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Seminar Calendar
for events the day of Thursday, September 10, 2015.

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Thursday, September 10, 2015

11:00 am in 241 Altgeld Hall,Thursday, September 10, 2015

Fourier Analysis and the zeros of the Riemann zeta-function

Micah Milinovich (University of Mississippi)

Abstract: I will show how the classical Beurling-Selberg extremal problem in harmonic analysis arises naturally when studying the vertical distribution of the zeros of the Riemann zeta-function and other L-functions. Using this relationship, along with techniques from Fourier analysis and reproducing kernel Hilbert spaces, we can prove the sharpest known bounds for the number of zeros in an interval on the critical line and we can also study the pair correlation of zeros. Our results on pair correlation extend earlier work of P. X. Gallagher and give some evidence for the well-known conjecture of H. L. Montgomery. This talk is based on a series of joint works with E. Carneiro, V. Chandee, and F. Littmann.

1:00 pm in 243 Altgeld Hall,Thursday, September 10, 2015

Some statistical properties of digits in continued fractions

Florin Boca (UIUC Math)

Abstract: Specifically, we will discuss results on generalized Gauss-Kuzmin statistics and on the distribution of partial sums of digits of a random irrational number.

2:00 pm in 243 Altgeld Hall,Thursday, September 10, 2015

Fixed points of completely contractive maps, ternary rings of operators and convolution operators on classical and quantum groups

Adam Skalski (Inst. of Math. Polish Academy of Sciences and University of Warsaw)

Abstract: The Choi-Effros product, granting the fixed point space of a unital completely positive map a unique C*-algebra (or von Neumann algebra) structure is the key tool in the construction of the abstract Poisson boundary, generalising the classical concept of a probabilistic-type boundary for a random walk. In this talk we will discuss how replacing the completely positive map by a completely contractive one leads instead to a construction of a (weak*-closed) ternary ring of operators and present some applications to the study of fixed point spaces of contractive convolution operators on classical and quantum locally compact groups. (Mainly based on joint work with Pekka Salmi, Matthias Neufang and Nico Spronk.)

2:00 pm in 241 Altgeld Hall,Thursday, September 10, 2015

On Fourier and Koshliakov kernels in Ramanujan's work

Nicolas Robles (UIUC Math)

Abstract: In 1918 Ramanujan observed that the reciprocity of two functions could be used in the theory of the Riemann zeta-function. Specifically, Ramanujan showed that reciprocal functions could produce intriguing explicit formulae involving the non-trivial zeros and the M\"{o}bius function. This mechanism can be used as well to prove highly generalized integral identities of the Riemann $\Xi$-function. Some of the existing theorems in the literature (due to Hardy, Ferrar, Koshliakov and Ramanujan) now follow as corollaries of these new integrals. This is joint work with A. Dixit, A. Roy and A. Zaharescu.

2:00 pm in 347 Altgeld Hall,Thursday, September 10, 2015

Menger Continua in Logic and Topology

Aristotelis Panagiotopoulos (UIUC Math)

Abstract: Menger continua are the n-dimensional analogues of the Hilbert cube. Many remarkable properties of the Hilbert cube transfer in a n-truncated manner to each Menger continuum. I will start by defining the Menger continua and I will present some of these remarkable properties. In the second part of the talk I will present some notions of “smallness” which appear in the intersection of homotopy theory with descriptive set theory. In the last part of the talk I will demonstrate a dualized model theoretic construction (category theory style) introduced by T.Irwin and S.Solecki under the name projective Fra\"iss\'e construction and I will show how this construction can provide a very canonical and useful construction of the Menger continua. This last part is joint work with S.Solecki.

4:00 pm in 245 Altgeld Hall,Thursday, September 10, 2015

Induced cycles and coloring

Maria Chudnovsky (Princeton University)

Abstract: The Strong Perfect Graph Theorem states that graphs with no induced odd cycle of length at least five, and no complements of one behave very well with respect to coloring. But what happens if only some induced cycles (and no complements) are excluded? Gyarfas made a number of conjectures on this topic, asserting that in many cases the chromatic number is bounded by a function of the clique number. In this talk we discuss recent progress on some of these conjectures. This is joint work with Alex Scott and Paul Seymour.