Department of

Mathematics


Seminar Calendar
for events the day of Thursday, October 26, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, October 26, 2017

11:00 am in 241 Altgeld Hall,Thursday, October 26, 2017

Pair correlation in Apollonian circle packings

Xin Zhang (UIUC Math)

Abstract: Montgomery and Dysonís Pair Correlation Conjecture says that the pair correlation of the non-trivial zeros of the Riemann zeta function agrees with the pair correlation of the eigenvalues of a random Hermitian matrix. Despite its huge influence, a resolution of this conjecture seems far away. Pair correlations for some other deterministic sequences can otherwise be determined rigorously. In this talk, I will show that the limiting pair correlation of the circles from a fixed Apollonian circle packing exists. A key feature in our work, which differs from previous work in literature, is that the underlying point process is fractal in nature. A critical tool in our analysis is an extended version of a theorem of Mohammadi-Oh on the equidistribution of expanding horospheres in the frame bundles of infinite volume hyperbolic spaces. This work is motivated by an IGL project that I mentored in Spring 2017.

11:30 am in Urbana Country Club,Thursday, October 26, 2017

Department of Mathematics Retiree's Luncheon

12:00 pm in 243 Altgeld Hall,Thursday, October 26, 2017

Siegel-Veech transforms are in L2.

Jayadev Athreya (University of Washington)

Abstract: Let H denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on R2 is in L2(H,μ), where μ is the Masur-Veech measure on H, and give applications to bounding error terms for counting problems for saddle connections. We will review classical results in the Geometry of Numbers which anticipate this result. This is joint work with Yitwah Cheung and Howard Masur.

2:00 pm in 243 Altgeld Hall,Thursday, October 26, 2017

A Spider's Web of Doughnuts

Daniel Stoertz (Northern Illinois University)

Abstract: A result in complex dynamics states that, if the Julia set of a holomorphic map of polynomial type is a Cantor set, then the fast escaping set of its Poincarť Linearizer at a repelling fixed point is a spider's web. Some work has been done to generalize this result into higher-dimensional dynamics with quasiregular mappings, though this work has required that the Cantor sets be tame. We will summarize the necessary background in complex dynamics and the dynamics of quasiregular mappings to state the result for wild (and therefore all) Cantor sets. We will then present topological results justifying interest, as well as outline the strategy toward proving this more general result. For a particular type of Cantor set, the resulting fast escaping set will be built out of tori, making a spider's web of doughnuts.

2:00 pm in 241 Altgeld Hall,Thursday, October 26, 2017

Combinatorial Proofs of Identities from Ramanujanís Lost Notebook

Hannah Burson (UIUC)

Abstract: In his lost notebook, Ramanujan stated at least 27 identities related to the Rogers-Fine identity. In this talk, I discuss a group of 6 such identities relating to Roger's false theta functions. We give a new combinatorial interpretation and proof of one identity.

3:00 pm in 345 Altgeld Hall,Thursday, October 26, 2017

Toroidal actions on fermionic q-Fock space

Joshua Wen (Illinois)

Abstract: One origin story for quantum toroidal algebras comes from the observation that some integrable representations of quantum affine algebras admit an alternate action of that same algebra: the toroidal algebras are what glue these two structures together. One of the first worked-out examples of this phenomenon is the case of fermionic q-Fock space (by Varagnolo-Vasserot and Takemura-Uglov-Saito), where level-1 representations of quantum affine $\mathfrak{sl_n}$ also admit a level-0 action. One can understand this level-0 action by diagonalizing a commutative subalgebra, which was done by Takemura-Uglov using nonsymmetric Macdonald polynomials. Iíll explain this story and work of Nagao relating it to equivariant K-theory of cyclic quiver varieties, wherein the above diagonalization comes for free via fixed-point classes.

4:00 pm in 245 Altgeld Hall,Thursday, October 26, 2017

Planar graphs and Legendrian surfaces

Emmy Murphy (Northwestern)

Abstract: Associated to a planar cubic graph, there is a closed surface in R^5, as defined by Treumann and Zaslow. R^5 has a canonical geometry, called a contact structure, which is compatible with the surface. The data of how this surface interacts with the geometry recovers interesting data about the graph, notably its chromatic polynomial. This also connects with pseudo-holomorphic curve counts which have boundary on the surface, and by looking at the resulting differential graded algebra coming from symplectic field theory, we obtain new definitions of n-colorings which are strongly non-linear as compared to other known definitions. There are also relationships with SL_2 gauge theory, mathematical physics, symplectic flexibility, and holomorphic contact geometry. During the talk we'll explain the basic ideas behind the various fields above, and why these various concepts connect.