Department of

Mathematics

Seminar Calendar
for events the day of Tuesday, April 8, 2014.

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events for the
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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, April 8, 2014

1:00 pm in 243 Altgeld Hall,Tuesday, April 8, 2014

Generalized Lorentzian Horospheres and Bartnik's Splitting Conjecture

Carlos Vega (CUNY)

Abstract: n the early 80's, Yau posed the problem of establishing the rigidity of the classical singularity theorems of Hawking and Penrose. This lead to the introduction of the Busemann function and its horospheres (level sets) in the Lorentzian setting. While a Lorentzian splitting theorem was successfully established by the end of the decade, the original rigidity question remained, and was formulated concretely in a splitting conjecture by Bartnik in '88. We will discuss a new approach to (generalized) Lorentzian horospheres, and a generalized splitting theorem, with applications to Bartnik's conjecture. This is joint work with G. Galloway.

1:00 pm in 347 Altgeld Hall,Tuesday, April 8, 2014

A Morse index theorem for elliptic operators on bounded domain.

Graham Cox (UNC Chapel Hill)

Abstract: The Maslov index is a symplecto-geometric invariant that counts signed intersections of Lagrangian subspaces. It was recently shown that the Maslov index can be used to compute Morse indices of Schrodinger operators on star-shaped domains. We extend these results to general selfadjoint, elliptic operators on domains with arbitrary boundary geometry, and discuss some applications. (Joint work with C. Jones and J. Marzuola)

1:00 pm in 345 Altgeld Hall,Tuesday, April 8, 2014

The differential field of transseries

Lou van den Dries (UIUC)

Abstract: Transseries originated some 25 years ago, mainly in Ecalle's work. Some 20 years ago I formulated some conjectures about the differential field of transseries, and started work on it, first joint with Matthias Aschenbrenner, and a few years later also in collaboration with Joris van der Hoeven. Around 2011 we finally saw a clear path towards a proof, sharpening the conjectures in the mean time. In the last few weeks we finished the job, with one proviso: as part of a division of labour and for lack of time, some final details have only been checked by some of us, but not yet by all three of us. In any case, I will motivate these conjectures, and give an account of their status.

2:00 pm in 241 Altgeld Hall,Tuesday, April 8, 2014

Centralizers of generic measure-preserving automorphisms (part 2)

Abstract: We continue working through the proof by Melleray and Tsankov of the fact that for the generic measure-preserving automorphism $T$ of a standard probability space $(X,\mu)$, the centralizer of $T$ inside $\text{Aut}(X,\mu)$ is as small as possible, i.e. is equal to the closure of the group generated by $T$.

3:00 pm in 243 Altgeld Hall,Tuesday, April 8, 2014

Springer Theory for D-modules

Sam Gunningham (University of Texas-Austin)

Abstract: The Springer correspondence relates unipotent conjugacy classes in a reductive algebraic group G (e.g. GL_n), with representations of its Weyl group W (e.g. S_n). More precisely to every irreducible representation of W, one can attach an equivariant local system on a unipotent conjugacy class. Lusztig was able to account for all such local systems using his notion of cuspidal sheaves, together with certain relative Weyl groups. In this talk I will give a new perspective on Springer Theory using tools from sheaf theory and category theory, and I will explain how to generalize the Springer correspondence to give a description of the derived category of conjugation equivariant D-modules on G.

3:00 pm in 241 Altgeld Hall,Tuesday, April 8, 2014

Perfect Transitive Triangle Tilings in Oriented Graphs

Theodore Molla   [email] (UIUC Math)

Abstract: If $G$ is a graph, directed graph or oriented graph, a perfect $H$-tiling of $G$ is a spanning subgraph consisting entirely of disjoint copies of $H$. In 1963 Corrádi and Hajnal proved that if $G$ is a graph on $n$ vertices, $n$ is divisible by 3 and $\delta(G) \ge 2n/3$, then G contains a perfect triangle tiling. We will discuss an analogue of this result for oriented graphs. For an oriented graph $G$ and any vertex $v \in V(G)$, let $d^0(v)$ be the minimum of the out-degree and in-degree of $v$. Define the minimum semidegree of $G$, denoted $\delta^0(G)$, to be the minimum of $d^0(v)$ over all vertices $v$. We will show that if $n$ is sufficiently large and divisible by $3$, then every oriented graph $G$ on $n$ vertices such that $\delta^0(G) \ge \frac{7n}{18}$ has a perfect transitive triangle tiling. This supports a conjecture of Treglown and is best possible: For any $n$ divisible by $3$ there exists an oriented graph $G$ on $n$ vertices such that $\delta^0(G) = \left\lceil \frac{7n}{18} \right\rceil - 1$ that does not contain a perfect transitive triangle tiling. This is joint work with József Balogh and Allan Lo.

4:00 pm in 245 Altgeld Hall,Tuesday, April 8, 2014

Building a Successful Company – With Mathematicians???

Allen Butler (Daniel H. Wagner Associates, Inc., Hampton, VA)

Abstract: In this talk, Dr. Butler will give an overview of his company, Daniel H. Wagner, Associates, from its historic beginnings in 1963 through its current role today. In 1963, Dr. Dan Wagner founded his eponymous company with two guiding principles in mind. First, Dan believed in hiring young mathematicians and training them to solve real-world problems. Second, Dan felt the quality of the writing in the technical reports and briefings provided to the clients was nearly as important as the technical content itself. To this end, and while the company size was small, Dan personally reviewed every scrap of paper that went out under the company name. Through the years, the company developed an impressive reputation for mathematical analysis applied to the budding field of Search Theory (find the lost H-bomb, find the sunken treasure, find the enemy submarine, etc.), and this continues to be an area of expertise today. At the same time, the company demonstrated the breadth of its capabilities by working in areas as diverse as DNA sequencing, retirement planning, crane anti-sway, speech recognition, speaker verification, and random number generation on GPUs.

5:00 pm in Ballroom, Illini Union,Tuesday, April 8, 2014

Department Awards Ceremony

Abstract: The Department of Mathematics Awards Ceremony will be held from 5 to 8 p.m. in the Ballroom of the Illini Union. Awards will be presented to undergraduate students, graduate students, faculty and staff.