Seminar Calendar
for events the day of Tuesday, January 24, 2012.

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Tuesday, January 24, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, January 24, 2012
 Del Edit Copy
Submitted by franklan.
 Michael Mandell (Indiana University)Localization sequences in THHAbstract: (Joint work with Andrew Blumberg, preprint arXiv:1111.4003.) For a discrete valuation ring R with quotient field F and residue field k, you have a cofibration sequence of K-theory spectra K(k) → K(R) → K(F). The corresponding sequence in THH is not a cofibration sequence, but both the cofiber of the map THH(k)→ THH(R) and the fiber of the map THH(R) → THH(F) have an interpretation in terms of the THH of Waldhausen categories. Thinking in terms of Waldhausen categories, we therefore get two cofibration sequences for THH, THH(k) → THH(R) → THH(F|R) (first constructed by Hesselholt and Madsen) and THH( Spec(R) on Spec(k) ) → THH(R) → THH(F) (generalizing to THH a well-known exact sequence in Hochschild homology). The first arises by looking at enrichments by connective spectra and the second by looking at enrichments in non-connective spectra.

Algebra, Geometry and Combinatoric
2:00 pm   in 345 Altgeld Hall,  Tuesday, January 24, 2012
 Del Edit Copy
Submitted by darayon2.
 Dave Anderson (University of Washington)Arc spaces and equivariant cohomologyAbstract: When an algebraic group acts on a smooth complex variety X, it also acts on the arc space of X, an infinite-dimensional space parametrizing germs of curves in X. In joint work with Alan Stapledon, we develop a new perspective on the equivariant cohomology of X, by replacing X with its arc space. Under certain hypotheses, these infinite-dimensional varieties allow us to obtain a geometric basis (over the integers!) for equivariant cohomology, as well as geometric representatives for cup products as intersections. I'll explain how this leads to a new invariant of singularities, and illustrate our approach with examples from toric varieties and flag varieties.

Graph Theory and Combinatorics
3:00 pm   in 241 Altgeld Hall,  Tuesday, January 24, 2012
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Submitted by west.
 Ping Hu (UIUC Math)Upper bounds on the size of $4$- and $6$-cycle-free subgraphs of the hypercubeAbstract: Erdős proposed the problem of determining ex$_Q(n;C_{2t})$, i.e. determining the maximum number of edges that a subgraph of the $n$-dimensional hypercube containing no $2t$-cycle can have. We modify slightly Razborov's flag algebra machinery to be suitable for the hypercube. We use this modified method to show that the maximum number of edges in a subgraph of the $n$-dimensional hypercube containing no $4$-cycle is at most $0.6068$ times the number of edges in the hypercube. For subgraphs containing no $6$-cycle, we improve the upper bound on the proportion of edges from $\sqrt{2}-1$ to $0.3755$. (Joint work with Jozsef Balogh, Bernard Lidicky, and Hong Liu.)

Mathematics Colloquium - Special Lecture 2011-12
4:00 pm   in 245 Altgeld Hall,  Tuesday, January 24, 2012
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Submitted by kapovich.
 Jonathan Chaika (University of Chicago)Interval exchange transformationsAbstract: Interval exchange transformations are invertible, piecewise order preserving isometries of the unit interval with finitely many discontinuities. Starting from rotations of the circle, which they generalize, this talk will present their connections to flows on flat surfaces, rational billiards and symbolic coding. Recent results on diophantine approximation for interval exchange transformations will be presented.