This seminar will happen on Tuesdays, roughly monthly, alternating between Stanford and Berkeley. There will be two lectures, with a break in between, and plenty of time for informal discussions. There will also be a dinner afterwards (subsidized for students). Drives and parking will be organized. For more information, please contact David Eisenbud (de@msri.org) or Tom Graber (graber@math.berkeley.edu) at Berkeley, or Jun Li (jli@math.stanford.edu) or Ravi Vakil (vakil@math.stanford.edu) at Stanford. Here is a map of Stanford's campus, with parking indicated.

Click here for the Stanford Algebraic Geometry Seminar, and here for the Berkeley Commutative Algebra and Algebraic Geometry Seminar.

Tuesday, October 26, at Stanford (Rm. 383-N): (poster)

Sergey Yuzvinsky (University of Oregon): Differential operators annihilating a polynomial (3:30-4:30 pm)

Abstract: In the talk we will discuss two themes whose relations have not yet been studied much. The first one concerns with first degree operators with polynomial coefficients. In the case where the annihilated polynomial is the product of linear forms this theme has a long history (started by K. Saito) and an unresolved conjecture. The second one concerns with operators of arbitrary degrees but with constant coefficients. The ideal of these operators annihilating an arbitrary homogeneous polynomial was studied by Macaulay. We will draw parallel between these two topics, recall how they interplay for finite reflection groups, and state a conjecture.

Rahul Pandharipande (Princeton): Gromov-Witten theory, Donaldson-Thomas theory, and the Hilbert scheme of points of the plane (5:00-6:00 pm)

Abstract: I will mainly speak about the local Gromov-Witten theory of curves in threefolds (joint with J. Bryan). The entire theory is determined in the TQFT formalism by two exact calculations. The local theory is connected to Donaldson-Thomas theory via a GW/DT correspondence. More recently, the local theory has been found to compute the quantum cohomology of the Hilbert scheme of points of the plane (joint with A. Okounkov).

Tuesday, November 30, at Berkeley (Rm. 939 Evans): (poster)

Mihnea Popa (Harvard): New intersection numbers on smooth varieties (3 pm (unusual time because of McMullen's lecture))

Abstract: Given a smooth projective variety X of dimension n, embedded in projective space by a very ample line bundle, the degree of the variety can be regarded as the intersection number D^n, where D is a hyperplane section of X. The Riemann-Roch theorem, together with some known facts about cohomology, shows that the Hilbert polynomial of D has leading term (1/n!)D^n. The Hilbert polynomial generalizes to the case of an arbitrary divisor D (codimension 1 subvariety) as the dimension of the set of divisors of zeros of rational functions with poles only on D, and only of order at most n. Ever since Riemann people have tried to understand the growth in this more general case as well. For curves things are easy, and a famous theorem of Zariski explains what to do, in a very geometric way, for surfaces. In this talk I will discuss the background of this problem and some recent work with Ein, Lazarsfeld, Mustata and Nakamaye in which we define asymptotic analogues of the usual intersection numbers in order to extend the results above.

Frank-Olaf Schreyer (Saarlandes): An experimental approach to the moduli space of numerical Godeaux surfaces (5:10 pm (unusual time because of McMullen lecture))

Abstract: Numerical Godeaux surfaces are minimal surface of general type with no holomorphic 1 or 2 forms and K^2=1. The first classical known example due to Godeaux, is the quotient X=Y/Z_5, of the Fermat quinitc w^5+x^5+y^5+z^5=0 in P^3 by the action by a fifth root of unity g: (w,x,y,z) --> (gw,g^2x,g^3y,g^4z). Other examples with different fundamental groups are known. Constructions via double planes branched along very particular curves have been given as well. The expected dimension of the moduli spaces is 8. However only in case of a large fundamental group the constructions known lead to locally complete families. The most difficult case is apparently the case of simply connected Godeaux surfaces, where we only know the existence of a two dimensional family constructed by Barlow. In the talk I will out line an approach based on Computer algebra, to construct (what I believe is ) the main family of Godeaux surfaces). The approach is a mixture of homological algebra, structure results, deformation theory and finite field experiments.

Tuesday, March 8, at Stanford (Rm. 383-N): (poster)

Persi Diaconis (Stanford): Counting Integer Points in polytopes (3:30-4:30)

Abstract: In practical statistical problems one needs to know the number of non negative integer points in the poloytope. Gorbner bases for toric ideals are one route. In joint work with Fu Liu we give extensions of the Gale Ryser theorem as a competing route.

Brian Osserman (Berkeley): Two Perspectives on Maps of Curves (5:00-6:00)

Abstract: A map f:C-->D of smooth algebraic curves is typically ramified at some points of C, which is to say that the map looks locally like z --> z^e for some e>1. Such points are called ramification points of f, and their images are called branch points. There are rich theories studying maps of curves in terms of their branch points, and, particularly when D is the projective line, in terms of their ramification points. We will compare and contrast these perspectives, discussing in particular their deformation theories, and we will conclude with an explanation of a ramified Brill-Noether theorem, valid in any characteristic.

Tuesday, April 12, at Berkeley (Rm. 939): (poster)

Mircea Mustata (Michigan): On some invariants of singularities (3:45-4:45)

Abstract: I will describe some invariants of singularities defined in positive characteristic via the Frobenius morphism. While elementary to define, they record information that in characteristic zero comes from the resolution of singularities, but also some more subtle information of an arithmetical nature. This is based on joint work with S. Takagi and K.-i. Watanabe.

Rob Lazarsfeld (Michigan): Asymptotic invariants of line bundles (5:00-6:00)

Abstract: It is a classical fact in algebraic geometry that hyperplane sections of projective varieties (or more generally ``ample'' divisors) satisfy many beautiful geometric, numerical and cohomological properties. On the other hand, various examples led to the traditional belief that the behavior of more general divisors was mired in pathology. However it has recently become clear that arbitrary effective (or ``big'') divisors display a surprising number of properties analogous to those of ample line bundles. The key is to study the divisors in question from an asymptotic perspective. I'll give an introduction to this circle of ideas, focusing on one invariant (the ``volume'') that measures the rate of growth of the number of sections of powers of a line bundle. The talk will center around examples and open problems.