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DEPARTMENT
COLLOQUIUM

Fall 2007

WEDNESDAY 4:30-5:30 pm
Tea served at 4:00 p.m.
Fine Hall 314


Information for the speakers

DATE
SPEAKER
TOPIC
Sep 19 James McKernan
MIT
Finite generation of the canonical ring.
One of the most fundamental invariants of any smooth projective variety is the canonical ring, the graded ring of all global pluricanonical holomorphic n-forms. We explain some of the recent ideas behind the proof of finite generation of the canonical ring and its connection with the programme of Iitaka and Mori in the classification of algebraic varieties.
Sep 26 Boaz Klartag
Princeton University
Approximately gaussian marginals of convex bodies
We will consider the following high dimensional phenomenon: Suppose X is a random vector in R^n, uniformly distributed in some convex set. We assume that n is a very large number. Then, with respect to an appropriate linear basis, the coordinates of X are random variables whose distribution is approximately gaussian. Moreover, these n coordinates are approximately independent in pairs, in triplets or in k-tuples, for k as large as a power of n. We will also discuss results and open problems regarding the speed of convergence to the normal distribution.
Oct 3 Alexandru Ionescu
University of Wisconsin
A uniqueness property of the Kerr spacetimes
In astrophysics, "no hair" theorems postulate that all physically relevant black hole solutions of the Einstein vacuum equations can be described by two parameters: the mass of the black hole and its angular momentum. Such theorems, which go back to work of Hawking, Carter, and Robinson, are known to hold under suitable assumptions, including real analiticity. I will talk about some recent work with Sergiu Klainerman on proving a conditional "no hair" theorem in the class of smooth manifolds. Our approach is based on certain analytical tools, such as Carleman estimates and uniqueness properties of solutions of nonlinear wave equations.
Oct 10 Hiraku Nakajima
Kyoto and IAS
Perverse coherent sheaves on a blowup surface
The blowup of a complex surface at a point is one of the most basic birational operation in algebraic geometry. It replaces a point by a projective line. In topology, it corresponds to making the connected sum with the projective plane with the opposite orientation. The relation between moduli spaces of coherent sheaves (vector bundles with singularities) on the original surface and on the blowup surface is of our interest. In this talk, I will explain that two moduli spaces can be understood via the `wall-crossing', i.e., a change of the stability parameters. Hence two moduli spaces are connected by a sequence of birational operations. (Joint work with Kota Yoshioka)
Oct 17 Jean Bourgain
IAS
The sum product phenomena and applications
Oct 24 David Eisenbud
UC Berkeley
Castelnuovo-Mumford regularity and Projections of Algebraic Varieties
Riemann Surfaces were first studied algebraically by first projecting them into the complex projective plan; later the same idea was applied to surfaces and higher dimensional varieties, projecting them to hypersurfaces. How much damage is done in this process? For example, what can the fibers of a general projection look like? This is pretty easy for smooth curves and surfaces (though there are still open questions), not so easy in the higher-dimensional case. I'll explain some of what's known, including recent work of mine with Roya Beheshti, Joe Harris, and Craig Huneke.
Nov 7 Bálint Virág
University of Toronto
Continuum limits of random matrices
The eigenvalue distribution of large random matrices arises naturally in many areas of mathematics. It can be understood by taking a continuum limit of the random matrix. For the sine point process, which is conjectured to describe the zeros of the Riemann zeta function, this limit is given in terms of Brownian motion in the hyperbolic plane. Other limits are 1-dimensional random Schrodinger operators. In fact, a large class of such operators show random matrix behaviour in parts of their spectra.
Nov 14 Isaac Held
NOAA and Princeton
Some dynamical problems central to understanding climate change
The talk will focus on two problems in climate change science: the poleward expansion of the subtropics, and the effect of warming ocean temperatures on tropical cyclones. Attempts at capturing the essence of these two problems in idealized dynamical frameworks will be described.
Nov 28 Andrew Granville
University of Montreal
Pretentiousness in analytic number theory
Inspired by the "rough classification" ideas from additive combinatorics, Soundararajan and I have recently introduced the notion of pretentiousness into analytic number theory. Besides giving a more accessible description of the ideas behind the proofs of several well-known difficult results of analytic number theory, it has allowed us to strengthen several results, like the Polya-Vinogradov inequality, the prime number theorem, etc. In this talk we will introduce these ideas and give some flavour of these developments.
Dec 5 Clifford Taubes
Harvard University
Contact structures in dimension 3 and the Seiberg-Witten equations
I hope to give some indication of how the Seiberg-Witten equations are used to study the dynamics of vector fields on 3-dimensional manifolds. One result of this research is a proof of Alan Weinstein's conjecture about the existence of closed integral curves of the Reeb vector field for a contact 1-form.
Dec 12


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