UIUC Dept. of Mathematics Seminar Calendar

     October 2009          November 2009          December 2009    
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              1  2  3    1  2  3  4  5  6  7          1  2  3  4  5
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Friday, January 16, 2009

Mathematics Colloquium - Special Lecture 2008-09
4:00 pm in 245 Altgeld Hall, Friday, January 16, 2009

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Benjamin Schmidt (University of Chicago)
Blocking geodesics in Riemannian manifolds
Abstract: To what extent does the collision of light in a Riemannian manifold determine the metric? I'll discuss some conjectures and supporting results that aim to characterize non-negatively curved symmetric spaces in terms of their geodesics.

Please join us for refreshments at 3:30 p.m. in Room 321 Altgeld Hall.

Wednesday, January 21, 2009

Mathematics Colloquium - Special Lecture 2008-09
4:00 pm in 245 Altgeld Hall, Wednesday, January 21, 2009

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Michael Zieve (Institute for Advanced Study)
Polynomial mappings
Abstract: I will present properties of polynomials mappings and generalizations. I will first describe all polynomials f and g for which there is a complex number c such that the orbits &ob;c, f(c), f(f(c)), ...&cb; and &ob;c, g(c), g(g(c)), ...&cb; have infinite intersection. I will also discuss a common generalization of this result and Mordell's conjecture (Faltings' theorem). After this I will move to polynomial mappings over finite fields, with connections to curves having large automorphism groups and instances of a positive characteristic analogue of Riemann's existence theorem.

Please join us for refreshments at 3:30 p.m. in Room 321 Altgeld Hall.

Friday, January 23, 2009

Mathematics Colloquium - Special Lecture 2008-09
4:00 pm in 245 Altgeld Hall, Friday, January 23, 2009

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Frank Vallentin (CWI Amsterdam)
Optimization in geometry: Kissing and coloring
Abstract: The theta function, introduced by Lovász, is an invariant of a finite graph defined as the optimal solution of a semidefinite program. It gives an upper bound of the size of a maximum independent set and a lower bound for the chromatic number of the graph.

In the talk I discuss two possible generalizations of the Lovász theta function for infinite graphs. In particular these generalizations allow to give new upper bounds for the kissing number and to give new lower bounds for the measurable chromatic number in several dimensions.

Both problems have a rich history and have been studied extensively by geometers and combinatorialists. The kissing number is the maximum number of non- overlapping unit balls that can simultaneously touch a central unit ball. The measurable chromatic number is the minimum number of colors one needs to color all points in n-dimensional Euclidean space so that any two points at distance 1 receive different colors and so that points receiving the same color form Lebesgue measurable sets.

(Based on joint work with C. Bachoc, G. Nebe, F.M. de Oliveira Filho.)

Please join us for refreshments at 3:30 p.m. in Room 321 Altgeld Hall.

Monday, January 26, 2009

Mathematics Colloquium - Special Lecture 2008-09
4:00 pm in 245 Altgeld Hall, Monday, January 26, 2009

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Tobias Kaiser (University of Regensburg)
The first order content of the Riemann Mapping Theorem
Abstract: One of the main topics in logic consists in describing the structure of all sets which can be obtained from a given family of sets or functions by applying first order operations (taking intersections, complements, cartesian products, projections, etc.) For example, all sets which are definable from the usual addition, multiplication and order on the reals are by Tarski's principle precisely the semialgebraic sets. These have strong finiteness properties and show therefore 'tame' geometric behaviour. But the semialgebraic structure is too small for many purposes from analysis (e.g. integration, solving ordinary differential equations, etc.). Expanding the semialgebraic structure by a family of sets or functions obtained by a certain procedure from analysis, the following question arises: does the 'tame' geometric behaviour of definable sets persist or do 'chaotic' phenomena occur?

O-minimal structures expand the semialgebraic structure and are axiomatically defined by a finiteness property. This finiteness property implies that the geometry of the definable sets is 'tame'. Many concepts from analysis can be captured in o-minimal structures. For example first order differential equations with semialgebraic raw data can be solved in an o-minimal expansion. We are interested in complex analysis, especially in one of its most important theorems, the Riemann Mapping Theorem: a simply connected proper domain in the complex plane can be mapped biholomorphically onto the unit ball. We show the following: for a large class of simply connected and proper domains in the complex plane which are semialgebraic, the biholomorphic map onto the unit ball is definable in an o-minimal structure. Related to the Riemann Mapping Theorem is the Dirichlet problem in the plane. We also show the o-minimal content of the Dirichlet problem, which allows us to solve an important partial differential equation in the o-minimal context.

Please join us for refreshments at 3:30 p.m. in Room 321 Altgeld Hall.

Tuesday, January 27, 2009

Mathematics Colloquium - Special Lecture 2008-09
4:00 pm in 245 Altgeld Hall, Tuesday, January 27, 2009

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Mirela Ciperiani (Columbia University )
Genus one curves over the rationals
Abstract: Let C be a genus one curve defined over Q. Assume that C has a point defined over each completion of Q. The local to global principle suggests that C should have a point defined over Q. In general, this is not the case. The motivating question is: What can be said about such curves C? A first approach is to analyze the fields of definition of the points of C. In joint work with A. Wiles we have shown that C has a point defined over some solvable extension of Q. Another approach is to see how many such curves C there are with a fixed Jacobian. By viewing these curves as cohomology classes we will describe the current understanding of the answer to this question from the point of view of Iwasawa theory.

Friday, January 30, 2009

Mathematics Colloquium - Special Lecture 2008-09
4:00 pm in 245 Altgeld Hall, Friday, January 30, 2009

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Vera Mikyoung Hur (MIT)
Dispersive properties of surface water waves
Abstract: I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a precise account of the formulation of the surface water-wave problem and discussion of its distinct features. They include the dispersion relation, its severe nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some open questions regarding long-time behavior and stability.

Thursday, February 19, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, February 19, 2009

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Allen Knutson (Cornell University)
Why do matrices commute? Algebraic geometry meets statistical mechanics
Abstract: The matrix equations M^2 = 0 are quadratic, so to derive the linear equation Trace(M)=0 from them requires nonalgebraic operations. Are there corresponding "surprising" equations implied by the matrix equation XY=YX? This question was posed in the '60s, and still nobody knows. Even the (normalized) volume of this space &ob;(X,Y) : XY=YX&cb; is very difficult to compute for large matrices, and until recently was only known to start 1,3,31,1145.

I'll talk about a bunch of related spaces of matrices, some of which are provably harder and some easier to understand than the commuting scheme &ob;(X,Y) : XY=YX&cb;, and the volumes of these spaces. Then I'll explain how physicists came up with the same set of numbers from a statistical mechanical model (making them much easier to compute), and why they are indeed the same.

Some of this work is joint with Paul Zinn-Justin.

Thursday, February 26, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, February 26, 2009

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Robin Thomas (School of Mathematics, Georgia Institute of Technology)
K_t minors in large t-connected graphs
Abstract: A graph G has a K_t minor if a graph isomorphic to K_t, the complete graph on t vertices, can be obtained from a subgraph of G by contracting edges. A deep theorem of Robertson and Seymour describes the structure of graphs with no K_t minor. The theorem is very powerful, but it is fairly complicated to state, and the condition it gives is necessary, but not sufficient, for the exclusion of a K_t minor.

We prove a necessary and sufficient condition under additional restrictions on the graph G. We prove that for every integer t there exists an integer N such that every t-connected graph on at least N vertices with no K_t minor has a set of at most t-5 vertices whose deletion makes the graph planar. This is best possible in the sense that neither t-connectivity nor the size of the deleted set can be lowered, and for t>7 some lower bound on the number of vertices is needed.

This is joint work with Sergey Norin.

Thursday, March 5, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, March 5, 2009

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K. Soundararajan (Stanford)
Quantum Unique Ergodicity and Number Theory
Abstract: A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic manifolds arising as a quotient of the upper half-plane by a discrete "arithmetic" subgroup of SL_2(R) (for example, SL_2(Z), and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equi-distributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms.

Thursday, April 2, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, April 2, 2009

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Ilpo Laine (University of Joensuu, Finland)
Tropical Nevanlinna theory and ultradiscrete equations
Abstract: In this talk (mostly based on joint work with C.C. Yang) we shall consider a max-plus semi-ring structure in the real line, -∞ included, endowed with tropical addition, equivalent to the usual maximum operation, and tropical multiplication, equivalent to the usual addition. In this framework, tropical Nevanlinna theory describes value distribution of continuous piecewise linear functions (called tropical meromorphic functions) of a real variable, whose one-sided derivatives are integers at every point. We also briefly describe what might be called tropical algebroid functions. In the main part of the talk, we describe tropical counterparts of certain classical results from the Nevanlinna theory such as the lemma of the logarithmic derivative, the Clunie lemma and the Valiron-Mohon'ko lemma. In the last part of the talk, we shall consider piecewise linear solutions of certain ultradiscrete equations, as an application of the preceding theory.

Tuesday, April 14, 2009

Mathematics Colloquium - Special Lecture 2008-09
4:00 pm in 245 Altgeld Hall, Tuesday, April 14, 2009

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Vigleik Angeltveit (University of Chicago)
Uniqueness of Morava K-theory
Abstract: Any cohomology theory is represented by what is called a spectrum. For each prime p and positive integer n there is a cohomology theory, or spectrum, called the n'th Morava K-theory K(n). When n=1 this is essentially mod p complex K-theory. I will discuss multiplicative structure on K(n). In particular I will discuss how many associative multiplications K(n) has. The main result is that there is essentially only one. To analyze this question I will discuss the notion of an A-infinity structure, a multiplication which is associative up to homotopy and higher homotopies in a precise sense. Please join us for cookies and coffee at 3:30 p.m. in 321 Altgeld Hall. Faculty host: Randy McCarthy.

Wednesday, April 15, 2009

Thursday, April 16, 2009

Monday, April 20, 2009

Mathematics Colloquium - Special Lecture 2008-09
4:00 pm in 245 Altgeld Hall, Monday, April 20, 2009

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Karl Schwede (University of Michigan)
Singularities in characteristic zero and singularities in characteristic p
Abstract: For more than 30 years, algebraic geometers and commutative algebraists have known about subtle links between singularities of algebraic varieties defined by a "resolution of singularities" in characteristic zero, and singularities defined by the "action of Frobenius" in characteristic p > 0. I will discuss: 1) what singularities of algebraic varieties are, 2) the history of both of these classes of singularities, 3) the precise dictionary that now links these two classes, and 4) some of the ways in which this dictionary has been used to inspire new questions and prove new theorems.
Please join us for cookies and coffee at 3:30 p.m. in 321 Altgeld Hall. Faculty host: Thomas Nevins

Thursday, April 23, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, April 23, 2009

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Mikael Rordam (University of Copenhagen)
C*-algebras and their invariants
Abstract: C*-algebras have successfully been studied via their invariants. The invariants of a C*-algebra includes first of all K-theory (reflecting the fact that C*-algebras are non-commutative generalizations of topological spaces). With some additional information (order on K-theory and including the trace simplex) one obtains the so-called Elliott invarariant of the C*-algebra, which rather surprisingly, has turned out to be a complete invariant for an extensive class of (simple, separable, nuclear) C*-algebras. Although extensive we know today that the class of thus classifiable C*-algebras is not exhaustive; there are (simple, separable, nuclear) C*-algebras that seem to defy classification. One can detect some of the excotic behaviour of these "unclassifiable" C*-algebras through some finer invariants, such as the semigroup of Murray-von Neumann equivalence classes of projections of the C*-algebra (the semigroup from which K₀ is defined) and the Cuntz semigroup.

We shall give a brief overview of the classification of C*-algebras. Then we shall describe the interplay between these C*-algebras and their invariant, in particular the finer ones mentioned above, and how the study of the invariants may be the key to a better understanding of the C*-algebras themselves. We shall focus on some very recent results on comparability of ordered abelian semigroups. Host: Marius Junge

Thursday, April 30, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, April 30, 2009

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Philippe Di Francesco (Saclay/University of Illinois)
TBA
Abstract: What is the link between Alternating Sign Matrices (ASM), Totally Symmetric Self-Complementary Plane Partitions (TSSCPP) and the equivariant cohomology of the variety of strictly upper triangular matrices with vanishing square? Two-dimensional Integrable lattice models from statistical physics provide the natural framework for this missing link. We show in particular how a physical model, involving densely-packed loop configurations on an infinite surface, is connected to all three subjects above. This model turns out to be integrable, and we�ll use this fact to reformulate all of the above in terms of polynomial solutions of the quantum Knizhnik-Zamolodchikov equation. Results include a proof of the Razumov-Stroganov sum rule, a new connection between ASM and TSSCPP, and the computation of the (multi)degree of the variety M² = 0. We also present generalizations to the commuting variety and to M^k = 0.

Thursday, September 10, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, September 10, 2009

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Michael Hill (University of Virginia)
On the Non-Existence of Kervaire Invariant One Manifolds
Abstract: The existence of smooth, Kervaire invariant one manifolds has been an open problem in algebraic topology for well over fifty years. These manifolds were first encountered by Pontryagin in the 1930s, and Kervaire and Milnor tied the existence of such manifolds to an ambiguity in enumerating the number of smooth structures on spheres. In this talk, I will discuss recent work with Hopkins and Ravenel in which we show that in dimensions greater than 126, there are no Kervaire invariant one manifolds, thereby reducing the problem to a single dimension to check.

Monday, September 14, 2009

ECE - Special Lecture
4:00 pm in B02 CSL, Monday, September 14, 2009

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Yuliy Baryshnikov (Bell Laboratories)
Caging and linking
Abstract: For a planar domain, a finite point configuration is said to be caging, if the set of Euclidean motions of the domain not hitting the point configuration is bounded. Caging configurations are important in robotics and are a popular topic in computational geometry. We argue that caging is best understood in topological terms, manifesting itself as nontrivial linking of certain closed curves (in the group of motions of Euclidean plane), which can be translated into effectively computable invariants of pairs of framed planar curves. [No knowledge of topology will be assumed.]

Dr. Baryshnikov will also present a Mathematics Colloquium Special Lecture Tuesday, Sept 15, at 4 p.m. in 245 Altgeld Hall.

Tuesday, September 15, 2009

Mathematics Colloquium - Special Lecture 2009-10
4:00 pm in 245 Altgeld Hall, Tuesday, September 15, 2009

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Yuliy Baryshnikov (Bell Laboratories)
Algebraic curves and frozen regions in planar tilings
Abstract: There are many examples of "Arctic circle" type phenomena in random tilings of planar regions such as domino tilings of Aztek diamonds or diabolo tilings of fortresses: regions with high correlations between tiling orientations are separated from those with nontrivial entropy (per area) by algebraic curves. Our (with Robin Pemantle) recent results on asymptotics of multivariate generating function allow to derive the shapes of these boundaries easily as the duals to certain curves given by hyperbolic polynomials.

Dr. Baryshnikov will also present an ECE Special Lecture Monday, Sept 14, at 4 p.m. in B02 CSL.

Thursday, September 24, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, September 24, 2009

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Gregory F. Lawler (University of Chicago)
Random Laplacian Motion
Abstract: I will discuss random curves whose evolution involves the solution of the Laplace equation with boundary values on the given curve. (This is an example of a "moving boundary" problem.) At the discrete level it is easy to define such models but it is hard to analyze them. At the continuum level, it is not obvious (or even known in some cases) how to give a mathematically precise definition of the curve. The interesting cases give irregular curves of nontrivial fractal dimension. I will survey some of the area discussing:

Discrete case: Laplacian random walks and the loop-erased walk
The role of dimension and why dimensions four and greater are not so interesting for this problem.
Continuous case in two dimensions: We know much more than we did ten years ago because of the invention of the Schramm-Loewner evolution (SLE) by the late Oded Schramm. Wendelin Werner was awarded a Fields Medal in 2006 primarily for work on SLE and related problems. In two dimensions, conformal invariance plays a very strong role.
The very open case of three dimensions.
Speculations about the self-avoiding walk and its limits.

This talk is intended for a general mathematical audience.

Thursday, October 15, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, October 15, 2009

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David Nicholls (University of Illinois at Chicago)
Boundary Perturbation Methods for Electromagnetic Scattering
Abstract: The scattering of electromagnetic waves by irregular obstacles arises in a wide array of applications including remote sensing, nondestructive testing, and imaging. In this talk we will discuss a class of highly accurate numerical methods for the simulation of rough-surface scattering based upon the classical approach of Lord Rayleigh from the early 1900's. While these methods feature rapid execution times and exponentially fast convergence, subtle cancellations in the underlying recursions render them unreliable for very challenging problems (e.g., non-smooth surfaces). We will show how a simple change of variables produces not only a robust and highly accurate numerical procedure applicable to the most difficult configurations, but also delivers a proof of the algorithm's convergence. Furthermore, we will illustrate how these ideas can be adapted to the multi-scale problem of high-frequency scattering where the wavelength of the electromagnetic radiation is many orders of magnitude smaller than the features of the scattering obstacle.

Tuesday, November 3, 2009

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Tuesday, November 3, 2009

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Oscar Garcia-Prada (CSIC)
Geometry of surface group representations
Abstract: Given a compact real surface S and a semisimple Lie group G, we consider the moduli space R(S,G) of representations of the fundamental group of S in G (sometimes called the character variety). This moduli space plays a central role in many problems in geometry, topology and physics. By considering a complex structure on the surface S (thus making it a Riemann surface), the moduli space of representations is in bijection with a moduli space of holomorphic objects, known as Higgs bundles. We explain this correspondence and show how to use it to study the topology of R(S,G). We give special attention to the case where G is the isometry group of a non-compact Hermitian symmetric space. In this situation, the moduli space has special components that can be regarded in some sense as generalizations of the Teichmueller space of S (which can be identified with a component of the character variety when G=PSL(2,R)).