary microstructure in arbitrary space dimension d is considered. We depart from

previous treatments by introducing an integral equation for the ``cavity'' strain field. This leads to new, exact series

expansions for the effective elastic tensor. The nth-order tensor coefficients of these expansions are explicitly

expressed as absolutely convergent integrals over products of certain tensor fields and a determinant involving n-point

correlation functions that characterize the random microstructure. These series expressions perturb about the optimal

structures that realize certain rigorous bounds (e.g., coated-inclusion assemblages or finite-rank laminates). Accurate

approximate relations for the effective elastic moduli of isotropic dispersions are obtained by truncating, after third-order

terms, the exact series expansions. Our third-order approximations are in very good agreement with benchmark

simulation data, always lie within rigorous bounds, and are superior to popular self-consistent approximations.

Topic: Convex Optimization: Interior-point methods and Applications April 20

Presenter: Robert Vanderbei, Princeton University

Abstract: Fifteen years ago, interior-point methods revolutionized the subject of linear programming (LP) by providing the first class of algorithms for LP that are efficient both in theory and in practice. A currently active area of research is the extension of these methods beyond the domain of linear programming. In this talk, we will start with a brief survey of the extension to smooth convex nonlinear optimization. Many real-world problems, especially within engineering, belong to this general class. We will present several examples such as digital filter design, structural design, and minimal surfaces, to name a few. We will offer a multimedia presentation of the optimal solution to many of these problems (technology permitting).

Topic: Representations of covering groups of SL(n) April 20

Presenter: Jeffrey D. Adams, University of Maryland

Topic: Mathematical models of language evolution April 21

Presenter: Martin Nowak, IAS

Topic: Stable Ergodicity April 22

Preseneter: Andrei Torok, Princeton University

Topic: Mixed Volumes and Valuations: A Tale of Two Topics April 22

Presenter: Maragaret Readdy, IAS

Abstract: The work we will discuss is directly influenced by Klain and Rota's recent book on geometric probability. As a

warning, this talk has everything to do with old mathematics and geometry and nothing to do with geometric probability.

Part I: Laplace proved the volume of the k-th slice of a unit cube is given by an Eulerian number. We generalize

Laplace's result by giving a combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube in

terms of a refinement of the Euleriannumbers.

Part II: We will discuss valuations and complex hyperplane arrangements. Namely, we present a new combinatorial

method to compute the characteristic polynomial of subspace arrangements using the theory of valuations. This method

applies to any subspace arrangement over an infinite field. Examples include complex subspace arrangements, the

Dowling divisor lattice and its interpolations. We also consider the effect of the Dowlingization transformation on the

characteristic polynomial of real subspace arrangements. This talk is intended for a general audience.

Topic: TBA April 22

Presenter: Robert Gompf, University of Texas

Topic: TBA April 26

Presenter: Michael Weinstein, A T & T

Topic: Monodromy in families of abelian varieties April 27

Presenter: Jeffrey Achter, University of Massachusetts

Topic: On the rank of elliptic curves April 29

Presenter: Joseph Silverman, Brown University

Topic: The what and the why of quantum mechanics April 30

Presenter: Sheldon Goldstein, Rutgers University