Department of

# Mathematics

Seminar Calendar
for Algebraic Geometry Seminar events the year of Wednesday, September 12, 2012.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2012           September 2012          October 2012
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4                      1       1  2  3  4  5  6
5  6  7  8  9 10 11    2  3  4  5  6  7  8    7  8  9 10 11 12 13
12 13 14 15 16 17 18    9 10 11 12 13 14 15   14 15 16 17 18 19 20
19 20 21 22 23 24 25   16 17 18 19 20 21 22   21 22 23 24 25 26 27
26 27 28 29 30 31      23 24 25 26 27 28 29   28 29 30 31
30


Thursday, March 15, 2012

11:00 am in 217 Noyes,Thursday, March 15, 2012

#### On the areas of rational triangles

###### Noam Elkies (Harvard Math)

Abstract: By a "rational triangle" we mean a plane triangle whose sides are rational numbers. By Heron's formula, there exists such a triangle of area $\sqrt{a}$ if and only if $a > 0$ and $x y z (x + y + z) = a$ for some rationals $x, y, z$. In a 1749 letter to Goldbach, Euler constructed infinitely many such $(x, y, z)$ for any rational $a$ (positive or not), remarking that it cost him much effort, but not explaining his method. We suggest one approach, using only tools available to Euler, that he might have taken, and use this approach to construct several other infinite families of solutions. We then reconsider the problem as a question in arithmetic geometry: $xyz(x+y+z) = a$ gives a K3 surface, and each family of solutions is a singular rational curve on that surface defined over $\mathbb{Q}$. The structure of the Neron-Severi group of that K3 surface explains why the problem is unusually hard. Along the way we also encounter the Niemeier lattices (the even unimodular lattices in $\mathbb{R}^{24}$) and the non-Hamiltonian Petersen graph.

Tuesday, April 3, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, April 3, 2012

#### Quantum Cohomology of Toric Varieties

###### Sheldon Katz   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: The structure of the quantum cohomology ring of a smooth projective toric variety was described by Batyrev and proven by Givental as a consequence of his work on mirror symmetry. This talk is in part expository since some details were never written down by Givental. I conclude with some open questions related to the quantum cohomology ring and the quantum product. An extension of these questions play a foundational role in the development of quantum sheaf cohomology which has been undertaken jointly with Donagi, Guffin, and Sharpe. Given a smooth projective variety X and a vector bundle E with $c_i(E)=c_i(X)$ for i=1,2, the quantum sheaf cohomology ring of string theory is supposed to be a deformation of the algebra $H^*(X,\Lambda^*E^*)$. If E=TX, quantum sheaf cohomology is the same as ordinary quantum cohomology.

Wednesday, April 4, 2012

3:00 pm in 145 Altgeld Hall,Wednesday, April 4, 2012

#### A construction of codes based on the Cartier operator

###### Alain Couvreur (INRIA Saclay and Ecole Polytechnique Paris)

Abstract: We present a new construction of codes from algebraic curves which is suitable to provide codes on small fields. The approach involves the Cartier operator and can be regarded as a natural generalisation of classical Goppa codes. As for algebraic geometry codes, lower bounds on the parameters of these codes can be obtained by algebraic geometric methods.

Tuesday, August 28, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, August 28, 2012

#### Maps in Kahler Geometry associated to Okounkov Bodies

###### Julius Ross (University of Cambridge)

Abstract: The Okounkov body is a convex body in Euclidean space that can be associated to a projective manifold with a given flag of submanifolds. This convex body generalises certain aspects of the familiar Delzant polytope for toric varieties, although the Okounkov body will not be polyhedral or rational in general. In this talk I will discuss some joint work with David Witt-Nystrom that involves the study of maps from a manifold to its Okounkov body coming from Kahler geometry that are similar to the moment map in toric geometry. I will start by introducing the Okounkov body and the kind of maps that one might like to have, and then give an inductive construction that works in a neighbourhood of the flag. This is acheived through a homogeneous Monge-Ampere equation associated to the degeneration to the normal cone of a divisor, that can be thought of as a kind of "tubular neighbourhood" theorem in complex geometry.

Tuesday, September 25, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, September 25, 2012

#### Refined Stable Pair Invariants on Local Calabi-Yau Threefolds

###### Sheldon Katz (Illinois Math)

Abstract: A refinement of the stable pair invariants of Pandharipande and Thomas is introduced, either as an application of the equivariant index recently introduced by Nekrasov and Okounkov or as "motivic" Laurent polynomial based on what we call the virtual Bialynicki-Birula decomposition, specializing to the usual stable pair invariants. We propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local $P^1$, based on the refined BPS invariants of the string theorists Huang, Kashani-Poor, and Klemm. We explicitly compute the invariants in low degree for local $P^2$ and local $P^1 \times P^1$ and check that they agree with the predictions of string theory and with our conjectured product formula. This is joint work with Jinwon Choi and Albrecht Klemm.

Tuesday, October 2, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, October 2, 2012

#### Abramovich-Vistoli vs. Alexeev/Kollar--Shepherd-Barron

###### Gabriele La Nave (UIUC Math)

Abstract: I will discuss why Kontsevich stable maps into DM stacks are stacky in nature and discuss Abramovich-Vistoli's theory of twisted curves and their consequent description of the compactification of the moduli space of "fibered surfaces" in contrast with Kollar--Shepherd-Barron MMP type of compactifications. I will then describe how to use these tools along with some toric geometry to give complete explicit description of the boundary of the moduli space of elliptic surfaces with sections.

Tuesday, October 9, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, October 9, 2012

#### Stability of finite Hilbert points

###### David Smyth (Harvard)

Abstract: The classical construction of the moduli space of stable curves via Geometric Invariant Theory relies on the asymptotic stability result of Gieseker and Mumford that the m-th Hilbert Point of a pluricanonically embedded curve is GIT-stable for all sufficiently large m. Several years ago, Hassett and Keel observed that if one could carry out the GIT construction with non-asymptotic linearizations, the resulting models could be used to run a log minimal model program for the space of stable curves. A fundamental obstacle to carrying out this program has been the absence of a non-asymptotic analogue of Gieseker's stability result, i.e. how can one prove stability of the m-th Hilbert point for small values of m? In recent work with Jarod Alper and Maksym Fedorchuk, we prove that the the m-th Hilbert point of a general smooth canonically or bicanonically embedded curve is GIT-semistabe for all m>1. For (bi)canonically embedded curves, we recover Gieseker-Mumford stability by a much simpler proof.

Tuesday, October 30, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, October 30, 2012

#### Hyperdeterminants of polynomials

###### Luke Oeding   [email] (University of California, Berkeley)

Abstract: Hyperdeterminants were brought into a modern light by Gelʹfand, Kapranov, and Zelevinsky in the 1990's. Inspired by their work, I will answer the question of what happens when you apply a hyperdeterminant to a polynomial (interpreted as a symmetric tensor). The hyperdeterminant of a polynomial factors into several irreducible factors with multiplicities. I identify these factors along with their degrees and their multiplicities, which both have a nice combinatorial interpretation. The analogous decomposition for the μ-discriminant of polynomial is also found. The methods I use to solve this algebraic problem come from geometry of dual varieties, Segre-Veronese varieties, and Chow varieties; as well as representation theory of products of general linear groups.

Tuesday, November 6, 2012

3:00 pm in Altgeld Hall,Tuesday, November 6, 2012

#### The birational geometry of the Hilbert scheme of points on surfaces and Bridgeland stability

###### Izzet Coskun (UIC)

Abstract: In this talk, I will discuss the cones of ample and effective divisors on Hilbert schemes of points on surfaces. I will explain a correspondence between the Mori chamber decomposition of the effective cone and the Bridgeland decomposition of the stability manifold. This is joint work with Daniele Arcara, Aaron Bertram and Jack Huizenga.

Tuesday, November 13, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, November 13, 2012

#### Affine Lie algebras and Rational Cherednik Algebras

###### Peng Shan (MIT)

Abstract: Varagnolo-Vasserot conjectured an equivalence between the category O of cyclotomic rational Cherednik algebras and a parabolic category O of affine Lie algebras. I will explain a proof of this conjecture and some applications on the characters of simple modules for cyclotomic rational Cherednik algebras and the Koszulity of its category O. This is a joint work with R. Rouquier, M. Varagnolo and E. Vasserot.

Tuesday, November 27, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, November 27, 2012

#### Syzygies and Boij--Soederberg Theory

###### Daniel Erman (University of Michigan)

Abstract: For a system of polynomial equations, it has long been known that the relations (or syzygies) among the polynomials provide insight into the properties and invariants of the corresponding projective varieties. Boij--Soederberg Theory offers a powerful perspective on syzygies, and in particular reveals a surprising duality between syzygies and cohomology of vector bundles. I will describe new results on this duality and on the properties of syzygies. This is joint work with David Eisenbud.

Tuesday, December 4, 2012

3:00 pm in 243 Altgeld Hall,Tuesday, December 4, 2012

#### Extremal effective divisors on the moduli space of curves

###### Dawei Chen (Boston College)

Abstract: The cone of effective divisors plays a central role regarding the birational geometry of a variety X. In this talk we discuss several approaches that verify the extremality of a divisor, with a focus on the case when X is the moduli space of curves.