Department of

Mathematics


Seminar Calendar
for events the day of Thursday, March 30, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, March 30, 2017

11:00 am in 241 Altgeld Hall,Thursday, March 30, 2017

Pseudorepresentations and the Eisenstein ideal

Preston Wake (University of California at Los Angeles)

Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, but also posed a number of questions: how big is the space of cusp forms that are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Carl Wang Erickson, we give an answer to these questions using the deformation theory of Galois pseudorepresentations. The answer is intimately related to the algebraic number theoretic interactions between the primes N and p, and is given in terms of cup products (and Massey products) in Galois cohomology.

12:30 pm in 464 Loomis Laboratory,Thursday, March 30, 2017

Non-local Virasoro algebras

Gabriele La Nave (Illinois Math)

Abstract: In recent years there have been various proposals explaining physical phenomena via the use of more or less explicitly non-local actions. This is the case of the recent work Guillou-Nunez-Schaposnik on 3-dimensional Bosonization or in the (mostly phenomenological) proposal of Hartnoll and Karch on the strange metal, where they hypothesize the presence of an anomalous dimension for the vector potential. The natural question that arises is to what extent can one construct generalization of the Virasoro algebra that accommodate for the needs of non-local operators. P. Phillips and I in recent work construct such generalizations, thus providing a mathematical foundation for the quest of such non-local CFT's.

3:00 pm in 347 Altgeld Hall,Thursday, March 30, 2017

The m=1 amplituhedron and cyclic hyperplane arrangements

Steven Karp (UIUC)

Abstract: The m=1 amplituhedron and cyclic hyperplane arrangements The totally nonnegative part of the Grassmannian Gr(k,n) is the set of k-dimensional subspaces of R^n whose Plücker coordinates are all nonnegative. The amplituhedron is the image in Gr(k,k+m) of the totally nonnegative part of Gr(k,n), through a (k+m) x n matrix with positive maximal minors. It was introduced in 2013 by Arkani-Hamed and Trnka in their study of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. Taking an orthogonal point of view, we give a description of the amplituhedron in terms of sign variation. We then use this perspective to study the case m=1, giving a cell decomposition of the m=1 amplituhedron and showing that we can identify it with the complex of bounded faces of a cyclic hyperplane arrangement. It follows that the m=1 amplituhedron is homeomorphic to a ball. This is joint work with Lauren Williams.

4:00 pm in 245 Altgeld Hall,Thursday, March 30, 2017

Projections and Curves in Infinite-Dimensional Banach Spaces

Bobby Wilson (MIT and MSRI)

Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections including the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss some related questions. This is joint work with Marianna Csornyei and David Bate.