Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, February 18, 2014.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2014          February 2014            March 2014
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4                      1                      1
5  6  7  8  9 10 11    2  3  4  5  6  7  8    2  3  4  5  6  7  8
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26 27 28 29 30 31      23 24 25 26 27 28      23 24 25 26 27 28 29
30 31


Tuesday, February 18, 2014

1:00 pm in 243 Altgeld Hall,Tuesday, February 18, 2014

#### Circumcenter of Mass and the generalized Euler line

###### Sergei Tabachnikov (Penn State Math)

Abstract: I shall define and study a variant of the center of mass of a polygon, called the Circumcenter of Mass. The Circumcenter of Mass is an affine combination of the circumcenters of the triangles in a non-degenerate triangulation of a polygon, weighted by their areas, and it does not depend on the triangulation. For an inscribed polygon, this center coincides with the circumcenter. The Circumcenter of Mass satisfies an analog of the Archimedes Lemma, similarly to the center of mass of the polygonal lamina. The line connecting the circumcenter and the centroid of a triangle is called the Euler line. Taking an affine combination of the circumcenters and the centroids of the triangles in a triangulation, one obtains the Euler line of a polygon. The construction of the Circumcenter of Mass extends to simplicial polytopes and to the spherical and hyperbolic geometries.

1:00 pm in 345 Altgeld Hall,Tuesday, February 18, 2014

#### Generalized Homogenous Metric Spaces in First Order Model Theory

###### Gabriel Conant (UIC)

Abstract: We develop an abstract framework for studying arbitrary first-order models of homogeneous metric spaces over countable distance sets. Examples of such structures are generalized Urysohn spaces, ultrametric spaces, and certain metrically homogeneous graphs. We characterize forking independence in the complete theory of such structures, and then show that they give a wide class of examples in the spectrum of first order theories without the strict order property. Finally, we discuss some connections to continuous model theory.

2:00 pm in Altgeld Hall 347,Tuesday, February 18, 2014

#### Occupation times of (refracted) spectrally negative Lévy processes

###### Jean-François Renaud   [email] (University of Quebec at Montreal)

Abstract: In this talk, I will present results on the distributions of various occupation times of (refracted) spectrally negative Lévy processes. To get semi-explicit expressions in terms of the $q$-scale functions of the underlying spectrally negative Lévy process(es), new analytical identities for scale functions are needed. Applications to actuarial ruin theory and exotic option pricing will also be presented. This talk is based on joint work with Ronnie Loeffen (U. of Manchester) and Xiaowen Zhou (Concordia U.).

2:00 pm in 241 Altgeld Hall,Tuesday, February 18, 2014

#### Generic representations of Abelian groups and extreme amenability (part 1)

###### Anush Tserunyan (UIUC Math)

Abstract: We start working through the paper "Generic representations of Abelian groups and extreme amenability" by Melleray and Tsankov. We cover the necessary background, discuss motivating questions and show that extreme amenability of a representation of a countable group in a Polish group is a $G_\delta$ property. The latter uses the equivalence of extreme amenability and a Ramsey type property called finite oscillation stability, which we take as a black box to be opened later.

3:00 pm in 241 Altgeld Hall,Tuesday, February 18, 2014

#### Fool's Solitaire on Joins and Cartesian Products with Complete Graphs

###### Sarah Loeb (UIUC Math)

Abstract: Peg solitaire is a game generalized to connected graphs by Beeler and Hoilman. In the game pegs are placed on all but one vertex. If $xyz$ form a 3-vertex path and $x$ and $y$ each have a peg but $z$ does not, then we can remove the pegs at $x$ and $y$ and place a peg at $z$. By analogy with the moves in the original game, this is called a jump. The goal of the peg solitaire game on graphs is to find jumps that reduce the number of pegs on the graph to 1. Beeler and Rodriguez proposed a variant where we instead want to maximize the number of pegs remaining when no more jumps can be made. Maximizing over all initial locations of a single hole, the maximum number of pegs left on a graph $G$ when no jumps remain is the fool's solitaire number $F(G)$. We determine the fool's solitaire number for the join of any graphs $G$ and $H$. For the cartesian product, we determine $F(G\ \square\ K_k)$ when $k \ge 3$ and $G$ is connected and show why our argument fails when $k=2$.

3:00 pm in 347 Altgeld Hall,Tuesday, February 18, 2014

#### Representation theorems for multinormed spaces

###### Vladimir Troitsky (University of Alberta)

Abstract: For $1\leq p\leq\infty$, a $p$-multinormed space is a vector space $E$ and a sequence of complete norms $\lVert\cdot\rVert_n$ on $E^n$ such that $\lVert Ax\rVert_m\leq\lVert A\rVert\lVert x\rVert_n$ for every $x\in E^n$ and $A\colon\ell_p^n\to\ell_p^m$. Multinormed spaces were introduced by G.Dales and M.Polyakov. They bear certain similarities to Operator Spaces. Similarly to Ruan's representation of Operator Spaces as subspaces of $B(H)$, there are representation theorems for multinormed spaces. Namely, every $\infty$-multinormed space can be represented as a subspace of a Banach lattice (this result is essentially due to Pisier), while every 1-multinormed space can be represented as a quotient of Banach lattice. This is a joint work with G.Dales and N.J.Laustsen.

4:00 pm in Altgeld Hall,Tuesday, February 18, 2014

#### Fusion products and a novel way to compute their characters

Abstract: We will introduce a graded tensor product of simple Lie algebras called the Fusion product and discuss the character of this module. This will be done through examples. Then we will see a novel way to compute the characters of Fusion products of $\mathfrak{sl}_2(\mathbb{C})$-modules using the quantum Q-system for $\mathfrak{sl}_2(\mathbb{C})$.