Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, December 7, 2004.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, December 7, 2004

11:00 am in Altgeld Hall 347,Tuesday, December 7, 2004

A Family Version of Lefschetz-Nielsen Fixed Point Theory

Vesta Coufal (Fort Lewis College)

1:00 pm in 241 Altgeld Hall,Tuesday, December 7, 2004

Traces of special values of modular functions

Jens Funke (New Mexico State University)

Abstract: The values of the famous j-invariant at quadratic irrationalities in the upper half plane are known as singular moduli and are of particular interest in number theory. Recently, Zagier realized the generating series of the traces of the singular moduli as a classical meromorphic modular form of weight 3/2. In this talk, we will discuss this and related results and give a generalization to modular functions on Riemann surfaces of arbitrary genus. Furthermore, we realize a certain generating series of arithmetic intersection numbers and Faltings heights as the derivative of Zagier's Eisenstein series of weight $3/2$. This recovers a result of Kudla, Rapoport and Yang. This is joint work with Jan Bruinier.

1:00 pm in 345 Altgeld Hall,Tuesday, December 7, 2004

Descriptive set theory and automorphism groups of countable models

Christian Rosendal (Caltech)

Abstract: (Joint work with Alexander S. Kechris) Automorphism groups of sufficiently homogeneous countable models have been known for some time to have strong properties relating their abstract and their permutation group structures. We use methods of descriptive set theory in order to obtain new stronger properties such as automatic continuity of homomorphisms (a strengthening of the small index property) and boundedness of actions on metric spaces. The main tool used is the existence of comeagre orbits for the diagonal action of the group on its finite powers, the verification of which depends on the model theory of the underlying structure.

2:00 pm in Everett 168,Tuesday, December 7, 2004

p-valued groups and the Cohomology of Profinite Groups

Charles Rezk (UIUC)

Abstract: I will talk about the notion of a "p-valued group", as described in Serre's Bourbaki Seminar article on "p-adic analytic groups" (the notions are almost the same). In particular, we'll show that the Morava stabilizer group has open subgroups which are p-valued, and conclude that such subgroups have finite cohomological dimension.

2:00 pm in 345 Altgeld,Tuesday, December 7, 2004

On the optimal placement of unreliable sensors matrices

Bruce Reznick   [email] (UIUC Math)

Abstract: This is a very preliminary presentation on a topic which the speaker knows very little about, aside from a few provocative calculations. It is easy to answer questions such as: "How can n `sensors' be placed on I = [0,1] so as to minimize the expected minimum distance from x in I to its closest sensor?" It's also easy to answer variations in which sensors are constrained to be at one or both endpoints. We start from the following variation: "What happens if the sensors `work' independently, with probability p?" (To avoid calamity, we put "monitors" at one or both endpoints that always work.) The true applications of this problem ought to be in R2, but we have no answers in that direction.

2:00 pm in 243 Altgeld Hall,Tuesday, December 7, 2004

No meeting this week.

3:00 pm in Altgeld Hall,Tuesday, December 7, 2004

R-transform

Mingchu Gao (UIUC)

Abstract: We will give the defifnition of R-transform of a distribution on the real line. We will discuss the properties of R-transforms. Finally, we may study the Cauchy transform of a distribution and the compution of R-transforms.

3:00 pm in 241 Altgeld Hall,Tuesday, December 7, 2004

Decomposition of products of regular graphs into isomorphic trees

Douglas B. West (UIUC Math)

Abstract: Let T be a tree with m edges. Ringel conjectured that the complete graph K2m+1 decomposes into copies of T; such a partition is a T-decomposition. Häggkvist posed the more general conjecture that every 2m-regular graph has a T-decomposition. Graham and Häggkvist conjectured that also every m-regular bipartite graph has a T-decomposition. Later work by Snevily and by Avgustinovitch obtained T-decompositions for various classes of 2m-regular graphs and m-regular bipartite graphs. We extend their ideas to enlarge the families of 2m-regular graphs and m-regular bipartite graphs that are known to have T-decompositions. The new families consist of various cartesian products of regular graphs. (This is joint work with Alexandr Kostochka.)

4:00 pm in 245 Altgeld Hall,Tuesday, December 7, 2004

Modeling Spontaneous Activity in Developing Neural Systems

John Rinzel (Center for Neural Science and Courant Institute of Mathematical Sciences, New York University)

Abstract: Many neural circuits are hyperexcitable and show spontaneous oscillations during early development. Such activity may be important for establishing meaningful architecture. We formulate and study models for the slow episodic population rhythms that are seen in chick embryonic spinal cord. Novel features: at this development stage even the neurotransmitter for inhibition leads to functional excitation, the inter-episode periods are very long (time scale, mins), the slow process that regulates the rhythm’s phases is depression of synaptic coupling rather than an intrinsic cellular property. We use mean field models for the population firing rate in a recurrent network of excitatory-coupled cells. Geometric singular perturbation methods are used to analyze the models. Theoretical predictions and experimental confirmations will be described.

J Tabak, W Senn, MJ O'Donovan, J Rinzel: Modeling of spontaneous activity in developing spinal cord using activity-dependent depression in an excitatory network. J Neuroscience 20:3041-3056, 2000.

J Tabak, J Rinzel, M O’Donovan: The role of activity-dependent network depression in the expression and self-regulation of spontaneous activity in the developing spinal cord. J Neuroscience 21: 8966-8976, 2001.