UIUC Dept. of Mathematics Seminar Calendar

     August 2005           September 2005          October 2005    
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                                               30 31

Wednesday, February 2, 2005

Graduate Student Algebraic Geometry Seminar
3:00 pm in 441 Altgeld Hall, Wednesday, February 2, 2005

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Bin Wang (Dept. of Mathematics, UIUC)
The Étale Slice Theorem
Abstract: We will consider a linearly reductive group acting on an affine variety. We will prove an equivariant Zariski main theorem and show that every closed orbit has an étale neighborhood admitting a slice decomposition.

Wednesday, February 9, 2005

Tuesday, March 1, 2005

Wednesday, March 2, 2005

Graduate Student Algebraic Geometry Seminar
3:00 pm in 441 Altgeld Hall, Wednesday, March 2, 2005

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Submitted by nevins.

Thomas Nevins (UIUC)
An introduction to rationality
Abstract: Algebraic varieties that are ``rational,'' that is, are birationally isomorphic to projective space, form a fundamental and yet poorly understand class. In this talk I will give an introduction to rational varieties: what is a rational variety? why should one study them? what are some basic examples and nonexamples? what are some fundamental open questions in their study? I'll strive to keep the talk low-tech---in particular, if you know a bit about algebraic curves or some basics of projective algebraic varieties, and are willing to take a few facts on faith, a substantial portion of the talk should be within your grasp. On the other hand, we'll also manage to touch on some topics of active current research.

Tuesday, March 8, 2005

Wednesday, March 9, 2005

Wednesday, March 16, 2005

Wednesday, March 30, 2005

Tuesday, April 5, 2005

Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, April 5, 2005

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Submitted by nevins.

Ravi Vakil (Stanford)
Murphy's Law in algebraic geometry: badly behaved deformation spaces
Abstract: We consider the question: ``How bad can the deformation space of an object be?'' (Alternatively: ``What singularities can appear on a moduli space?'') The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space can be arbitrarily ugly.'' Hence many of the most important moduli spaces in algebraic geometry are arbitrarily singular, justifying a philosophy of Mumford. More precisely, every singularity of finite type over $\mathbb{Z}$ (up to smooth parameters) appears on the Hilbert scheme of curves in projective space, and the moduli spaces of: smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces have very ample canonical bundle, the stable sheaves are torsion of rank 1, the singularities are normal and Cohen-Macaulay, etc. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior along various associated subschemes. Similarly one can give a surface over $\mathbb{F}_p$ that lifts to $p^7$ but not $p^8$. (Of course the results hold in the holomorphic category as well.)

Tuesday, April 12, 2005

Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, April 12, 2005

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Submitted by nevins.

Alex Yong (UC Berkeley)
On smoothness and Gorensteinness of Schubert varieties
Abstract: The study of singularities of Schubert varieties in the flag manifold involves interesting interplay between algebraic geometry, representation theory and combinatorics. Although all Schubert varieties are Cohen-Macaulay, few are smooth. An explicit combinatorial characterization of the smooth ones was given by Lakshmibai and Sandhya (1990). The singular locus of an arbitrary Schubert variety was determined around 2001 by several authors. Gorensteinness is a measurement of the ``pathology'' of the singularities of an algebraic variety; it logically sits between smoothness and Cohen-Macaulayness. We explicitly characterize which Schubert varieties are Gorenstein, analogous to Lakshmibai and Sandhya's theorem. Here is the geometric interpretation: a Schubert variety is Gorenstein if and only if it is Gorenstein at the generic points of the singular locus. We also compute the canonical sheaf of a Gorenstein Schubert variety as a line bundle in terms of the Borel-Weil construction. I will discuss the geometric corollaries and questions that arise in this work. This is a joint project with Alexander Woo.

Tuesday, April 19, 2005

Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, April 19, 2005

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Submitted by nevins.

Kiran Kedlaya (MIT)
Some p-adic differential equations in algebraic geometry and number theory
Abstract: It has been known for over a century that certain special differential equations (e.g., Picard-Fuchs equations) occupy a distinguished role in algebraic geometry, by regulating the variation of periods (integrals of algebraic differential forms). It emerged from the work of Dwork on zeta functions of varieties over finite fields that similar differential equations (in fact, often the very *same* equations) play a similar role in controlling zeta functions! Since then, much work has gone into trying to explain Dwork's insights by constructing a form of de Rham cohomology for varieties over finite fields; this project continues to the present. I'll describe some recent progress in this area, and perhaps point out some surprising offshoots: some consequences in the theory of p-adic Galois representations (which lead to advances in the theory of modularity of Galois representations, as initiated by Wiles), and some "practical" applications (e.g., in cryptography).

Tuesday, April 26, 2005

Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, April 26, 2005

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Submitted by nevins.

Bruce Reznick (UIUC)
The structure of non-negative ternary sextic forms
Abstract: A real form is called "psd" if it only takes non-negative values; it is called "sos" is if can be written as a sum of squares of real forms. Clearly, every sos form is psd. In 1888, Hilbert proved that there are psd forms that not sos -- this is the motivation of his 17th Problem -- and the simplest case occurs with degree 6 in 3 variables. The description of the "extremal" psd ternary sextics has been an open question since then. Famous examples are due to Motzkin, Robinson and Choi-Lam. The speaker has been working on this problem for about 25 years and (at last) is close to proving that the extremal forms are simply the squares of cubic forms, together with the psd forms with exactly ten zeros, when singular zeros are counted appropriately. Of the 10 zeros, 8 can be taken in general position.

Tuesday, June 14, 2005

Thursday, June 16, 2005

Tuesday, June 21, 2005

Tuesday, June 28, 2005

Thursday, June 30, 2005

Tuesday, July 5, 2005

Tuesday, July 12, 2005

Thursday, July 14, 2005

Tuesday, July 19, 2005

Thursday, July 21, 2005

Tuesday, July 26, 2005

Thursday, July 28, 2005

Tuesday, August 30, 2005

Tuesday, September 6, 2005

Thursday, September 8, 2005

Tuesday, September 13, 2005

Thursday, September 15, 2005

Tuesday, September 20, 2005

Thursday, September 22, 2005

Tuesday, September 27, 2005

Thursday, September 29, 2005

Tuesday, October 4, 2005

Thursday, October 6, 2005

Tuesday, October 18, 2005

Commutative Ring Theory/Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, October 18, 2005

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Mahdi Majidi-Zolbanin (U of I at Springfield, Math)
In codimension 2, Gorenstein implies complete intersection
Abstract: We give a positive answer (in a special case) to a conjecture of R. Hartshorne asserting that certain quotients of regular local rings have to be complete intersections. More precisely we show that if (R,m) is a regular local ring of dimension at least five, p is a prime ideal of codimension two, and the ring $\Gamma(V,\widetilde(R/p))$ is Gorenstein, where V is the open set Spec(R/p)-{m}, then R/p is a complete intersection. The proof is based on a splitting criterion for vector bundles of small rank on punctured spectrum U of a regular local ring (R,m), in terms of vanishing of their intermediate cohomology modules H^i(U,E) for 1 < i < n-2, where n is the dimension of the regular local ring. This is the local version of a splitting criterion by N. Mohan Kumar, C. Peterson, and A. Prabhakar Rao for vector bundles of small rank on projective spaces.

Thursday, October 20, 2005

Tuesday, October 25, 2005

Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, October 25, 2005

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Submitted by amustata.

Adrian Clingher (Stanford University)
On a Family of Lattice Polarized K3 Surfaces
Abstract: In this talk, I will discuss a family of K3 surfaces with lattice polarization of type H+E_8+E_8. By a version of Global Torelli Theorem this type of surfaces are classified by two modular invariants much in the same way elliptic curves over complexes are classified by the J-invariant. I will present a method of computing these invariants based on an explicit construction of a Shioda-Inose structure. I will also discuss an extension of this method to H+E_8+E_7 polarizations as well as the relevance of these surfaces from the point of view of the F-theory/heterotic string duality in eight dimensions.

Thursday, October 27, 2005

Tuesday, November 1, 2005

Thursday, November 3, 2005

Tuesday, November 8, 2005

Algebraic Geometry Seminar
3:00 pm in 243 Altgeld Hall, Tuesday, November 8, 2005

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Submitted by amustata.

Andrei Caldararu (University of Wisconsin)
Toward computing the product in orbifold Hochschild cohomology
Abstract: I'll describe recent progress in generalizing to orbifolds Kontsevich's Theorem on Complex Manifolds (which relates Hochschild cohomology of a smooth variety to polyvector field cohomology). I shall present a generalization of the Hochschild-Kostant-Rosenberg theorem to orbifolds, and give a conjectural formula for the required correction, analogous to the root-A-hat that appears in Kontsevich's theorem. If time allows, I'll conclude by explaining how one can hope to apply the Duflo theorem to get a complete proof of the conjectural results I am mentioning. The talk is based on work in progress, partly joint with Simon Willerton.

Thursday, November 10, 2005

Graduate Student Algebraic Geometry Seminar
2:00 pm in 345 Altgeld Hall, Thursday, November 10, 2005

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Jason McCullough (UIUC)
An Introduction to Green's Conjecture
Abstract: This talk will build on Anca's talk covering linear series and canonical curves. Roughly stated, Green's conjecture relates two invariants associated to canonical curves, namely an invariant derived from Betti numbers of the coordinate ring of the curve and the clifford index, which contains a lot of geometric information. I will define these ideas, state and discuss Green's conjecture and try to convey the current status of the problem. This talk should be accessible to those with a little algebraic geometry.

Tuesday, November 15, 2005

Algebraic Geometry Seminar
3:00 pm in 3:00 pm 243 Altgeld Hall, Tuesday, November 15, 2005

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Submitted by amustata.

Aleksey Zinger [email] (SUNY Stony Brook)
Reduced Genus-One Gromov-Witten Invariants and Applications
Abstract: I will describe a "part" of the standard GW-invariant, which under ideal conditions counts genus-one curves without any genus-zero contribution. In contrast to the standard GW-invariant, the resulting reduced GW-invariant has the expected behavior with respect to certain embeddings. These invariants have applications to computing the standard genus-one GW-invariants of complete intersections as well as some enumerative genus-one invariants of sufficiently positive complete intersections.

Tuesday, November 29, 2005