Seminars & Events for Algebraic Geometry Seminar

September 15, 2009
4:30pm - 6:30pm
Transversality and noncommutative geometry

Birationally commutative graded algebras solve the moduli problem for "point modules" over a graded ring. They have been a fruitful source of counterexamples, examples, and intuition in noncommutative ring theory. We investigate when a large subclass of birationally commutative algebras is noetherian. Formally, these are idealizer subrings of twisted homogeneous coordinate rings. In the process, we give a (purely algebro-geometric) generalization of the Kleiman-Bertini theorem.

Speaker: Susan J. Sierra, Princeton University
Location:
Fine Hall 322
October 6, 2009
4:30pm - 6:30pm
Homotopy Theoretic methods on Chow varieties

The homotopy theoretic method has been applied to the algebraic cycle theory for a long period of time. In particular, it can be applied to compute topological invariants of Chow varieties. In this talk I will discuss this method in calculating the Euler Characteristic of Chow varieties. The calculation in a direct and simple way (this result has been obtained by Blaine Lawson and Stephen Yau in a different way). This technique also can be applied to Chow varieties with certain group actions and other cases. Furthermore, I will also talk about the application of the method on $l$-adic Euler-Poincare Characteristic of Chow varieties over arbitrary algebraic closed field.

Speaker: Wenchuan Hu, IAS
Location:
Fine Hall 322
October 20, 2009
4:30pm - 6:30pm
Algebraic curves with CM

Is every Abelian variety isogenous with the Jacobian of an algebraic curve?  We will study also several other questions in arithmetic geometry and show various implications.  We will mention some solutions to these problems.

Speaker: Frans Oort, University of Utrecht
Location:
Fine Hall 322
October 27, 2009
4:30pm - 6:30pm
BGG correspondence and the cohomology of compact Kaehler manifolds

The cohomology algebra of the sheaf of holomorphic functions on a compact Kaehler manifold can be naturally viewed as a module over the exterior algebra of a vector space. A well-known result of Bernstein-Gelfand-Gelfand gives a correspondence between such "exterior" modules and linear complexes of modules over the symmetric algebra, i. e. the polynomial ring. I will explain how one can use a modern view on this correspondence, together with the Generic Vanishing theory developed by Green and Lazarsfeld via Hodge-theoretic methods, in order to understand subtle algebraic structures of the cohomology algebra. As a bonus, homological and commutative algebra tools can be applied on the polynomial ring side to obtain new inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kaehler manifolds.

Speaker: Mihnea Popa, University of Illinois at Chicago (UIC)
Location:
Fine Hall 322
November 10, 2009
4:30pm - 6:30pm
Rational Simple Connectedness

Rational simple connectedness is an analog of simple connectedness for complex varieties having important applications: every 2-parameter family of rationally simply connected varieties has a rational section, and a 1-parameter family has so many rational sections that they approximate every power series section to arbitrary order. Unfortunately the condition is quite difficult to verify and is known to hold only for homogeneous spaces and also for some projective hypersurfaces satisfying a list of hypotheses. My new approach for verifying this condition works by studying a canonically defined foliation on the moduli space of rational curves on the variety.

Speaker: Matt DeLand, Stony Brook University
Location:
Fine Hall 322
November 13, 2009
3:00pm - 4:30pm
Analogue of the Narasimhan-Seshadri theorem in higher dimensions and holonomy

Joint Columbia-Courant-Princeton University Algebraic Geometry Seminar

Speaker: Vikraman Balaji, Chennai Mathematical Institute
Location:
Fine Hall 214
November 13, 2009
4:30pm - 6:30pm
Mirror symetry for del Pezzo surfaces

Joint Columbia-Courant-Princeton University Algebraic Geometry Seminar

Speaker: Tony Pantev, University of Pennsylvania
Location:
Fine Hall 214
November 17, 2009
4:30pm - 6:30pm
Smoothing surface singularities via mirror symmetry

We use the Strominger-Yau-Zaslow interpretation of mirror symmetry to describe deformations of surface singularities in terms of counts of holomorphic curves and discs on a mirror surface. In particular we prove Looijenga's conjecture on smoothability of cusp singularities. This is joint work with Mark Gross and Sean Keel, and builds on work of Gross-Siebert and Gross-Pandharipande-Siebert.

Speaker: Paul Hacking, University of Massachusetts, Amherst
Location:
Fine Hall 322
November 24, 2009
4:30pm - 6:30pm
Rigidity properties of Fano varieties

From the point of view of the Minimal Model Program, Fano varieties constitute the building blocks of uniruled varieties. Important information on the biregular and birational geometry of a Fano variety is encoded, via Mori theory, in certain combinatorial data corresponding to the Neron–Severi space of the variety. It turns out that, even when there is actual variation in moduli, much of such combinatorial data remains unaltered, provided that the singularities are "mild" in an appropriate sense. The talk is based on joint work with C. Hacon.

Speaker: Tommase deFernex, University of Utah
Location:
Fine Hall 322
December 8, 2009
4:30pm - 6:30pm
Rational curves on hypersurfaces

This talk is on the geometry of spaces of rational curves on Fano hypersurfaces. I will talk about some of the known results on the dimension, irreducibility, and the Kodaira dimension of these spaces. I will also discuss the problem of bounding the dimension of the cones of non-free rational curves on general hypersurfaces.

Speaker: Roya Beheshti Zavareh, Washington University, St. Louis
Location:
Fine Hall 322
March 2, 2010
4:30pm - 6:30pm
Rational simple connectedness and Serre's "Conjecture II"

In the early 1960's Serre formulated two conjectures about Galois cohomology. The first was proved by Steinberg shortly thereafter, but the second remains open. I will discuss the proof of Serre's Conjecture II in the "geometric case": every principal homogeneous space for a bundle of simply connected, semisimple groups over a surface has a rational section. Due to the work of many people — Merkurjev and Suslin, E. Bayer and Parimala, Chernousov, Gille—the geometric case further reduces to the "split, geometric case" i.e., the bundle of groups is constant. And this case was proved by de Jong, X. He and myself using "rational simple connectedness." No background in Galois cohomology or rational connectedness will be assumed.

Speaker: Jason Starr, Stony Brook University
Location:
Fine Hall 322
March 22, 2010
3:00pm - 5:00pm
Introduction to DB singularities
Speaker: Sándor Kovács, University of Washington
Location:
Fine Hall 801
March 23, 2010
4:30pm - 6:30pm
DB pairs and vanishing theorems
Speaker: Sándor Kovács, University of Washington
Location:
Fine Hall 322
March 24, 2010
3:00pm - 5:00pm
Smoothing of surface singularities and symplectic 4-manifolds
Speaker: Jonathan Wahl, University of North Carolina
Location:
Fine Hall 801
March 31, 2010
3:00pm - 5:00pm
Noncommutaive Hodge structure and applications

In this talk we will look at some classical problems. - rationality - from a new prospective. Our point of view will be based on Homological Mirror Symmetry. Examples will be discussed at the end.

Speaker: Ludmil Katzarkov, University of California - Irvine
Location:
Fine Hall 801
April 27, 2010
4:30pm - 6:30pm
Cohomology groups of structure sheaves
Speaker: János Kollár, Princeton University
Location:
Fine Hall 322