Seminar Calendar
for events the day of Thursday, September 20, 2012.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, September 20, 2012

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, September 20, 2012
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Submitted by ford.
 Harold Diamond (UIUC Math)Chebyshev bounds for Beurling generalized numbersAbstract: This is a semi-expository talk. It begins with a survey of Beurling generalized numbers, a structure that is similar to rational integers, except for having only multiplicative structure. We seek conditions on the counting function of g-numbers that allow us to deduce analogs of the Chebyshev upper and lower prime bounds. An early conjecture of the speaker is shown to be inadequate, and further conditions are given for which the bounds hold. The results are proved to be optimal in their class.

Math/Theoretical Physics Seminar
12:00 pm   in 464 Loomis Laboratory,  Thursday, September 20, 2012
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Submitted by katz.
 Sheldon Katz (Illinois Math)Supermanifolds and Berezin IntegrationAbstract: This is an expository talk on the mathematical formalism of supermanifolds, which provides a rigorous way to analyze the "anticommuting coordinates" of physics used in the study of fermions. I will also introduce Berezin integration, a mathematical formalism which gives a precise meaning to fermionic integration as it arises in physics. I would like to say something as well about supersymmetry but I won't promise due to time constraints.

2:00 pm   in 241 Altgeld Hall,  Thursday, September 20, 2012
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Submitted by collier3.
 Mike DiPasquale (UIUC Math)Bezout, Cayley-Bacharach, and PascalAbstract: We introduce some basic constructions of algebraic geometry in the process of exploring the geometry of curves in the complex projective plane. In particular we will discuss Bezout's theorem and the Cayley-Bacharach theorem for plane cubics, pointing out the special case of Pascal's 'mystic hexagon.' The object is to communicate the power of algebraic machinery in proving some beautiful geometric theorems.

Women's Seminar
3:00 pm   in 147 Altgeld Hall,  Thursday, September 20, 2012
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Submitted by funk3.
 Sarah Loeb and Sogol Jahanbekam (UIUC Math)Combinatorics at MightyAbstract: Results in Chromatic-Paintability and the Paintability of Complete Bipartite Graphs by Sarah Loeb Introduced independently by Schauz and by Zhu, the Marker-Remover game is an on-line version of list coloring. The game is played on a graph $G$ with a token assignment $f$ giving each $v \in V(G)$ a nonnegative number of tokens. On each round Marker marks a subset $M$ of the remaining vertices, which uses up a token on each vertex in $M$. Remover deletes from the graph an independent subset of vertices in $M$. Marker wins by marking a vertex that has no tokens. Remover wins if the entire graph is removed. The paint number, or paintability, of a graph $G$ is the least $k$ such that Remover has a winning strategy when $f(v) = k$ for all $v \in V(G)$. We show that if $G$ is $k$-paintable and $|V(G)| \le \frac{t}{t-1} k$, then the join of $G$ with $\overline{K}_t$ is $(k+1)$-paintable. As a corollary, the paint number of $G$ equals to its chromatic number when $|V(G)| \le \chi(G) + 2 \sqrt{\chi(G)-1}$. This strengthens a result of Ohba. We also explore the paintability of complete bipartite graphs. Extending a result of Erd\H{o}s, Rubin, and Taylor, $K_{k,r}$ is $k$-paintable if and only if $r < k^k$. For $j \ge 1$ we provide an upper bound on the least $r$ such that $K_{k+j,r}$ is not $k$-paintable. 1,2,3-Conjecture and 1,2-Conjecture for sparse graphs by Sogol Jahanbekam We apply the Discharging Method to prove the 1, 2, 3-Conjecture and the 1, 2-Conjecture for graphs with maximum average degree less than 8 3. As a result, the conjectures hold for planar graphs with girth at least 8.

Commutative Ring Theory
3:00 pm   in 243 Altgeld Hall,  Thursday, September 20, 2012
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Submitted by s-dutta.
 Javid Validashti (UIUC Math)On Syzygies and Singularities of Tensor Product SurfacesAbstract: On Syzygies and Singularities of Tensor Product Surfaces. Let $U \subseteq H^0({\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}}(2,1))$ be a basepoint free four-dimensional vector space. We study the associated bigraded ideal $I_U \subseteq \textsf{k}[s,t;u,v]$ from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation for the image of the projective surface in $\mathbb{P}^3$ parametrized by generators of $U$ over $\mathbb{P}^1 \times \mathbb{P}^1$. This problem arises from a real world application in geometric modeling, where one would like to understand the implicit equation and singular locus of a parametric surface. This talk is based on a joint work with H. Schenck and A. Seceleanu.

Fall Department Faculty Meeting
4:00 pm   in 245 Altgeld Hall,  Thursday, September 20, 2012
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Submitted by seminar.
 Fall Department Faculty Meeting