UIUC Dept. of Mathematics Seminar Calendar

      June 2012              July 2012             August 2012     
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
                 1  2    1  2  3  4  5  6  7             1  2  3  4
  3  4  5  6  7  8  9    8  9 10 11 12 13 14    5  6  7  8  9 10 11
 10 11 12 13 14 15 16   15 16 17 18 19 20 21   12 13 14 15 16 17 18
 17 18 19 20 21 22 23   22 23 24 25 26 27 28   19 20 21 22 23 24 25
 24 25 26 27 28 29 30   29 30 31               26 27 28 29 30 31

Tuesday, February 7, 2012

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, February 7, 2012

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Dick Bishop (UIUC)
Comparison geometry, total curvature, pursuit-evasion
Abstract: (Joint work with S. Alexander and R. Ghrist.) After a review of comparison geometry, total curvature, and pursuit-evasion in CAT(0) spaces, the properties of pursuit flow, particularly its invariant conics (see Figure), for an evader tracking a line in the plane will be examined. A comparison for a pursuer of an evader tracking a geodesic in a CAT(0) space is given. More generally, any evader curve of finite total curvature in a CAT(0) can be developed to a curve in the plane with the same total curvature on all corresponding segments and there is a comparable pursuit flow in the plane.

Tuesday, March 13, 2012

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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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 17, 2012

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, April 17, 2012

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Matthew Wright (Huntington University)
Hadwiger Integration and Applications
Abstract: Integration of integer-valued functionals with respect to Euler characteristic has surprising applications in problems arising in sensor networks, as Rob Ghrist demonstrated in the Trjitzinsky Lectures. This integration theory makes use of Euler characteristic as a topological invariant for counting objects detected by the network. However, Euler characteristic is only one of n+1 Euclidean-invariant valuations on "tame" subsets of n-dimensional Euclidean space. Integration of integer-valued functionals with respect to any of these valuations is straightforward. We can extend this integration theory to real-valued functionals to obtain what we call Hadwiger integrals. The Hadwiger integrals provide various notions of the size of a functional. This talk will explain the theory of Hadwiger integration and discuss some of its challenges and potential applications.

Tuesday, April 24, 2012

Differential Geometry Seminar
1:00 pm in 243 Altgeld Hall, Tuesday, April 24, 2012

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Steven Rayan (U Toronto Math)
Combinatorics of the moduli space of L-twisted Higgs bundles at genus 0
Abstract: An L-twisted Higgs bundle on a compact Riemann surface is a vector bundle together E with an L-valued Higgs field, that is, an endomorphism taking values along a fixed line bundle L.� (Ordinary Higgs bundles arise by choosing the canonical line bundle for L.)� The Betti numbers of the moduli space of L-twisted Higgs bundles on P^1, with fixed numerical invariants, can be determined by Hitchin's localization calculation: the Poincar\'e series of the moduli space is the (weighted) sum of Poincar\'e series of certain subvarieties of the nilpotent cone.� These subvarieties are precisely moduli spaces of holomorphic chains: these are chains of vector bundles where the maps are L-twisted Higgs fields.� Some of the difficulty in classifying these chains is avoided in the case of P^1, over which the situation becomes very combinatorial.� I will calculate Betti numbers for certain low values of the rank of E and degree of L, in order to verify some conjectural numbers coming from Mozgovoy's twisted version of Chuang, Diaconescu, and Pan's ADHM formula.� I will also make some conjectures about properties of the Betti numbers, including in the case of arbitrary genus.

Thursday, May 3, 2012

Geometry Seminar
3:00 pm in 347 Altgeld Hall, Thursday, May 3, 2012

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Joseph Rosenblatt (UIUC)
Partitions for Optimal Approximations
Abstract: The Riemann integral can be approximated using partitions and a rule for assigning weighted sums of the function at points determined by the partition. Approxi- mation methods commonly used include endpoint rules, the midpoint rule, the trapezoid rule, Simpson�s rule, and other quadrature methods. The rate of approximation depends mostly on the rule being used and the smoothness of the function, but fine structure in this rate of approximation depends on choosing an optimal partition. We discuss how one chooses an optimal partition of points, what is the resulting rate of approximation as the number of points tends to infinity, and how to determine the characteristic distribution of the points in these optimal partitions.

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 11, 2012

Tuesday, September 25, 2012

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

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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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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.

Tuesday, October 16, 2012

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, October 16, 2012

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Kenneth Stolarsky (UIUC)
Distance geometry, reproducing kernels, and numerical integration
Abstract: How should n points be placed on the surface of a d-dimensional sphere to minimize or maximize a given function of their mutual Euclidean distances? Some fundamental problems of numerical integration have recently been given a new framework based on the not so recent concept of reproducing kernels. It turns out that this framework provides new insights into distance geometry by viewing in a certain way the Euclidean distance between points as the essential part of a reproducing kernel. This talk will be based on work by J. Brauchart, J. Dick, F. Hickernell, and F. Pillichshammer. It will have significant overlap with the seminar given by the speaker on October 4.

Tuesday, October 30, 2012

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, October 30, 2012

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Matthew Wright (Huntington University)
Hadwiger Integration of Random Fields
Abstract: Hadwiger integrals provide various notions of the "size" of a function, analogous to the notions of size that the intrinsic volumes provide for a set. A random field is a stochastic process that we can think of as a random function; its value at each point its domain is a random variable. We can combine these concepts to consider the Hadwiger integrals of a random field. In this talk, I will provide some necessary background information about Hadwiger integrals and random fields. I will then compute the expected Hadwiger integrals of a family of random fields known as Gaussian-related random fields. These random fields have possible applications in sensor networks, signal processing, and other areas. I will discuss these applications and opportunities for future work that will make this integration theory more useful in practice.

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.

Tuesday, November 6, 2012

Differential Geometry Seminar
1:00 pm in 243 Altgeld Hall, Tuesday, November 6, 2012

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Submitted by clein.

Chih-Chung Liu (UIUC Math)
The Analysis of Vortex Equations
Abstract: I will introduce the notion of vortices, pairs of connections and smooth sections solving a set of PDE's on a vector bundle called vortex equations. These equations characterize the minimum of certain gauge invariant functionals known as the Yang-Mills Higgs functional. A natural variation of the study of classical vortex equations is to introduce a parameter $s$ and let $s \to \infty$, a process known as the "adiabatic limit". I will present the results on the controls of the vortices in suitable Sobelev norms over $s$ and the limiting behaviors. The results provide an application on the dynamics of vortices given by the "kinetic energy" of vortices, or a certain "$L^2$ metric. As $s \to \infty$, we show that this metric degenerates to a familiar $L^2$ metric on the space of holomorphic maps to projective space. The results are joint work with Steven Bradlow and Gabriele La Nave.

Tuesday, November 13, 2012

Geometry/Differential Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, November 13, 2012

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Andy Sanders (U Maryland)
Domains of discontinuity of almost-Fuchsian groups
Abstract: An almost-Fuchsian group is a quasi-Fuchsian group which preserves an embedded minimal disk in hyperbolic 3-space such that the quotient of this disk is a closed minimal surface all of whose principal curvatures lie in the interval (-1, 1). The hyperbolic Gauss map from the minimal disk defines a di ffeomorphism onto each component of the domain of discontinuity. We will explain how a study of the Gauss map imposes constraints on the structure of the domain of discontinuity. In particular, we will explain how this structure can be used to show that no geometric limit of almost-Fuchsian groups can be doubly degenerate.

Tuesday, November 27, 2012

Tuesday, December 4, 2012