Department of

Mathematics

Seminar Calendar
for events the day of Thursday, January 26, 2006.

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events for the
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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 26, 2006

1:00 pm in 241 Altgeld Hall,Thursday, January 26, 2006

Small Gaps Between Products of Two Primes

Sid Graham (Central Michigan)

Abstract: The techniques that Goldston, Pintz, and Yildirim recently used to prove the existence of short gaps between primes can be applied to other sequences. For example, one can apply these techniques to the sequence of numbers that are products of exactly two primes. Using this, we can prove that there are infinitely many integers n such that at least two of the numbers n, n+2, n+6 are products of exactly two primes. The same can be done for more general linear forms; e.g., there are infintely many n such at least two of 42n+1, 44n+1, 45n+1 are products of exactly two primes. This in turn leads to simple proofs of Heath-Brown's theorem that d(n)=d(n+1) infinitely often and of Schlage-Puchta's theorem that $\omega(n)=\omega(n+1)$ infinitely often. With other choices of linear forms, we can sharpen this to d(n)=d(n+1)=24 and $\omega(n)=\omega(n+1)=3$ infinitely often. This is joint work with D. Goldston, J. Pintz, and C. Yildirim.

2:00 pm in 443 Altgeld Hall,Thursday, January 26, 2006

Some Category Theory

Shivi Bansal (UIUC Math)

Abstract: We will start the semester gently by talking about some general ideas like triangulated categories and their truncations. The goal is to lead up to perverse sheaves and hopefully discuss some applications in topology/geometry.

2:00 pm in 241 Altgeld Hall,Thursday, January 26, 2006

Iwasawa Theory

Tim Kilbourn (UIUC Math)

Abstract: (RAP Elliptic Curves and Iwasawa Theory, Part 1) This week and next week will be a review of results from Iwasawa Theory.

3:00 pm in 241 Altgeld Hall,Thursday, January 26, 2006

Floer homologies for fibered 3-manifolds

Michael Usher (Princeton)

Abstract: In 1999, D. Salamon proved a result equating the (3-dimensional) Seiberg-Witten invariant of the mapping torus $Y_{\phi}$ of a symplectomorphism $\phi:\Sigma\to\Sigma$ of a Riemann surface with the Lefschetz number of an associated symplectomorphism $\Phi: S^d\Sigma\to S^d\Sigma$ of the symmetric product of $\Sigma$. This talk will concern a $4$-dimensional analogue of that result: the symplectic Floer homologies of Salamon's symplectomorphisms $\Phi$ fit into a TQFT for Lefschetz fibrations on $4$-manifolds in which the $4$-dimensional Seiberg-Witten invariant appears naturally. Time permitting, we will also indicate reasons to believe that these Floer homology groups interpolate between Hutchings-Thaddeus' periodic Floer homology and the monopole Floer homology of the mapping torus, providing a conjectural approach to relating these groups.

4:00 pm in 245 Altgeld Hall,Thursday, January 26, 2006

Calculus of the embedding functor and spaces of knots

Ismar Volic (University of Virginia)

Abstract: I will describe how calculus of the embedding functor can be used for the study of spaces of knots. In particular, I will give the complete description of the rational homotopy type of spaces of knots in the Euclidean space of dimension four or more in terms of a certain simple algebra of chord diagrams. For classical knots, i.e. knots in the Euclidean space of dimension three, I will indicate how embedding calculus gives a new point of view on the theory of finite type (or Vassiliev) invariants. Parts of the talk will describe joint work with Greg Arone and Pascal Lambrechts. Host: R. McCarthy

7:00 pm in 245 Altgeld Hall,Thursday, January 26, 2006

Fermat's Last Tango showing and Pi Mu Epsilon induction

Robert Smith   [email] (Miami University of Ohio)

Abstract: The UIUC chapter of the national mathematics honor society Pi Mu Epsilon is being reactivated, with remarks and an induction of new members by Robert Smith. This will be followed by a showing of the film "Fermat's Last Tango" from the Clay Math Institute and refreshments. All faculty and students are welcome.