UIUC Dept. of Mathematics Seminar Calendar

       May 2012              June 2012              July 2012      
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Wednesday, January 25, 2012

Mathematics Colloquium - Special Lecture 2011-12
4:00 pm in 245 Altgeld Hall, Wednesday, January 25, 2012

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Submitted by kapovich.

Philipp Hieronymi (Department of Mathematics, University of Illinois)
Tame geometry
Abstract: This talk is an introduction into the study of well-behaved expansions of semialgebraic geometry. I will focus on the classification of such geometries and my contribution to it. In particular, I will discuss new tameness phenomena outside the setting of local finiteness and I will described how a new result that certain classical structures are not tame at all, sheds new light on the question what tameness actually means.

Thursday, January 26, 2012

Graduate Geometry and Topology Seminar
2:00 pm in 241 Altgeld Hall, Thursday, January 26, 2012

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Mee Seong Im (UIUC Math)
The Hamiltonian reduction of a certain affine variety
Abstract: I will discuss certain theories in symplectic geometry and in algebraic geometry which give us various ways to view the same complex manifold. More specifically, the Hamiltonian reduction of the cotangent bundle of a certain variety can be thought of as the symmetric product of the complex plane while the GIT quotient of the same cotangent bundle but which is twisted by a character of the general linear group can be thought of as a certain Hilbert scheme. These varieties are related by the Hilbert-Chow morphism in the sense that one is a desingularization of the other. I will end with an analogous construction in which a notion of noncommutativity appears in the algebro-geometric quotient. Lots of examples will be provided throughout my talk.

Tuesday, February 14, 2012

Algebraic Geometry
3:00 pm in 243 Altgeld Hall, Tuesday, February 14, 2012

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Wenbo Niu [email] (Purdue)
Asymptotic Regularity of Ideal Sheaves
Abstract: Let $I$ be an ideal sheaf on $\mathbb{P}^n$ . Associated to $I$ there are three elementary invariants: the invariant $s$ which measures the positivity of $I$, the minimal number $d$ such that $I(d)$ is generated by its global sections, and the Castelnuovo-Mumford regularity reg $I$. In general one has $s \leq d\leq \mbox{reg }I$. If we consider the asymptotic behavior of the regularity of $I$, that is the regularity of $I^p$ when $p$ is sufficiently large, then we could have a clear picture involving these invariants. We will talk about two main theorems in this direction. The first one is the asymptotic regularity of $I$ is bounded by linear functions as $sp\leq \mbox{reg }I^p\leq sp+e$, where $e$ is a constant. The second one is that if $s=d$, i.e., $s$ reaches its maximal value, then for $p$ large enough reg $I^p=dp+e$ for some positive constant $e$.

Wednesday, February 29, 2012

Thursday, March 8, 2012

Joint Group Theory/Differential Geometry/Ergodic Theory
1:00 pm in 347 Altgeld Hall, Thursday, March 8, 2012

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Alex Wright (University of Chicago)
Arithmetic and Non-Arithmetic Teichmüller Curves
Abstract: Teichmüller curves are isometrically immersed curves in the moduli space of Riemann surfaces. Their study lies at the intersection of dynamics, Teichmüller theory, and algebraic geometry. I will begin by summarizing known results on Teichmüller curves, pointing out some similarities to the study of lattices, for example in PU(n,1). I will then move on to new research involving abelian square-tiled surfaces, Schwarz triangle mappings, and the Veech-Ward-Bouw-Moller Teichmüller curves.

Tuesday, March 13, 2012

Algebraic Geometry
3:00 pm in 243 Altgeld Hall, Tuesday, March 13, 2012

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Tom Nevins [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)
Derived equivalence of quantum symplectic varieties
Abstract: Singular symplectic varieties and their resolutions of singularities lie at the crossroads of algebraic and symplectic geometry, representation theory, and integrable systems. Central examples include the nilpotent cone of a complex semisimple Lie algebra and its resolution by the cotangent bundle of the flag variety (the Springer resolution); the nth symmetric product of the affine plane and its resolution by the Hilbert scheme of points; and a Kleinian surface singularity and its minimal resolution. A singular variety and its resolution never have equivalent geometry (as encoded, for example, in their derived categories). Replacing a symplectic variety by a quantization, however---an algebro-geometric analog of passing to a Fukaya-type category---one miraculously finds that such equivalences are common. I'll discuss singular symplectic varieties and their resolutions, examples, quantization, and a general criterion for such geometric equivalences that extends classical results (for example, Beilinson-Bernstein localization). Time permitting, I'll also discuss some additional features of these quantizations that parallel emerging structures in the (much more complicated) world of Fukaya categories. This is based on joint work with K. McGerty.

Thursday, March 15, 2012

Joint number theory / algebraic geometry seminar
11:00 am in 217 Noyes, Thursday, March 15, 2012

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Noam Elkies (Harvard Math)
On the areas of rational triangles
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

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

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Submitted by katz.

Sheldon Katz [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)
Quantum Cohomology of Toric Varieties
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

Tuesday, April 10, 2012

Algebraic Geometry
3:00 pm in 243 Altgeld Hall, Tuesday, April 10, 2012

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Submitted by nevins.

Sarah Kitchen (Albert-Ludwigs-Universität Freiburg)
Koszul categories and mixed Hodge modules
Abstract: In this talk, I will report on joint work with Pramod Achar. We considered the following problem: Can we generate a Koszul category from the category of mixed Hodge modules on a smooth complex variety X (constructible along an affine stratification S) by a general procedure, which gives a grading on the category of S-constructible rational perverse sheaves on X? We were motivated by the fact that in their paper on Koszul Duality, Beilinson, Ginzburg and Soergel (BGS) produce their grading from mixed Hodge modules in a way specific to the Bruhat stratification of a flag variety, whereas their approach to l-adic perverse sheaves was more general. I will explain how to "winnow" the category of mixed Hodge modules to come up with the desired Koszul category, and how to obtain a grading on S-constructible perverse sheaves from this, plus the relationship to the grading obtained by BGS.

Wednesday, April 11, 2012

Tuesday, April 17, 2012

Thursday, April 26, 2012

Tuesday, August 28, 2012

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

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Submitted by choi29.

Julius Ross (University of Cambridge)
Maps in Kahler Geometry associated to Okounkov Bodies
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 4, 2012

Tuesday, September 11, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, September 11, 2012

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Andrew Arana (UIUC Philosophy and Math)
Transfer in algebraic geometry - Part II
Abstract: The focal question of this talk is to investigate the value of transfer between algebra and geometry, of the sort exemplified by the Nullstellensatz. Algebraic geometers frequently talk of such transfer principles as a "dictionary" between algebra and geometry, & claim that these dictionaries are fundamental to their practice. We'll first need to get clear on what such transfer consists in. We'll then investigate what how such transfer might improve how knowledge is gathered in algebraic geometric practice. --- This talk is a continuation of the talk given in the Logic seminar last week. Enough of a survey of what has come before will be given so that people who missed the first talk, can still attend this talk with profit.

Thursday, September 20, 2012

Tuesday, September 25, 2012

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

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Submitted by katz.

Sheldon Katz (Illinois Math)
Refined Stable Pair Invariants on Local Calabi-Yau Threefolds
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

Tuesday, October 9, 2012

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

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Submitted by katz.

David Smyth (Harvard)
Stability of finite Hilbert points
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.

Friday, October 12, 2012

Special Seminar
2:00 pm in 241 Altgeld Hall, Friday, October 12, 2012

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Jordan Ellenberg (University of Wisconsin - Madison)
New developments in homological stability and FI-modules
Abstract: In topology and algebraic geometry one often encounters phenomena of _stability_. A famous example is the cohomology of the moduli space of curves $M_g$; Harer proved in the 1980s that the sequence of vector spaces $H_i(M_g, \mathbb{Q})$, with $g$ growing and $i$ fixed, has dimension which is eventually constant as $g$ grows with $i$ fixed. In many similar situations one is presented with a sequence $\{V_n\}$, where the $V_n$ are not merely vector spaces, but come with an action of $S_n$. In many such situations the dimension of $V_n$ does not become constant as $n$ grows -- but there is still a sense in which it is eventually "always the same representation of $S_n$" as n grows. The preprint http://arxiv.org/abs/1204.4533 shows how to interpret this kind of "representation stability" as a statement of finite generation in an appropriate category; we'll discuss this set-up and some applications to the topology of configuration spaces, the representation theory of the symmetric group, and diagonal coinvariant algebras. As a sample result, we explain how to show that the $i$th Betti number of the configuration space of $n$ (ordered) points on a manifold $M$ is, for large enough $n$, not constant, but rather a polynomial in $n$. This talk will be aimed at a broad audience at the level of a typical colloquium. (joint work with T. Church, B. Farb, and R. Nagpal)

Tuesday, October 30, 2012

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

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Luke Oeding [email] (University of California, Berkeley)
Hyperdeterminants of polynomials
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.

Thursday, November 1, 2012

Tuesday, November 6, 2012

Thursday, November 8, 2012

Women'sSeminar
2:00 pm in 345 Altgeld Hall, Thursday, November 8, 2012

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Mee Seong Im (UIUC Math)
Moduli problems through representations of quivers
Abstract: Representations of quivers are prominent in many areas of mathematics: in symplectic geometry, representation theory, and mathematical physics to name a few. One interesting fact connecting algebraic geometry and quivers is M Reineke's two page proof that every projective variety is a quiver Grassmannian [arXiv: 1204.5730]. The aim of this talk is for the general audience as I will give a number of examples and use them as a basis as we discuss moduli problems and representations of quivers. I intend to mostly follow King's paper Moduli of Representations of Finite Dimensional Algebras. Minimal mathematical background is linear algebra, and questions are encouraged.

Commutative ring theory seminar
3:00 pm in 243 Altgeld Hall, Thursday, November 8, 2012

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Youngsu Kim (Purdue University)
On the Equality of Ordinary and Symbolic Powers of Ideals
Abstract: Symbolic powers of ideals are central objects in commutative algebra and algebraic geometry. For example, in a polynomial ring over an algebraically closed field, the $n$-th symbolic power $P^{(n)}$ of a prime ideal P is the set of functions that vanish to order at least $n$ along $V(P)$. In this talk, we discuss criteria for the equality of ordinary and symbolic powers. In particular, we are interested to know if the equality of $P^n$ and $P^{(n)}$ for all $n \leq $ a certain bound implies the equality for all $n$ (joint work with A. Hosry and J. Validashti).

Tuesday, November 13, 2012

Tuesday, November 27, 2012

Tuesday, December 4, 2012

Tuesday, December 11, 2012

Topology Seminar
11:00 am in 243 Altgeld Hall, Tuesday, December 11, 2012

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John Harper (Purdue University)
TQ-homology completion of nilpotent structured ring spectra
Abstract: An important theme in current work in homotopy theory is the investigation and exploitation of enriched algebraic structures on spectra that naturally arise, for instance, in algebraic topology, algebraic K-theory, and derived algebraic geometry. Such structured ring spectra or ``geometric rings'' are most simply viewed as algebraic-topological generalizations of the notion of ring from algebra and algebraic geometry.

This talk will describe recent progress, in joint work with M. Ching, on developing standard tools of the homotopy theory of spaces in this new algebraic-topological context of structured ring spectra, with a special emphasis on recovering algebraic and topological structures from associated homology objects.

Tuesday, December 18, 2012

Algebraic Geometry
3:00 pm in 343 Altgeld Hall, Tuesday, December 18, 2012

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Mee Seong Im (UIUC Math)
Invariants and semi-invariants of filtered quivers
Abstract: Invariants and semi-invariant polynomials are important in classical invariant theory and in geometric invariant theory (GIT). More precisely, the affine quotient of a variety under the action of a group G is defined by those polynomials invariant under the action of the group while semi-invariant polynomials are used in the construction of GIT quotients. A.D. King in one of his celebrated papers connected the GIT quotient construction and the geometry of quiver variety construction. Quiver varieties are nice and ubiquitous in the sense that they arise in representation theory, mathematical physics, cluster algebras, etc.

I will define the notion of a filtered quiver as such examples arise naturally in mathematics. For example, consider a Borel acting on its Lie algebra; how would one strategically produce invariants and semi-invariant polynomials? In another instance, how should one study and manage the B-orbits on the cotangent bundle of the Grothendieck-Springer resolution? Filtered quiver varietes are a generalization of quiver varieties, yet mathematically, they seem to appear as a closed subvariety in an open subset in one of the irreducible components of quivers with relations. I will explain some of my results on how one produces invariants for any acyclic and cyclic filtered quivers. I will then explain how one produces semi-invariants for any acyclic and cyclic filtered quivers. Although there are many interesting examples, I will focus on the B-orbits on Lie(B) and the cotangent bundle of the Grothendieck-Springer resolution. If there is time, I will discuss how filtered quivers are embedded in quivers with relations.