Departmental Colloquium
Spring 2001
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 11 |
Patrick Guidotti
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PDEs in partly unbounded cylinder like domains occur frequently in applications. One of their apparent features is the natural anisotropy between bounded and non bounded directions. We shall illustrate how one can take advantage of this structure to simplify their analysis considering two different situations. Firstly we introduce the concept of semiclassical fundamental solution of a PDE and show how it can be utilized for several different purposes. Secondly we consider a two dimensional Free Boundary Problem with initial domain degeneration.
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January 18 |
John J. Millson
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Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. I will construct a one-parameter family of W -equivariant, unitary flat connections on h with values in any finite-dimensional g -module V and simple poles on the root hyperplanes. The fundamental group of the quotient by W of the complement of the root hyperplanes in h will be called the braid group of type g. The corresponding monodromy representation of the braid group of type g is a deformation of the action of (a finite extension of) W on V. I will discuss the resulting representations for the Cartan powers of the vector representation for the classical simple Lie algebras. This is a joint work with Valerio Toledano Laredo.
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January 22 |
Chandashekhar Khare
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The absolute Galois group of Q is one of the central objects of study in algebraic number theory. To vary a well-known quote, one may say that this group knows everything about number fields and one has only to persuade it to tell us. I will talk of recent attempts of studying this group through its representations.
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January 25 |
Peter J. Mucha
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Sedimenting suspensions of monodisperse rigid spherical particles in a state of creeping flow are considered in the dilute limit, where only low-order particle-particle interactions are kept to simplify the dynamical equations. In this limit, O(N) codes can easily simulate more than 10^5 particles. Both periodic boundary conditions and systems with side walls are considered, and found to be fundamentally different. Results are compared with the Caflisch-Luke scaling for the particle velocity fluctuations. Estimates for the velocity fluctuations are semi-analytically obtained for side-wall systems, and found to agree with both simulations and dilute experiments. Results are further compared with recent experiments, and the roles of the external boundary conditions and backflow details are addressed.
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January 30 |
Robert Bauer
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I will explain how to construct geometric heat flows on manifolds through the use of stochastic parallel translation and give applications of this construction to the regularity of heat flows and spectral theory of geometric Laplacians. Some applications are: stochastic characterizations of the Yang-Mills and Yang-Mills heat equation, a new proof of non-explosion for the Yang-Mills heat equation with small initial condition, calculations of the mean of random holonomy and determination of the spectrum of the horizontal Laplacian on the Hopf fibration over complex and quaternionic projective space.
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February 1 |
Hassan Allouba
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Stochastic PDEs (SPDEs) form one of the hottest and most difficult fields in Probability theory and its interactions with PDEs and Stochastic Analysis. In this talk, I will describe two approaches which are effective in the study of existence, uniqueness, as well as qualitative behavior questions for SPDEs: Change of measure and SDDEs. The change of measure theorem generalizes the well known Girsanov theorem in the one parameter setting to that of SPDEs. This theorem takes on an added significance in the SPDEs setting, for it applies well to different types of equations and allows us, among other things, to transfer hard results from simpler to more complex SPDEs. I will give examples of applications to a class of equations containing the stochastic Allen-Cahn and others. The Stochastic Differential Difference Equations (SDDEs) approach starts by discretizing space in the corresponding SPDEs and then looking at limits as the spatial lattice size goes to zero. This approach has several advantages over the usual direct one. In addition to its numerical flavor, it provides for rich interplay between random walks and SPDEs, and allows us to prove otherwise hard results for these equations. It also gives us an interesting and "natural" class of solutions to SPDEs. Time permitting, I will discuss new applications to a class of Burgers-related stochastic equations.
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February 5 |
Christopher Hacon
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One useful way to organize our understanding of algebraic varieties, that is, solution sets of systems of polynomials, is by counting the number of linearly independent ( m -multiple-valued) top-degree holomorphic differential forms on the variety. These numbers, denoted as P_m, are called plurigenera of the variety. Another useful number is the number of linearly independent holomorphic one-forms, called the irregularity and denoted by q. It turns out that knowing even a few of these numbers completely determines the type of the variety we are dealing with, and so determines the important properties of all varieties X with the given numerical invariants P_m(X) and q(X). The focus usually is on varieties X with small P_m, since those with ``maximal P_m growth'' are essentially unclassifiable. A turning point in modern classification theory is a theorem of Kawamata which roughly says that if all P_m 's are <= 1 then X is a fibration over a q -dimensional torus. In this talk we will introduce and motivate the study of algebraic varieties via these numerical invariants, leading toward refinements and effective versions of Kawamata's Theorem. Our goal will be the following results conjectured by Kollár: Theorem 1: Let X be smooth with P_2(X)=1. Then X maps surjectively to a complex projective torus of dimension q(X). (In particular dim( X ) >= q(X).) Theorem 2: Let X be smooth with P_2(X)=1 and dim( X )= q(X). Then X is birationally equivalent to a complex torus. (Both theorems are joint work with A. J. Chen)
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February 6 |
Jennifer Schultens
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The tunnel number of a knot is the minimal number of disjoint properly embedded arcs that must be removed from the complement of the knot in order to obtain a handlebody (i.e., a 3-dimensional fattening of a wedge of circles). We will compare and contrast the tunnel number with some of the other notions of complexity of a knot. Then we will discuss its apparently erratic behaviour under the operation of connected sum.
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February 13 |
Ken Bromberg
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In the Sullivan dictionary between Kleinian groups and rational maps, the space of complete hyperbolic structures on a 3-manifold is the analogue of the Mandelbrot set. We will describe some similarities between the two spaces, but will also see that the space of hyperbolic structures exhibits behavior not found in the Mandelbrot set. In particular, a component of the interior of the space of hyperbolic structures can ``self-bump''. This is joint work with John Holt.
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February 15 |
Dan Barbasch
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Unitary representations of reductive groups play a role in many branches of mathematics such as automorphic forms, analysis and mathematical physics. The classification of the irreducible unitary dual is a basic problem of representation theory which is still far from being solved. One of the basic philosophies is that the unitary dual should be parametrized by the orbits of a group on some space. On the one hand there is the "orbit method" (initiated by Kirillov) which parametrizes the unitary dual in terms of coadjoint orbits. On the other hand, conjectures of Arthur about the residual spectrum suggest that the unitary dual should be parametrized in terms of orbits in the dual group. In this talk I will illustrate some of these ideas in the context of the spherical dual which is easier to describe.
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March 1 |
Stanislav Smirnov
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We will study critical site percolation on triangular lattice in the plane. We will discuss proofs of Cardy's formula (physical prediction for the probability of rectangle crossings), its conformal invariance, and construction the (conformally invariant) continuum scaling limit.
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March 6 |
Selman Akbulut
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We will discuss some geometric structures on smooth 4-manifolds as a tool to understand their topology. For example, every closed smooth 4-manifold can be decomposed as a union of two Stein surfaces, and a Stein surface is nothing more than a Lefschetz fibration on a 2-disk.
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March 22 |
Eriko Hironaka
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For a monic integer polynomial f(x) the Mahler measure of f(x) is the product of the roots outside the unit circle. In 1933, Lehmer posed the following problems: Can the Mahler measure for noncyclotomic irreducible monic integer polynomials be bounded away from 1? If yes, is the lower bound given by L(x) = x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1? Despite extensive computer searches, and attempts from a wide range of areas in number theory and geometry, both problems remain open. In this talk I will describe some partial results involving Alexander polynomials of knots and growth rates of Coxeter reflection groups.
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March 29 |
Stephen Rudich
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Archival event entry migrated from the legacy colloquium website.
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April 5 |
Eric M. Friedlander
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Algebraic K-theory has been closely linked to the study of algebraic cycles in algebraic geometry ever since Grothendieck first formulated $K_0^{alg}(X)$. Inspired by algebraic K-theory, Atiyah, Hirzebruch, and Adams quickly followed Grothendieck by first formulating and then applying topological K-theory, $K_{top}^*(T)$. In the past 30 years, algebraic K-theorists have sought applications of their work to algebraic geometry and inspiration from algebraic topology. This lecture will mention some of this historical context before describing much newer results intertwining algebraic K-theory and algebraic cycles using techniques of algebraic topology.
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