Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, November 8, 2005.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, November 8, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, November 8, 2005

#### Analytic number theory and elliptic curves

###### Matt Young (American Institute of Mathematics)

Abstract: The Birch and Swinnerton-Dyer conjecture provides a way for methods of analytic number theory to produce results on ranks of families of elliptic curves by studying properties of their L-functions. In particular, we are interested in questions such as: How many L-functions in a given family vanish (or not) at the central point? What is the average (analytic) rank of the family? In this talk I will discuss recent progress on these problems.

1:00 pm in 345 Altgeld Hall,Tuesday, November 8, 2005

#### Borel Conjecture(s)

###### Tomek Bartoszynski (National Science Foundation)

Abstract: In 1919 Borel came up with a notion of a strong measure zero set of real numbers, and conjectured that if a set has strong measure then in fact it is countable. This conjecture was quickly disproved (assuming Continuum Hypothesis), but also it was shown by Richard Laver in 1976, to be consistently true. Similarly, given a family of "small" sets of reals, the corresponding Borel Conjecture is the statement that being a member of this family is equivalent to being countable. For some families of sets such as "Lebesgue measure zero" or "the first category" the Borel Conjecture is clearly false, for others such as "universally Lebesgue measure zero" or "universally first category" it is not so trivially false. In this talk I will give a user friendly introduction to this subject focusing on the recent results.

1:00 pm in 347 Altgeld Hall,Tuesday, November 8, 2005

#### Group Invariant CR Mappings and the Szego Limit Theorem

###### John D'Angelo   [email] (UIUC Math)

Abstract: We consider holomorphic mappings invariant under unitary representations of finite groups, and thereby discover analogues of two classical things: The Szego Limit Theorem, and the Fibonacci numbers. I promise to keep the talk completely elementary and accessible to second year graduate students. I will pose one open problem.

2:00 pm in 241 Altgeld Hall,Tuesday, November 8, 2005

#### Fundamental Theorem of Algebra

###### David Ross (U. Hawaii; visiting UIUC Math)

Abstract: I will work through George Leibman's recent proof of the Fundamental Theorem of Algebra, which uses nonstandard analysis.

2:00 pm in 152 Henry,Tuesday, November 8, 2005

#### Continuation of Lurie's thesis and multpile cover formulas in Gromov-Witten theory

###### Josh Mullet (UIUC Math)

Abstract: I will continue with Lurie's thesis and discuss some issues that Maarten raised last time. I also plan to begin discussing multiple cover formulas. These are formulas that aid one in understanding the enumerative significance of Gromov-Witten invariants.

2:00 pm in 243 Altgeld Hall,Tuesday, November 8, 2005

#### Inverse Pedal Curves of Conic Sections

###### Prof. Vincent Matsko (Quincy University)

Abstract: Conic sections are typically inverted when either they are centered at the origin, or have a focus at the origin. When conics are situated elsewhere in the plane, the geometry is more subtle. It happens that the inverse of the pedal curve of a conic is also a conic. How do these conics relate to each other? What happens when the process of taking an inverse pedal is iterated? Some unexpected results are obtained. Liberally sprinkled with computer-generated curves, this talk is accessible to undergraduates.

3:00 pm in 243 Altgeld Hall,Tuesday, November 8, 2005

#### Toward computing the product in orbifold Hochschild cohomology

###### Andrei Caldararu (University of Wisconsin)

Abstract: I'll describe recent progress in generalizing to orbifolds Kontsevich's Theorem on Complex Manifolds (which relates Hochschild cohomology of a smooth variety to polyvector field cohomology). I shall present a generalization of the Hochschild-Kostant-Rosenberg theorem to orbifolds, and give a conjectural formula for the required correction, analogous to the root-A-hat that appears in Kontsevich's theorem. If time allows, I'll conclude by explaining how one can hope to apply the Duflo theorem to get a complete proof of the conjectural results I am mentioning. The talk is based on work in progress, partly joint with Simon Willerton.

3:00 pm in 241 Altgeld Hall,Tuesday, November 8, 2005

#### On Ramsey numbers for sparse uniform hypergraphs

###### Alexandr V. Kostochka (UIUC Math)

Abstract: For a k-uniform hypergraph G, the Ramsey number R(G,G) is the least N such that in every 2-coloring of edges of a complete n-vertex k-uniform hypergraph, there is a monochromatic copy of G. A family F of k-uniform hypergraphs is f(n)-Ramsey if there is a positive constant c such that R(G,G) <= c f(|V(G)|) for every G\in F.

Burr and Erdös conjectured that for every d, the families M(d) of graphs with maximum degree d and D(d) of d-degenerate graphs are n-Ramsey. Recall that a graph is d-degenerate if each subgraph has a vertex of degree at most d. Chvátal, Rödl, Szemerédi and Trotter proved the first conjecture. The second is open. However, Kostochka and Rödl proved that D(d) is n2-Ramsey, and then Kostochka and Sudakov proved that the family is n1+\epsilon-Ramsey for every positive \epsilon.

In this talk, we prove that for every \epsilon > 0 and every k and d, the family D(d,k) of k-uniform hypergraphs with maximum degree at most d is n1+\epsilon-Ramsey. We also give some examples showing that the graph and hypergraph cases behave differently. This is joint work with Vojtech Rödl.

4:00 pm in 2369 Beckman Institute,Tuesday, November 8, 2005

#### Hilbert Series of Subspace Arrangements (d'apres Harm Derksen)

###### Robert Fossum (UIUC Math)

Abstract: Subspace Arrangements arise when segmenting data, such as images. Derksen has established a closed formula for the Hilbert Function of the vanishing ideal of a subspace arrangement that is useful in the analysis of the arrangements. In particular in segmenting images. The first few lectures in this series will go over Derksen's proof. Later applications will be made.