Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, December 10, 2013.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    November 2013          December 2013           January 2014
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1  2    1  2  3  4  5  6  7             1  2  3  4
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Tuesday, December 10, 2013

11:00 am in 241 Altgeld Hall,Tuesday, December 10, 2013

#### A mock modular form for the partition function

###### Nick Andersen (UIUC Math)

Abstract: It is well-known that the values of the partition function p(n) appear as the coefficients of a modular form. In my talk, I will show how we can construct a mock modular form whose "shadow" encodes the partition function, and whose coefficients are given by inner products of the partition generating function and other modular forms of weight -1/2. On the way, we will encounter Rademacher's exact formula for p(n). I will also show how this mock modular form is one member of an infinite basis for the space of mock modular forms of weight 5/2 on the full modular group. This is joint work with Scott Ahlgren.

1:00 pm in 345 Altgeld Hall,Tuesday, December 10, 2013

#### VC-minimality and valued fields

###### Joseph Flenner (University of Saint Francis)

Abstract: We present some recent results on VC-minimal theories, especially in relation to the model theory of valued fields. Adaptations of Holly's theorems on algebraically closed valued fields are described for the more general VC-minimal setting. We also look at some efforts towards the (still open) question of which valued fields have VC-minimal theories.

1:00 pm in 243 Altgeld Hall,Tuesday, December 10, 2013

#### Finite rigid sets and homologically non-trivial spheres in the curve complex of a surface

###### Nathan Broaddus (Ohio State Math)

Abstract: Aramayona and Leininger have provided a "finite rigid subset" $X(S)$ of the curve complex $C(S)$ of a surface $S$, characterized by the fact that any simplicial injection $X(S) \to C(S)$ is induced by a unique simplicial automorphism $C(S) \to C(S)$. We prove that, in the case of the sphere with $n>4$ marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a Mod$(S)$-module generator for the reduced homology of the curve complex $C(S)$, answering in the affirmative a question posed by Aramayona and Leininger. For the surface $S$ with genus $g>2$ and $n=0$ or $n=1$ marked points we find that the finite rigid set $X(S)$ of Aramayona and Leininger contains a proper subcomplex whose reduced homology class is a Mod$(S)$-module generator for the reduced homology of $C(S)$ but which is not itself rigid. This is joint work with J. Birman and W. Menasco.

1:00 pm in 347 Altgeld Hall,Tuesday, December 10, 2013

#### The zero set of a random Fourier polynomial on the sphere

###### Erik Lundberg   [email] (Purdue University)

Abstract: We start with a question motivated by the fundamental theorem of algebra: How many zeros of a random polynomial are real? We discuss three Gaussian ensembles that lead to three different answers. Of these, we emphasize the harmonic analyst’s model of choice which has the highest expected number of zeros (a fraction of the maximum) and reduces to a random trigonometric polynomial. The real section of the zero set of a polynomial in several variables is much more complicated. Hilbert’s sixteenth problem asks to study the possible arrangements of the connected components, and is especially concerned with the case of many components. I will describe a probabilistic approach to studying the topology, volume, and arrangement of the zero set (in real projective space) for a Gaussian ensemble of homogeneous polynomials. Again we will emphasize the harmonic analyst's model for random polynomials which is built out of a basis of spherical harmonics (eigenfunctions of the spherical Laplacian). This work is joint with Antonio Lerario.

3:00 pm in 241 Altgeld Hall,Tuesday, December 10, 2013

#### Flag algebras and its applications

###### Bernard Lidicky   [email] (UIUC Math)

Abstract: Flag algebras is a method, recently developed by Razborov, designed for attacking problems in extremal graph theory. There are recent applications of the method also in discrete geometry or permutation patterns. The aim of talk is to give a gentle introduction to the method and show some of its applications. The talk is based on a joint works with J. Balogh, P. Hu, H. Liu, O. Pikhurko, B. Udvari, and J. Volec.

3:00 pm in 243 Altgeld Hall,Tuesday, December 10, 2013

#### Almost purity theorem with applications to the homological conjectures - Part I

###### Kazuma Shimomoto (Meiji University)

Abstract: I will talk about almost purity theorem proved by Davis and Kedlaya with applications to the homological conjectures in local algebra. The almost purity theorem originates from p-adic Hodge theory by Faltings. I will also talk about its brief history and then construct a big Cohen-Macaulay algebra under some special condition.

4:00 pm in 243 Altgeld Hall,Tuesday, December 10, 2013

#### An Introduction to Boij-S\"oderberg Theory

###### Matt Mastroeni (UIUC Math)

Abstract: Let $k$ be a field. The aim of the talk is to give sufficient background on free resolutions and graded Betti numbers over the polynomial ring $k[x_1, \dots, x_n]$ in order to state the Boij-S\"oderberg Conjectures, which were proved in 2008 by Eisenbud and Schreyer. I will also explain how this answers the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan and give an example illustrating the conjectures. Time permitting, I might say a few words about the proof of the Boij-S\"oderberg Conjectures, but the details will be reserved for a future talk.