Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, January 25, 2005.

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Questions regarding events or the calendar should be directed to Tori Corkery.
    December 2004           January 2005          February 2005
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30 31


Tuesday, January 25, 2005

11:00 am in Altgeld Hall,Tuesday, January 25, 2005

#### No meeting this week

Abstract: Recover from the Jacob Lurie series.

1:00 pm in 241 Altgeld Hall,Tuesday, January 25, 2005

#### Integers covered by systems of congruences with distinct moduli.

###### Kevin Ford (UIUC)

Abstract: A covering system is a finite set of arithmetic progressions with distinct moduli >1 whose union is all the integers. Erdos conjectured that there are covering systems whose smallest moduli is arbitrarily large. We consider the following related problem: If N is a large integer and K>1, what is the densest union of arithmetic progressions with distinct moduli in [N,KN]? We show that if K is a bit smaller than a power of N, then the desity of the union of progressions cannot be much more than 1-1/K. In particular, a covering system with distinct moduli in [N,KN] cannot exist. This is joint work with Michael Filaseta, Sergei Konyagin, Carl Pomerance and Gang Yu.

1:00 pm in 345 Altgeld Hall,Tuesday, January 25, 2005

#### Generics, connected components, and algebraic groups over p-adically closed fields.

###### Anand Pillay (UIUC Math)

Abstract: I will discuss p-adic analogues of my conjectures on type-definable connected components (joint with Onshuus). I may also discuss theories of genericity and connected components in a general context.

2:00 pm in 243 Altgeld Hall,Tuesday, January 25, 2005

#### Two short presentations

###### Mr. Kyung-won Hwang, and Prof. John E. Wetzel (UIUC Department of Mathematics)

Abstract: Mr. Kyung-won Hwang
Title: A convex set in Z^k.
R. Graham, M. Simonovits, and V. T. Sos in 1980 posed the following question. Suppose S is a convex subset of Z^k, and let A_i be subsets of S whose pairwise intersections are non-empty and convex. If the A_i form a maximum such family, must the intersection of the A_i be non-empty? In this paper we answer this question in the negative by exhibiting a set S and a family of subsets of S for every dimension k>1. Then we will fix this question by asking a modified question, the answer to which we believe is in the affirmative. This is joint work with Naeem Shiekh.

Abstract: Prof. John E. Wetzel
Title: The worm problem - a status report
We report on some failed fits and starts on Leo Moser's well-known covering problem (especially having to do with the conjecture of long-standing that a 30 degree unit sector is a cover for all unit arcs in the plane); and we conclude with some positive recent developments hot off the press.

3:00 pm in 241 Altgeld Hall,Tuesday, January 25, 2005

#### Induced Ramsey numbers of P3 with other graphs

###### Naeem Sheikh (UIUC Math)

Abstract: The induced Ramsey number IR(G,H) equals n if there is a graph F on n vertices such that every 2-colouring of its edges with red and blue results in either a red copy of G as an induced subgraph of F, or an induced blue H, and no graph with fewer than n vertices has this property. The talk will present a few results on induced Ramsey numbers of P3 with other graphs. We prove that IR(P3,G) <= |V(G)| + |E(G)| and then show that this bound is sharp when every component of G is a complete graph. We will also show a better (and sharp) bound when G is a complete multipartite graph. (This is joint work with A. Kostochka.)

4:00 pm in 341 Altgeld Hall,Tuesday, January 25, 2005

#### Free groups and partial isomorphism extensions

###### Slawomir Solecki (UIUC Math)

4:00 pm in 245 Altgeld Hall,Tuesday, January 25, 2005

#### Generalization's of Tsen's theorem

###### Thomas Graber (University of California at Berkeley)

Abstract: Tsen's theorem says roughly that polynomials of low degree in many variables with coefficients in the field of meromorphic functions on a compact Riemann surface admit solutions. I will discuss joint work with Harris, Mazur, and Starr which suggests that this result is best understood in the context of the geometry of rational curves.