UIUC Dept. of Mathematics Seminar Calendar

      April 2003              May 2003              June 2003      
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Tuesday, May 6, 2003

Topology Seminar
11:00 am in 345 Altgeld Hall, Tuesday, May 6, 2003

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Submitted by cpfrench.

Daniel Christensen (University of Western Ontario)
Simplicial models of quantum gravity
Abstract: In this talk I will give an overview of the spin foam approach to quantum gravity, developed by Baez, Barrett, Crane, Reisenberger, Smolin and others. States for quantum geometry are given by labelled simplicial complexes, and transition amplitudes are defined using fun diagrammatic methods in SU(2) representation theory. After giving the background, I will discuss the computational difficulties that come up when trying to do calculations in this theory and will describe new algorithms due to Greg Egan, John Baez and myself which have allowed us to make the first computations. No previous exposure to the subject will be assumed.

String Theory RAP
12:00 pm in 341 Altgeld Hall, Tuesday, May 6, 2003

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Submitted by katz.

Sheldon Katz [email] (UIUC Math)
String theory, moduli spaces of pointed Riemann surfaces, and the genus expansion
Abstract: I will cover interactions and how this encodes the genus of the worldsheet, and will connect this to moduli spaces of Riemann surfaces and the expression of path integrals as a sum over the genus of the worldsheet. Depending on one's perspective, this allows algebraic geometry to be used as a tool to calculate path integrals exactly, or inspires mathematicians to organize enumerative data into generating functions, as has revolutionized intersection theory and enumerative geometry (e.g. the Witten conjectures a.k.a. Kontsevich's theorem, or Gromov-Witten theory). I reserve the right to not get through all this!

Geometric Potpourri Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, May 6, 2003

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Submitted by seminar.

Two short presentations are scheduled
Abstract: Speaker: Dr. Wacharin Wichiramala, UIUC Department of Mathematics.

Title: Admissible arcs are drapeable. Circumscribed arcs are drapeable?

Abstract: Wetzel conjectured that admissible arcs and circumscribed arcs are drapeable. For admissible arcs, he settled the case of simple arcs and suggested a few ways to prove the general case. For circumscribed arcs, he gave the main idea how to attack. In this talk, we complete the first problem and give an attempt to prove the second problem. The arguments contain some messy, mind-twisting, bone-crushing analytical approximations that might not be suitable for audiences with high blood pressure. (Please bring 4 handouts from last seminar with you.)

Speaker: Prof. John E. Wetzel, UIUC Department of Mathematics.

Title: Projections of a regular tetrahedron

Abstract: We establish the following recent elegant result of Michael Eastwood and Roger Penrose:

Four points a, b, c, d in the complex plane are the orthogonal projections of the vertices of a regular tetrahedron if and only if

(a + b + c + d)^2 = 4(a^2 + b^2 + c^2 + d^2)

The principal tool is a similar result for a cube known as the Gauss's Fundamental Theorem of Axonometry, stated by Gauss without proof and proved in 1844 by J. L. Weisbach.

Joint Group Theory Seminar and RAP "Metric Spaces of Non-Positive Curvature"
3:00 pm in Altgeld Hall 347, Tuesday, May 6, 2003

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Submitted by kapovitc.

Mark Sapir (Vanderbilt University)
Diagram groups and directed 2-complexes: homotopy and homology
Abstract: This is a joint work with Victor Guba (Vologda, Russia). A diagram group G is the fundamental group of the space of positive paths with fixed ends of a directed 2-complex. Unlike the ordinary loop space, that space of paths turns out to be a K(G,1). Its universal cover is a CAT(0) cubical complex. We show how to compute all integer homology groups of a diagram group. We construct FP_\infty (diagram) groups with virtually arbitrary rational Poincare series. This shows, in particular, that the class of FP_\infty infinitely dimensional groups is large. We find concrete examples of finitely presented (even FP_\infty) diagram groups containing all countable diagram groups as subgroups. We also discuss representations of diagram groups by homeomorphisms. Finally, we show that all diagram groups are totally orderable.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, May 6, 2003

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Submitted by west.

Jeong-Hyun Kang (UIUC Math)
Covering Euclidean n-space by translates of a convex body
Abstract: Rogers [1957] proved that for every closed convex body C in Rⁿ, there is a covering of Rⁿ by translates of C that has density at most O(nlnn). However, a covering with low global density can have high multiplicity, where the multiplicity is the maximum number of copies of C covering a single point. Erdös and Rogers [1962] showed that, for sufficiently large n, there is a covering of Rⁿ by translates of C that has density at most O(nlnn) and multiplicity at most O(nlnn). In this talk, we give a combinatorial proof of this using the Local Lemma.

We also give an application of this theorem. Let G be the graph on Rⁿ in which points are adjacent if their distance is 1 in the l_p norm. Kang and Füredi proved that G has chromatic number between (1.067)ⁿ and \sqrt(p/(2\pi n))(5(ep)^1/p)ⁿ. We apply the theorem above to obtain an upper bound of c(nlnn)5ⁿ on the chromatic number of G, independent of p. This simplifies the previous upper bound argument and improves the upper bound when p is not too large. (These results are joint work with Zoltán Füredi.)