UIUC Dept. of Mathematics Seminar Calendar

       May 2012              June 2012              July 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  5                   1  2    1  2  3  4  5  6  7
  6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
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 20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
 27 28 29 30 31         24 25 26 27 28 29 30   29 30 31

Tuesday, April 24, 2012

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

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Submitted by sba.

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.

Tuesday, November 6, 2012

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

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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.