UIUC Dept. of Mathematics Seminar Calendar

     August 2012           September 2012          October 2012    
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                        30

Tuesday, September 18, 2012

Topology Seminar
11:00 am in 243 Altgeld Hall, Tuesday, September 18, 2012

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Submitted by franklan.

Sean Tilson (Wayne State University)
Power operations and the Kunneth Spectral Sequence
Abstract: Power operations have been constructed and successfully utilized in the Adams and Homological Homotopy Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. It was thought that such results were not specific to the spectral sequence, but rather that they arose because highly structured ring spectra are involved. In this talk, we show that while the Kunneth Spectral Sequence enjoys some nice multiplicative properties, the obvious algebraic operations are zero (other than the square). Despite the negative results we are able to use old computations of Steinberger's with our current work to compute operations in the homotopy of some relative smash products.

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, September 18, 2012

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Submitted by phierony.

Mia Minnes (UC San Diego)
Algorithmic Randomness via Random Algorithms
Abstract: Algorithmic randomness defines what it means for a single mathematical object to be random. This active area of computability theory has been particularly fruitful in the past several decades, both in terms of expanding theory and increasing interaction with other areas of math and computer science. Randomness can be equivalently understood in terms of measure theory, descriptive complexity, and martingales. In this context, we present a novel definition of betting strategies that uses probabilistic algorithms also studied in complexity theory. This definition leads to new characterizations of several central notions in algorithmic randomness and addresses Schnorr's critique, a longstanding philosophical question in algorithmic randomness. Moreover, these techniques suggest new approaches for tackling one of the biggest open questions in the field (KL = ML?). This is joint work with Sam Buss.

Probability Seminar
2:00 pm in Altgeld Hall 347, Tuesday, September 18, 2012

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Todd Kemp (UCSD Math)
Liberation of Random Projections
Abstract: Given two subspaces of a finite dimensional space, there is a minimal dimension their intersection can have; when this dimension is minimized the subspaces are said to be in general position. Easy 19th Century mathematics shows that any two subspaces are ``almost surely'' in general position in many senses. One modern precise meaning we can give to this statement is as follows: perform a Brownian motion on the unitary group (of rotations), applied to one of the subspaces. Then for any fixed positive time, the Brownian rotated subspaces are almost surely in general position, regardless of starting configuration. What happens if the ambient space is an infinite dimensional Hilbert? While there is no unitarily invariant measure, it is still possible to make sense of the unitary Brownian motion and its action on some (but not all) subspaces. However, the easy techniques for proving the general position theorem are unavailable. Instead, one can apply stochastic analysis and free probability techniques to to analyze a spectral measure associated to the problem. In this talk, I will discuss probabilistic and PDE techniques that come into play in proving the general position theorem in infinite dimensions. This is joint work with Benoit Collins.

Algebra, Geometry and Combinatorics
2:00 pm in 345 Altgeld Hall, Tuesday, September 18, 2012

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Submitted by bwyser.

Ben Wyser [email] (UIUC Math)
Geometry and Combinatorics of K-Orbits on the Flag Manifold
Abstract: The orbits of a symmetric subgroup on a flag manifold ("K-orbits") are of importance in the representation theory of real Lie groups, and have been studied extensively from this perspective. The closures of such orbits are generalizations of Schubert varieties, and any geometric and/or combinatorial question one has about Schubert varieties can equally well be posed about these more general orbit closures. However, while the geometry and combinatorics of Schubert varieties have been studied exhaustively, even apart from their role in representation theory, K-orbits and their closures have received far less attention from these perspectives. I will discuss the K-orbit analogue of a story which is well understood in the case of Schubert varieties. Namely, I will describe how one can compute representatives for the torus-equivariant cohomology classes of K-orbit closures, and how these formulas can be interpreted as Chern class formulas for certain types of degeneracy loci. This is in parallel with the well-known story, due to Lascoux-Schutzenberger, Fulton, Pragacz, Graham, et. al., of double Schubert polynomials as representatives for the torus-equivariant classes of Schubert varieties, and their interpretation as Chern class formulas for the classes of degeneracy loci associated to flagged vector bundles.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, September 18, 2012

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Submitted by lidicky.

Matthew Yancey [email] (UIUC Math)
Color-Critical Graphs With Few Edges
Abstract: A graph $G$ is $k$-critical if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$-colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. We give a bound on $f_k(n)$ that is exact for every $n=1\,({\rm mod }\, k-1)$. It is also exact for $k=4$ and every $n\geq 6$. The result improves the classical bounds by Gallai and Dirac and subsequent bounds by Krivelevich and Kostochka and Stiebitz. It establishes the asymptotics of $f_k(n)$ for every fixed $k$. We also present some applications of the result, in particular, a simple proof of the Grotzsch Theorem that every triangle-free planar graph is $3$-colorable. This is joint work with Alexandr Kostochka.