Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, October 22, 2019.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 22, 2019

1:00 pm in Altgeld Hall,Tuesday, October 22, 2019

Orbit equivalence relations of some classes of non-locally compact Polish groups

Joseph Zielinski

Abstract: By results of A.S. Kechris, whenever a locally compact Polish group acts continuously on a Polish space, the orbit equivalence relation of the action is essentially countable—that is, Borel reducible to the orbit equivalence relation of an action of a countable group. It is unknown if this characterizes the locally compact Polish groups. S. Solecki, after proving an analogous characterization for smooth actions of compact Polish groups, showed this to be true in the case where the group, G, is the additive group of a separable Banach space. The characterization also holds for abelian pro-countable groups, by results of M. Malicki. We discuss recent work on this problem, including an extension of this characterization to some important classes of Polish groups. This is joint work with A.S. Kechris, M. Malicki, and A. Panagiotopoulos.

1:00 pm in 347 Altgeld Hall,Tuesday, October 22, 2019

The Dirichlet problem for elliptic operators having a BMO antisymmetric part

Linhan Li (UMN Math)

Abstract: In this talk, we are going to introduce our result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO antisymmetric part. In particular, the coefficients of the operator are not necessarily bounded. Our method relies on kernel estimates and off-diagonal estimates for the semigourp e^{-tL}, solution to the Kato problem, and various estimates for the Hardy norms of certain commutators. This is a joint work with S. Hofmann, S. Mayboroda, and J. Pipher.

2:00 pm in 243 Altgeld Hall,Tuesday, October 22, 2019

River landscapes and Optimal Channel Networks

József Balogh (Illinois Math)

Abstract: We study tree structures termed Optimal Channel Networks (OCNs) that minimize the total gravitational energy loss in the system, an exact property of steady-state landscape configurations that prove dynamically accessible and strikingly similar to natural forms. Here, we show that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent. We also study the natural river trees in an arbitrary graph in terms of forbidden sub-structures.

The talk is based on: P. Balister, J. Balogh, E. Bertuzzo, B. Bollobas, G. Caldarelli, A. Maritan, R. Mastrandrea, R. Morris, A Rinaldo: River landscapes and Optimal Channel Networks, Proceeding of the National Academy of Sciences U.S.A., 115, 6548--6553 (2018).

This paper is based on a true collaboration among mathematicians, theoretical computer scientists and physicists.

3:00 pm in 243 Altgeld Hall,Tuesday, October 22, 2019

Structure of local cohomology modules associated with projective varieties

Wenliang Zhang (UIC)

Abstract: Let R be a polynomial ring over a field and I be an ideal of R. The local cohomology modules H^j_I(R) are rarely finitely generated as R-modules. However, they have finite length when viewed as objects in the category of D-modules in characteristic 0 (or in the category of F-modules in characteristic p). Computing the actual length (in the appropriate category) has been an open problem; it is also an open problem just to determine whether they are simple objects (in the appropriate category). In this talk, I will explain a solution to this problem when the ideal I is a homogeneous prime ideal and the projective variety Proj(R/I) has mild singularities in characteristic p. This is a joint work with Nicholas Switala.