Department of

Mathematics


Seminar Calendar
for events the day of Thursday, March 3, 2016.

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Thursday, March 3, 2016

11:00 am in 241 Altgeld Hall,Thursday, March 3, 2016

Lattice point count and continued fractions

Michael Magee (Yale University)

Abstract: In this talk I’ll discuss a lattice point count for a thin semigroup inside $\mathrm{SL}_2(\mathbf{Z})$. It is important for applications that one can perform this count uniformly throughout congruence classes and for arbitrary moduli. The approach to counting is dynamical - with input from both the real and finite places. At the real place one brings ideas of Dolgopyat concerning oscillatory functions into play. At finite places, a rapid mixing property is supplied by expansion of Cayley graphs and injected into the thermodynamical formalism. The expansion of the relevant Cayley graphs was first established by Bourgain and Gamburd (for prime places) and extended to arbitrary moduli by Bourgain and Varjú. Until recently it was only known how to apply these expansion results in the thermodynamical setting for squarefree moduli. I’ll discuss a decoupling method (developed in work with Bourgain and Kontorovich) that allows arbitrary moduli to be treated. This talk is based on joint works with Oh and Winter, and with Bourgain and Kontorovich.

12:00 pm in Altgeld Hall 243,Thursday, March 3, 2016

Equations in free associative algebras and group algebras of hyperbolic groups.

Olga Kharlampovich (Hunter College of CUNY)

Abstract: We will talk about Diophantine problems for groups and rings and show that equations are undecidable in a free associative algebra and in a group algebra of a free group (or torsion free hyperbolic group) over a field of characteristic zero. This is a surprising result that shows that answers to first-order questions in a free non-abelian group and in its group algebra are completely different. (Joint results with A. Miasnikov)

12:30 pm in 464 Loomis Laboratory,Thursday, March 3, 2016

Quantum Entanglement of Local Operators

Masahiro Nozaki (U Chicago Physics)

Abstract: We have investigated the time evolution of (Renyi) entanglement entropies for locally excited states which are defined by acting with local operators on the ground state when the subsystem is given by a half of the total space. We have found that they approach finite constants in the free field theory. They depend on the details of local operators. We expect that they characterize local operators from the viewpoint of quantum entanglement. We also found the sum rule which those entropies obey. We also found that these results are interpreted in terms of the relativistic propagation of quasi-particles. We also have investigated these quantities in strongly coupled theory. In holographic field theory, (Renyi) entanglement entropy does not approach constant even in late time. We also obtained some interesting features of those quantities. We would like to talk about our recent works as long as time allows.

2:00 pm in 243 Altgeld Hall,Thursday, March 3, 2016

Quasiconformal non-parametrizability of almost smooth spheres

Vyron Vellis (University of Jyvaskyla)

Abstract: In 1996, Semmes constructed a metric on the unit 3-sphere $S^3$ that resembles the Euclidean metric geometrically (linearly locally contractible) and measure theoretically (3-regular) but is not quasiconformal to the Euclidean metric. Semmes's surprising result was extended by Heinonen and Wu in every dimension $n>3$. Semmes' metric is Riemannian outside of a Cantor set on $S^3$ while the metric of Heinonen and Wu is Riemannian outside of a co-dimension 3 subset of $S^n$. In this talk, we improve these results by constructing a metric on $S^n$, for each $n>2$, that is linearly locally contractible, $n$-regular and Riemannian outside of a single point, but is not quasiconformal to the Euclidean metric. This is a joint work with Pekka Pankka.

2:00 pm in 241 Altgeld Hall ,Thursday, March 3, 2016

Geometry of translated Chebyshev polynomials

Michael Oyengo (UIUC Math)

Abstract: We show that translated Chebyshev polynomials of the first kind have their roots on certain ellipses centered at the origin. The converse is also true, that for every ellipse centered at the origin, there exists a sequence of translated Chebyshev polynomials with roots on the ellipse. We will show that certain sequences of translated Chebyshev polynomials interlace on a fixed ellipse centered at the origin. Other properties of families of translated Chebyshev polynomials will also be discussed.

4:00 pm in 245 Altgeld Hall,Thursday, March 3, 2016

Tarski-type questions for group rings

Olga Kharlampovich (CUNY Graduate Center and Hunter College)

Abstract: We consider some fundamental model-theoretic questions that can be asked about a given algebraic structure (a group, a ring, etc.), or a class of structures, to understand its principal algebraic and logical properties. These Tarski-type questions include: elementary classification and decidability of the first-order theory. We describe solutions to Tarski's problems in the class of group algebras of free groups. We will show that unlike free groups, two groups algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will also show that for any field, the theory of a group algebra of a torsion free hyperbolic group is undecidable and for a field of zero characteristic even the diophantine problem is undecidable. (These are joint results with A. Miasnikov)