Events for the week of November 02 2009

Abstract. Motivated by a result of Bertolini?Darmon on Stark?Heegner points, we study the special value of the L-function of a rational elliptic curve, base-changed to a real quadratic field. The conjecture of Birch and Swinnerton-Dyer predicts that such value, appropriately normalized, is a perfect square. If time permits, we will reformulate this prediction in terms of a conjecture about Petersson inner products of automorphic forms on definite quaternion algebras.

Abstract. The hierarchical ($BV, L^2$) multiscale image representation of Tadmor, Nezzar and Vese, [2004], gives rise to a novelintegro-differential equation (IDE) for a multiscale image representation.To this end, one integrates in inverse scale space a succession ofrefined, recursive `slices' of the image, which are balanced by a typicalcurvature term at the finer scale. The importance of the IDE lies in thefact that even though its motivation comes from a variational problem, wecan manipulate the IDE suitable to our image processing needs. We proposedifferent forms of the IDE with filtering, tangential smoothing, anddeblurring. I will also present (time permitting) an IDE based on Chan, Esedoglu's (BV, L^1) image decomposition with some new results.

Abstract. Damien Gaboriau and Sorin Popa showed that the free group on two generators admits continuum many measure preserving, ergodic, orbit inequivalent actions on standard Borel probability spaces. I will sketch a proof which does not use property (T) -- and discuss some questions this argument raises.

Abstract. We'll introduce a family of rings whose Grothendieck groups can be naturally identified with weight spaces of the positive half of the universal enveloping algebra of a simple Lie algebra. Induction and restriction functors descend to multiplication and comultiplication in the universal enveloping algebra.

Abstract. We discuss how the machinery behind Miller's recent (classical) proof of the G_0 dichotomy can be modified to grant Komjath-Laczkovich-style limsup dichotomies.  In particular, this gives a classical proof of Gao-Jackson-Kieftenbeld's "sequential" generalization of Silver's theorem concerning the number of equivalence classes of a coanalytic equivalence relation.  Time permitting, we will also discuss obstacles preventing a similar "sequential" generalization of the G_0 dichotomy itself.

Abstract. This talk will bring ideas from number theory, e.g., the Prime Number Theorem, Bertrand~Rs Postulate, and Cebotar\"ev~Rs Density Theorem, to study properties of infinite groups. In particular, we will introduce the notion of quantifying the extent to which a finitely generated group is residually finite. This asymptotic study connects word and subgroup growth via a function called the normal divisibility function that measures the size of the smallest finite quotient that a fixed group element maps to nontrivially. In this talk we will investigate such behavior for examples that include free groups and $SL(n, \mathcal{O}_K)$, where $K$ is a number field. This talk uses ideas and methods from combinatorial group theory, number theory, the theory of profinite groups, and topology. Part of this talk covers joint work with B. McReynolds.

Abstract. The most common version of Hilbert's Fifth Problem (H5) asks whether every locally euclidean topological group (i.e. a topological group whose identity has an open neighborhood homeomorphic to some R^n) can be given the structure of a real analytic manifold so that the group multiplication and inversion become real analytic maps; briefly, whether every locally euclidean topological group is a Lie group. A positive solution to the H5 was given by Gleason, Montgomery, and Zippin in the 1950s. Shortly after, Jacoby claimed to have proven that every locally euclidean "local group" was locally isomorphic to a Lie group. Roughly speaking, a local group is a hausdorff space which has a partial group-like operation defined on it which is continuous. However, Jacoby's proof was discovered to be flawed in the '90s, leaving some important theorems whose truth rests on this local version of the H5 on shaky foundation. By modifying techniques used by Hirschfeld in a proof of the H5 using nonstandard analysis, I was able to give a correct proof of the local H5. In this talk, I will try and sketch a part of the proof of the local H5. No background knowledge in Lie theory or nonstandard analysis will be necessary and all relevant terms will be defined.