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Seminar Calendar
for events the day of Thursday, February 24, 2005.

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Thursday, February 24, 2005

12:00 pm in 464 Loomis,Thursday, February 24, 2005

Higgs bundles and Calabi-Yau threefolds

Tony Pantev (U Penn, Math)

1:00 pm in Altgeld Hall 347,Thursday, February 24, 2005

The intersection form and geodesic currents on free groups (continued)

Ilya Kapovich   [email] (UIUC)

Abstract: The notion of a geometric intersection number between free homotopy classes of closed curves on surfaces plays a pivital role in Thurston's treatment of the Teichmuller space and of the dynamics of surface homeomorphisms. In particular, Bonahon proved that this notion extends to a symmetric and bilinear notion of intersection number between two geodesic currents on a hyperbolic surface. We investigate to what extend these ideas are applicable in the free group context. Thus we define and study an Out(F_n)-equivariant "intersection form" on the product of the (non-projectivized) Culler-Vogtmann outer space and the space of geodesic currents on a free group. We also find an obstruction, arising from non-symmetric behaviour of generic stretching factors of free group automorphisms, to the existence of a symmetric notion of an intersection number between two geodesic currents on a free group. Time permitting, we will discuss an embedding of the Culler-Vogtmann outer space into the projectivized space of geodesic currents, based on the use of Patterson-Sullivan conformal densities.

1:00 pm in 241 Altgeld Hall,Thursday, February 24, 2005

Digital expansions, exponential sums and central limit theorems.

Michael Drmota (Technical University, Wien)

Abstract: The purpose of this talk is to present recent results on the distribution of the values of the ($q$-ary) sum of digits function $s_q(n)$, where we mainly focus on two problems, the joint distribution $(s_{q_1}(n),s_{q_2}(n))$ of two different expansions and on the distribution of the sum-of-digits function of squares: $s_q(n^2)$.
Interestingly there are strong relations to "classical analytic number theory". For example, one can use Baker's theorem on linear forms (and exponential sums) to prove that the pairs $(s_{q_1}(n),s_{q_2}(n))$, $n

2:00 pm in 241 Altgeld Hall,Thursday, February 24, 2005

Henn's calculation of the cohomology of S_2 at p=3, IV

Charles Rezk (UIUC Math)

2:00 pm in Altgeld Hall 243,Thursday, February 24, 2005

A limit theorem for the Riesz transform of singular measures

Prabu Janakiraman (UIUC)

Abstract: Let R be the vector Riesz transform on \bR^n and \nu a singular measure supported on a k dimensional Lipschitz surface. The precise asymptotic behavior of the weak-type (1,1) inequality \lambda m\{x\in\bR^n: |R\nu(x)|>\lambda\} as \lambda \rightarrow \infty will be stated and proved.

2:00 pm in 345 Altgeld Hall,Thursday, February 24, 2005

Anti-self-dual connections from a symplectic viewpoint

Katrin Wehrheim (Institute of Advanced Study)

Abstract: The space of connections over a Riemann surface is symplectic, and the gauge action of bundle isomorphisms can be viewed as Hamiltonian group action. The corresponding symplectic vortex equation is the anti-self-duality equation for connections over the product of two surfaces. So it is natural to consider anti-self-dual connections with Lagrangian boundary conditions. I will explain this framework and show that the boundary behaviour of these connections is governed by a holomorphic part of the equation. In a joint project with Dietmar Salamon we use these connections to define a new Floer theoretic invariant for 3-manifolds with boundary. This is an intermediate object between the instanton Floer homology of a homology sphere and a symplectic Floer homology arising from a Heegard splitting. It might thus be used to prove a conjecture of Atiyah and Floer relating these.

3:00 pm in 243 Altgeld Hall,Thursday, February 24, 2005

Tight Closure and Its Applications (III)

Jinjia Li   [email] (UIUC Math)

Abstract: We will discuss the notion of integral closure of an ideal and the tight closure proof of the Briancon-Skoda Theorem in characteristic p.

3:00 pm in 345 Altgeld Hall,Thursday, February 24, 2005

Weak Convergence of Refelcted Stable Processes

Panki Kim   [email] (UIUC Math)

Abstract: A possible way of studying properties of some stochastic process in a open set $D$ with ``rough'' boundary is first to consider a sequence of stochastic processes in smooth open sets $D_k$ increasing to $D$. Then one could consider properties of that stochastic process in $D$ through the weak limit. In this talk, we will discuss weak convergence of reflected stable processes, which were introduced recently by Bogdan, Burdzy and Chen. One of the powerful tools in studying weak convergence of Markov processes is Mosco convergence. Some generalization of classical approximation theory (Trotter-Kato) and Mosco convergence will be introduced.