Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, October 20, 2016.

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events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, October 20, 2016

11:00 am in 241 Altgeld Hall,Thursday, October 20, 2016

#### Finding integers from orbits of thin subgroups of SL(2, Z)

###### Xin Zhang (Illinois Math)

Abstract: Let $\Lambda < SL(2,\mathbb{Z})$ be a finitely generated, non-elementary Fuchsian group of the second kind, and $v,w$ be two primitive vectors in $\mathbb{Z}^2-0$. We consider the set $\mathcal{S}=\{\langle v\gamma,w\rangle_{\mathbb{R}^2}:\gamma\in\Lambda\}$, where $\langle\cdot,\cdot\rangle_{\mathbb{R}^2}$ is the standard inner product in $\mathbb{R}^2$. Using Hardy-Littlewood circle method and some infinite co volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if $\Lambda$ has parabolic elements, and the critical exponent $\delta$ of $\Lambda$ exceeds 0.995371, then a density-one subset of all admissible integers (i.e. integers passing all local obstructions) are actually in $\mathcal{S}$, with a power savings on the size of the exceptional set (i.e. the set of admissible integers failing to appear in $\mathcal{S}$). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when $\Lambda$ is free, finitely generated, has no parabolics and has critical exponent $\delta>0.999950$.

2:00 pm in 243 Altgeld Hall,Thursday, October 20, 2016

#### Recent advances in semicommutative Calderón-Zygmund theory: BMO spaces

###### Jose Conde (UA Barcelona)

Abstract: We consider BMO spaces of von Neumann algebra valued functions in measure spaces which are more general than the usual euclidean setting. Since von Neumann algebras are not UMD spaces, the classical vector valued techniques are not useful and we are forced to find probabilistic ways of defining the spaces. Such a point of view yields new results both in the commutative and in the noncommutative setting. Based on joint work with Javier Parcet.

3:00 pm in 243 Altgeld Hall,Thursday, October 20, 2016

#### Rees-like Algebras and the Eisenbud-Goto Conjecture

###### Jason McCullough (Rider University)

Abstract: Regularity is a measure of the computational complexity of a homogeneous ideal in a polynomial ring. There are examples in which the regularity growth is doubly exponential in terms of the degrees of the generators, but better bounds were conjectured for "nice" ideals. Together with Irena Peeva, I discovered a construction that overturns some of the conjectured bounds for "nice" ideals - including the Eisenbud-Goto conjecture. Our construction involves two new ideas that we believe will be of independent interest: Rees-like algebras and step-by-step homogenization. I'll explain the constructions and some of their consequences.

4:00 pm in 245 Altgeld Hall,Thursday, October 20, 2016

#### On the topology of black holes and beyond

###### Greg Galloway (University of Miami)

Abstract: There is a widely held belief in physics that a true astrophysical black hole, formed from the gravitational collapse of some stellar object, can be described by a certain exact solution to the Einstein equations discovered by Kerr in the 60's. This belief is based largely on a series of powerful results which shows that the Kerr solution is the unique solution to the vacuum (source-free) Einstein equations with certain prescribed properties. A basic step in the proof is Hawking's theorem on the topology of black holes which asserts that, under physically natural conditions, the surface of a black hole (cross-section of the event horizon) must be topologically a 2-sphere. Various developments in string theory have generated a great deal of interest in gravity in higher dimensions and, in particular, in higher dimensional black holes. The remarkable discovery of Emparan and Reall of a 4+1 dimensional vacuum black hole solution to the Einstein equations with nonspherical horizon topology raised the question as to what horizon topologies are allowable in higher dimensions. In this talk we review Hawking's theorem on the topology of black holes in 3+1 dimensions and present a generalization of it to higher dimensions. The latter is a geometric result which places restrictions on the topology of black holes in higher dimensions. We shall also discuss more recent work on the topology of space exterior to a black hole. This is closely connected to the Principle of Topological Censorship, which roughly asserts that the topology of the region outside of all black holes (and white holes) should be simple. All of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry. This talk is based primarily on joint work with Rick Schoen and with Michael Eichmair and Dan Pollack.

5:00 pm in 350 Altgeld Hall,Thursday, October 20, 2016

#### Job market experiences - academic and government-related

###### Kelly Yancey (Institute for Defense Analyses)

Abstract: All graduate students are welcome at this informal question-and-answer session about succeeding in the job market for academic and government-related institutions. Kelly received her PhD from our Department in 2013, and completed a postdoc at the University of Maryland before moving to a research position at the Institute for Defense Analyses.