Department of

Mathematics


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

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

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

Non-vanishing of Dirichlet series without Euler products

Bill Banks (U Missouri-Columbia Math)

Abstract: This talk explores the question: To what extent does the Euler product expansion of the Riemann zeta function account for the non-vanishing of the Riemann zeta function in the half-plane $\{\Re(s)>1\}$? We exhibit a family of Dirichlet series that are closely related to the Riemann zeta function and are nonzero in $\{\Re(s)>1\}$, but do not possess an Euler product.

12:00 pm in 243 Altgeld Hall,Thursday, November 30, 2017

Green metric, Ancona inequalities and Martin boundary for relatively hyperbolic groups

Ilya Gekhtman (Yale University)

Abstract: Generalizing results of Ancona for hyperbolic groups, we prove that a random path between two points in a relatively hyperbolic group (e.g. a nonuniform lattice in hyperbolic space) has a uniformly high probability of passing any point on a word metric geodesic between them that is not inside a long subsegment close to a translate of a parabolic subgroup. We use this to relate three compactifications of the group: the Martin boundary associated with the random walk, the Bowditch boundary, associated to an action of the group on a proper hyperbolic space, and the Floyd boundary, obtained by a certain rescaling of the word metric. We demonstrate some dynamical consequences of these seemingly combinatorial results. For example, for a nonuniform lattice G in hyperbolic space H^n, we prove that the harmonic (exit) measure on the boundary associated to any finite support random walk on G is singular to the Lebesgue measure. Moreover, we construct a geodesic flow and G invariant measure on the unit tangent bundle of hyperbolic space projecting to a finite measure on T^1H^n/G whose geodesic current is equivalent to the square of the harmonic measure. The axes of random loxodromic elements in G equidistribute with respect to this measure. Analogous results hold for any geometrically finite subgroups of isometry groups of manifolds of pinched negative curvature, or even proper delta-hyperbolic metric spaces.

12:30 pm in 222 Loomis,Thursday, November 30, 2017

Eigenstate Thermalization and Locality and Random Matrices

Anatoly Dymarsky (University of Kentucky)

Abstract: Eigenstate Thermalization Hypothesis (ETH) is a set of properties which explain the emergence of equilibrium statistical mechanics for an isolated quantum chaotic system. It is believed to be a characteristic feature, and even used as a working definition, of quantum chaos. At the technical level ETH can be understood as an ansatz for the matrix elements of certain observables. We start by showing how ETH can help define a novel order parameter which would distinguish chaotic and non-chaotic phases. We then proceed by constraining the ETH ansatz in case of the systems with local interactions. For the systems exhibiting diffusive transport we find a new stringent bound limiting applicability of Random Matrix Theory to describe the observables satisfying the ETH.

2:00 pm in 241 Altgeld Hall,Thursday, November 30, 2017

Exponential sums and the Linear Independence Conjecture

William Banks   [email] (University of Missouri)

Abstract: The Linear Independence Conjecture (LIC) for the Riemann zeta-function asserts that the positive ordinates of the nontrivial zeros of zeta(s) are linearly independent over the rationals. In this talk, I will describe some ongoing joint work with N. Ng, G. Martin and M. Milinovich in which we estimate certain exponential sums in order to obtain evidence for the truth of the LIC.

3:00 pm in 345 Altgeld Hall,Thursday, November 30, 2017

Integrability of generalized Toda lattice systems: Part 2

Matej Penciak (University of Illinois)

Abstract: In this continuation of my first talk I will recall the description of the Toda lattice phase space, and Hamiltonians. After this I will describe the quantization of the system. The quantized Hamiltonians will turn out to be restrictions of differential operators to so called Whittaker functions on the simply connected Lie group associated to $\mathfrak{g}$.

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

Modeling the evolution of drug resistance in cancer

Jasmine Foo (School of Mathematics, University of Minnesota-Twin Cities)

Abstract: Despite the effectiveness of many therapies in reducing tumor burden during the initial phase of treatment, the emergence of drug resistance remains a primary obstacle in cancer treatment. Tumors are comprised of highly heterogeneous, rapidly evolving cell populations whose dynamics can be modeled using evolutionary theory. In this talk I will describe some mathematical models of the evolutionary processes driving drug resistance in cancer, and demonstrate how these models can be used to provide clinical insights. In particular I will describe several branching process models of tumor evolution and analyze the timing tumor recurrence. These models will be applied to study the impact of dosing schedules and the tumor microenvironment on the emergence of drug resistance in lung cancer.