UIUC Dept. of Mathematics Seminar Calendar

      July 2012             August 2012           September 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  6  7             1  2  3  4                      1
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    2  3  4  5  6  7  8
 15 16 17 18 19 20 21   12 13 14 15 16 17 18    9 10 11 12 13 14 15
 22 23 24 25 26 27 28   19 20 21 22 23 24 25   16 17 18 19 20 21 22
 29 30 31               26 27 28 29 30 31      23 24 25 26 27 28 29
                                               30

Thursday, September 27, 2012

Graduate Geometry Topology Seminar
2:00 pm in 241 Altgeld Hall, Thursday, September 27, 2012

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Submitted by collier3.

Bill Karr (UIUC Math)
Geodesics on Surfaces of Revolution in Minkowski Space
Abstract: I will introduce some basic definitions from Lorentzian geometry and a notion of angle in the tangent space to a Lorentzian manifold. Then, I'll explain my REGS project about geodesics on surfaces of revolution in Minkowski space using a spacetime version Clairaut's relation from classical differential geometry.

Thursday, October 18, 2012

Tuesday, October 23, 2012

Differential Geometry
1:00 pm in 243 Altgeld Hall, Tuesday, October 23, 2012

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Submitted by lanave.

Gabriele La Nave (UIUC Math)
Macroscopic dimension and fundamental groups of manifolds with positive isotropic curvature.
Abstract: There is a long history in Comparison Geometry of drawing topological consequences from curvature properties. This usually involves studying consequences of curvature signs on either the nature of geodesics or the (non) existence of harmonic sections of special bundles. Micallef and Moore, around the late 80's, introduced the notion of manifolds with positive isotropic curvature --a notion of curvature which stems from complexifying the tangent bundle and its metric and which lies between positive sectional curvature and positive scalar curvature-- because best suited to the study of manifolds via the use of their minimal surfaces. Gromov introduced the notion of macroscopic dimension exactly to study questions related to the geometry/topology of manifolds with some sort of positive curvature and such notion turns out to be very useful in studying the topology of manifolds, especially in regards to questions pertaining the fundamental group. In the talk I will show how recent techniques of Donaldson's can be brought to bear in the study of the macroscopic dimension of manifolds with (uniformly) positive isotropic curvature

Wednesday, October 24, 2012

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Wednesday, October 24, 2012

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Ilya Kapovich [email] (UIUC Math)
Groups as geometric objects
Abstract: Geometric group theory is a vibrant and rapidly developing area of mathematics that lies at the juncture of group theory, low-dimensional topology, differential geometry and several other subjects. A crucial idea in the area is to view a finitely generated group as a geometric and not just as an algebraic object. One of the key tools for realizing this goal is the notion of the Cayley graph of a group, which is a metric space associated to a finitely generated group together with a finite generating set. Geometric group theory studies the connections between large-scale geometric properties of groups on one side and their algebraic and algorithmic properties on the other side. In this introductory talk we will explore the basic ideas and notions of the subject and demonstrate how the above connections manifest themselves in a number of representative results.

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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Submitted by clein.

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, December 4, 2012

Differential Geometry
2:00 pm in 243 Altgeld Hall, Tuesday, December 4, 2012

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Submitted by palbin.

Jesse Gell-Redman (University of Toronto)
Harmonic maps of conic surfaces
Abstract: Given a diffeomorphism between two compact Riemannian surfaces without boundary and negative curvature, there is a unique map between them that: 1) minimizes the Dirichlet energy, i.e. the $L^2$ norm of the derivative, and 2) is homotopic to the original diffeomorphism. This minimizer is a diffeomorphism. These facts are due to Eells-Sampson and Schoen-Yau, and have applications to Teichmueller theory. The same is true for maps between surfaces with conical singularities. We discuss the proof of this result when the cone angles are less than $2\pi$. The linear theory involved is Melrose's b-calculus, and we will give some background and explanation of its use in this context.

Tuesday, March 12, 2013

Differential Geometry
1:00 pm in Altgeld Hall, Tuesday, March 12, 2013

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Submitted by palbin.

David Sher (CRM Montreal)
The determinant of the Laplacian on moduli space
Abstract: The determinant of the Laplacian is a global spectral invariant of a Riemannian manifold which has important applications in PDE and spectral theory. One such application is to the inverse spectral problem, which aims to recover geometric information about a manifold from the spectrum of its Laplacian. In the late 1980s, Osgood, Phillips, and Sarnak used the determinant to show that any set of isospectral metrics on a closed surface is compact. They also proved an analogous result for planar domains; further generalizations, to isospectral sets of flat metrics on surfaces with boundary, were obtained more recently by Y.-H. Kim. In both the original proof and the generalizations, it is crucial to understand the behavior of the determinant of the Laplacian on various spaces of constant curvature metrics. Although the generalizations have been quite successful, the precise behavior of the determinant as a function on the moduli space of flat metrics on a surface with boundary is still not well-understood, and many interesting questions remain unanswered. In this talk, I will first define the determinant and then explain the results of Osgood-Phillips-Sarnak, as well as subsequent work of Khuri and of Kim, in order to motivate the study of the determinant on moduli space. I will conclude by briefly discussing recent work (arXiv 1210:1631) which analyzes the behavior of the determinant on particular degenerating families of flat metrics on surfaces with boundary.

Differential Geometry
2:00 pm in 243 Altgeld Hall, Tuesday, March 12, 2013

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Submitted by sba.

Christina Sormani (CUNY, Courant Institute)
THE TETRAHEDRAL PROPERTY AND A NEW GROMOV-HAUSDORFF COMPACTNESS THEOREM
Abstract: We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff �sense to a countably H^m rectiﬁable metric space of the same dimension. The tetrahedral property depends only on distances between points in spheres, yet we show it provides a lower bound on the volumes of balls. The proof is based upon intrinsic ﬂat convergence and a new notion called the sliced ﬁlling volume of a ball.

Tuesday, March 26, 2013

Joint Group Theory/Differential Geometry Seminar
1:00 pm in Altgeld Hall 243, Tuesday, March 26, 2013

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Submitted by kapovich.

Oleg Bogopolski (University of Duesseldorf)
On residual properties of groups
Abstract: The residual finiteness, the conjugacy separability, and the subgroup separability (LERF) are important properties of groups, which help to solve some algorithmic problems in groups and which are important in the theory of 3-manifolds. After a short introduction to these properties I will concentrate on a new residual property which is called subgroup conjugacy separability. A group G is called subgroup conjugacy separable (SCS) if for any two nonconjugate finitely generated subgroups of G, there exists a finite quotient of G, where the images of these subgroups are nonconjugate. We show that fundamental groups of orientable surfaces are SCS. The interesting point in the proof is the use of a hyperbolic metric for surfaces of genus at least 2. This is a joint work with K-U. Bux.

Thursday, April 4, 2013

Differential Geometry/Group Theory
1:00 pm in 347 Altgeld Hall, Thursday, April 4, 2013

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Submitted by nmd.

Dylan Thurston (Indiana University)
Conformal embeddings of graphs
Abstract: Given a branched topological covering $f$ of the sphere by itself, can it be realised as a rational map? William Thurston gave a criterion in 1982: If $f$ has a hyperbolic orbifold, it can be realised as a rational map iff there is not an invariant multi-curve satisfying certain conditions. This condition is hard to apply in practice, since it involves checking infinitely many multi-curves, but is nevertheless useful. We give a sufficient combinatorial condition in the other direction: if there is a metric graph $G$ so that $f^{-1}(G)$ conformally embeds inside $G$, then $f$ can be realised as a rational map. Here, a map $p: G \to H$ between metric graphs is a conformal embedding if, for all $y \in H$, \[ \sum_{f(x)=y} |f'(x)| < 1. \] This is joint-work-in-progress with Kevin Pilgrim.

Tuesday, April 9, 2013

Tuesday, April 23, 2013

Differential Geometry/Group Theory
1:00 pm in 243 Altgeld Hall, Tuesday, April 23, 2013

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Submitted by clein.

Sam Taylor (U Texas Math)
Right-angled Artin groups in Out($F_n$)
Abstract: Right-angled Artin groups have been used extensively to explore the structure of other geometrically significant groups. After reviewing some results that support this claim, we introduce a construction that produces homomorphisms from right-angled Artin groups into Out($F_n$), the outer automorphism group of a free group, which are quasi-isometric embeddings. These homomorphisms are analogous to those constructed by Clay, Leininger, and Mangahas into the mapping class group of a surface. Time permitting, we will also discuss the analogs of subsurface projections that are used in the proof as well as their relation to other notions of subfactor projection that are found in the literature.