UIUC Dept. of Mathematics Seminar Calendar

    December 2005           January 2006          February 2006    
 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
  4  5  6  7  8  9 10    8  9 10 11 12 13 14    5  6  7  8  9 10 11
 11 12 13 14 15 16 17   15 16 17 18 19 20 21   12 13 14 15 16 17 18
 18 19 20 21 22 23 24   22 23 24 25 26 27 28   19 20 21 22 23 24 25
 25 26 27 28 29 30 31   29 30 31               26 27 28

Thursday, January 26, 2006

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

Number Theory Seminar

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.

Submitted by ahlgren

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

Differential Geometry Seminar

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.

Submitted by ekerman

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

Mathematics Colloquium - Special Lecture 2005-06

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

Submitted by seminar

Mathematics

Small Gaps Between Products of Two Primes

Sid Graham (Central Michigan)

Some Category Theory

Shivi Bansal (UIUC Math)

Iwasawa Theory

Tim Kilbourn (UIUC Math)

Floer homologies for fibered 3-manifolds

Michael Usher (Princeton)

Calculus of the embedding functor and spaces of knots

Ismar Volic (University of Virginia)

Fermat's Last Tango showing and Pi Mu Epsilon induction

Robert Smith [email] (Miami University of Ohio)