The Berkeley-Stanford Algebraic Geometry Colloquium covers the full range of topics in algebraic geometry, and is intended for a broader audience than a typical research seminar. Graduate students and researchers in nearby fields are particularly welcome. Each meeting features (approximately) two speakers chosen for their contributions to the field and their expository ability. There will be a dinner afterwards.
Organizers: David Eisenbud, Dagan Karp, Martin Olsson, and Mauricio Velasco (Berkeley) and Jun Li, Dragos Oprea, Sam Payne, and Ravi Vakil (Stanford).
Tuesday October 2 (in Berkeley): (poster)
Frank-Olaf Schreyer (Saarbruecken): Over-constrained mechanisms and curves of genus 7 (3:45-4:45 pm, 939 Evans)
I will describe the algebraic geometry behind certain kinds of ``mechanisms'' -- those made from rigid rods, hinged together and attached to the ``ground'' at some points.
A mechanism is called over-constrained if the dimension of the variety of movements it can make exceeds the ``expected dimension'', computed from the number of objects that can move, and the number of constraints satisfied, much as one would compute the ``expected dimension'' of a variety as the dimension of the ambient space minus the number of equations.
I will present a general approach to the discovery of over-determined mechanisms, and study a specific kind of ``platform robot'' -- basically a table that sits on top of rods of various lengths attached to the floor and hinged so that they can meet floor and table at any angle -- that is connected to algebraic curves of genus 7.
Andrew Sommese (Notre Dame): Recent results in numerical algebraic geometry (5:00-6:00 pm, 939 Evans)
Following an overview of Numerical Algebraic Geometry, some recent work will be presented:
a) work with J. Hauenstein and C. Wampler on regeneration, a new equation-by-equation approach to solving large polynomial systems
b) work with D. Bates, J. Hauenstein, and C. Wampler on adaptive multiprecision and Bertini, our new software for doing numerical algebraic geometry computations.
c) work with D. Bates, C. Peterson, and C. Wampler on numerical computation of the geometric genus of curve components of algebraic sets.
Tuesday November 27 (in Stanford): (poster)
William Fulton (Michigan): On the Equivariant Cohomology of Homogeneous Varieties (3:45-4:45, 383-N)
Although equivariant cohomology was constructed by Borel in 1960, it has taken most algebraic geometers a long time to respond to it. For Grassmannians and other homogeneous varieties, working equivariantly gives Schubert calculus a rich and powerful combinatorial structure. I will not assume familiarity with equivariant cohomology, but hope to describe a few of the recent results in this active area.
Aaron Bertram (Utah): Vanishing and stability (5:00-6:00, 383-N)
In this talk I will discuss how Bridgeland stability conditions for algebraic surfaces can be used to prove classical vanishing theorems. Although Bridgeland stability involves esoteric notions such as the dervied category of coherent sheaves (or "D-branes," as the physicists like to call them), this talk is designed for a broad audience. It is my intention to show how, once again, a crazy idea coming from string theory sheds new light on old problems in mainstream algebraic geometry. This is joint work with Daniele Arcara.
Tuesday January 22 (in Stanford): (poster)
Lothar Goettsche (ICTP): Instanton counting and K-theoretic Donaldson invariants (3:45-4:45, 383-N)
This is an overview talk of joint work with H. Nakajima and K. Yoshioka. The Donaldson invariants can be computed in Algebraic Geometry as intersection numbers on moduli spaces of vector bundles on compact algebraic surfaces. K-theoretic refinement of the Donaldson invariants is given by the holomorphic Euler characteristics of line bundles on these moduli spaces. We expect that these K-theoretic Donaldson invariants have properties very similar to those of the usual Donaldson invariants.
The Nekrasov partition function (which has also a K-theoretic version) is the generating function of equivariant Donaldson invariants of R^4. By the Nekrasov conjecture (proved by several authors) it has nice regularity properties and is related to modular forms.
We express the (K-theoretic) Donaldson invariants of rational algebraic surfaces in terms of the Nekrasov partition function, and use this to give a generating function for these invariants in terms of elliptic functions. I will also talk about applications and generalizations of these results.
Jacob Lurie (MIT): Elliptic Cohomology (5:15-6:15, 380-C (different room!))
This talk will be an introduction to the theory of elliptic cohomology, aimed at an audience of algebraic geometers. I'll begin by briefly reviewing what a cohomology theory is, and will then describe some of Quillen's ideas relating cohomology theories with formal groups. Cohomology theories which are associated to formal groups of elliptic curves turn out to be particularly interesting, and can be studied using ideas from many areas of mathematics (number theory, algebraic topology, representation theory of loop groups, mathematical physics, etcetera). The main goal of this talk will be to explain the Hopkins-Miller theory of "topological modular forms" (which can be regarded as a "universal" elliptic cohomology theory). If time permits, I will explain how elliptic cohomology relates to the theory of the Witten genus.
Tuesday February 26 (in Berkeley): (poster)
Jerzy Weyman (Northeastern): The existence of pure resolutions (3:45-4:45 pm, 939 Evans)
I will report on one half of the recent proof of Boij-Soderberg
conjectures on Betti numbers of graded modules, proved in a joint
paper "The Existence of Pure Free Resolutions" with Eisenbud and
Floystad, arXiv:0709.1529 . The other half was done by Eisenbud and
Schreyer in the paper "Betti Numbers of Graded Modules and Cohomology
of Vector Bundles" arXiv:0712.1843.
A minimal free resolution of a graded module M of finite length
over a polynomial ring A = K[X_1, ... ,X_n ] is pure if the
i-th term F_i of the resolution is a free module generated in one
degree d_i. We show using representation theory of general linear
group that for every $n$ and for every sequence
{\underline d}= (d_0,\ldots ,d_n )
there exists a pure resolution with the i-th term
generated in degree d_i for i=0,\ldots ,n.
Hiraku Nakajima (Kyoto): (Conjectural) triply graded link homology groups of the Hopf link and Hilbert schemes of points on the plane (5:00-6:00 pm, 939 Evans)
pizza: 6:15-7:15, place TBA
Mikhail Kapranov (Yale): Noncommutative differential operators, unparametrized paths and Hodge structures (7:15-8:15, 939 Evans)
A noncommutative differential operator (NCDO) on a manifold X is a
compatible system of linear differential operators acting in all
vector bundles with connections on X. The ring of such operators can be
seen as a highly noncommutative version of the mildly noncommutative
ring of usual differential operators: the partial derivatives are replaced
by formal covariant derivatives which no longer commute and account for the
curvature data.
The talk will explain the relation of NCDO with the space of formal germs of
unparametrized paths. In particular, we will make precise the statement
that a connection is uniquely defined, up to a formal germ of isomorphism, by
all the higher covariant derivatives of the curvature evaluated at one point.
This relation allows us to give a ``gauge-theoretic' description of
the category of mixed Hodge structures.