UIUC Dept. of Mathematics Seminar Calendar

    December 2011           January 2012          February 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    1  2  3  4  5  6  7             1  2  3  4
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Tuesday, January 24, 2012

Topology Seminar
11:00 am in 243 Altgeld Hall, Tuesday, January 24, 2012

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Submitted by franklan.

Michael Mandell (Indiana University)
Localization sequences in THH
Abstract: (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

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Submitted by darayon2.

Dave Anderson (University of Washington)
Arc spaces and equivariant cohomology
Abstract: 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 hypercube
Abstract: 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.)