Applies functional analysis to physical problems; scattering theory, statistical mechanics, and quantum field theory.

Seminars

Harmonic Analysis and PDE learning seminar

Kerchof Hall 326

Mark Lewers will talk about the paper "Lp theory for outer measures and two themes of Carleson United" by Y.Do and C.Thiele

" class="addtocalendar" target="_new">Add to Google CalendarBoyu Zhang (Harvard) - A monopole invariant for foliations without transverse measure

The question about existence and uniqueness (up to deformation) of taut foliations on a three manifold, in a given homotopy class of distributions, has been studied for decades. Obstructions of existence of taut foliation on rational homology spheres have been obtained by considering the perturbations of the foliation to contact structures. Recently, by showing that the perturbed contact structure is unique in many cases, Vogel and Bowden constructed examples of taut foliations that are homotopic as distributions but can not be deformed to each other through taut foliations. In this talk I will introduce a new approach to this problem. Instead of perturbing the foliation to a contact structure, we directly study a symplectization of the foliation, and that leads to a canonically defined class in the monopole Floer homology. Then I will demonstrate how to apply this idea to the questions of existence and uniqueness of taut foliations.

Add to Google CalendarYen Do (UVA) - Variational estimates for the bilinear iterated Fourier integral

317 Kerchof Hall, Charlottesville, VA, United States

Abstract: We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Joint work with C. Muscalu and C. Thiele.

Add to Google CalendarMath Club - David Sherman (UVA) - Variations on Kuratowski’s 14-set theorem

Kuratowski’s 14-set theorem from 1920 says that in a topological space, the number of distinct sets that can be generated from a fixed set by taking closures and complements (in any order) is 14. I’ll present this — the proof is surprisingly simple — and some variations that I published in the American Mathematical Monthly a few years ago. Although the topic is “point-set topology,” the methods are algebraic, with cameos by logicians and an obsessed trucker. I will start at the beginning, explaining basic topological notions for the real line.

Add to Google CalendarBob Oliver (U Paris 13) Local structure of finite groups and of their p-completed classifying spaces

I will discuss the close connection between the homotopy theoretic properties of the p-completed classifying space BG_p of a finite group G and the p-local group theoretic properties of G. One way in which this arises is in the following theorem originally conjectured by Martino and Priddy: for finite groups G and H, BG_p is homotopy equivalent to BH^p if and only if G and H have the same p-local structure (the same conjugacy relations among p-subgroups). Another involves a description, in terms of the p-local properties of G, of the group Out(BG_p) of homotopy classes of self equivalences of the space BG_p.

After describing the general results, I’ll give some examples and applications of both of these, especially in the case where G and H are simple Lie groups over finite fields.

Add to Google CalendarDaniel Halpern-Leistner (Columbia) - Magic windows and representations of generalized braid groups on the derived category of a GIT quotient

Abstract: One consequence of the homological mirror symmetry conjecture predicts that many varieties will have ``hidden symmetries" in the form of autoequivalences of their derived categories of coherent sheaves which do not correspond to any automorphism of the underlying variety. In fact the fundamental groupoid of a certain "complexified Kaehler moduli space" conjecturally acts on the derived category. When the space in question is the cotangent bundle of a flag variety, actions of this kind have been studied intensely in the context of geometric representation theory and Kahzdan-Lusztig theory. We establish the conjectured group action on the derived category of any variety or orbifold which arises as a symplectic or hyperkaehler reduction of a linear representation of a compact Lie group. Our methods are quite explicit and essentially combinatorial -- leading to explicit generators for the derived category of certain GIT quotients and an explicit description of the complexified Kaehler moduli space. The method generalizes the work of Donovan, Segal, Hori, Herbst, and Page which studies grade restriction rules in specific examples associated to ``magic windows."

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MATH 8450

Applies functional analysis to physical problems; scattering theory, statistical mechanics, and quantum field theory.