UIUC Dept. of Mathematics Seminar Calendar

     January 2012          February 2012            March 2012     
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
  1  2  3  4  5  6  7             1  2  3  4                1  2  3
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    4  5  6  7  8  9 10
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Tuesday, February 21, 2012

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, February 21, 2012

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Submitted by jathreya.

David Aulicino (Maryland)
Teichmueller Disks with Completely Degenerate Kontsevich-Zorich Spectrum
Abstract: The moduli space of genus $g$ Riemann surfaces is the space of all complex structures on a closed orientable surface of genus $g$ up to orientation preserving diffeomorphisms. The Teichmueller geodesic flow is the flow on the cotangent bundle of the Teichmueller space of surfaces defined by the direction of minimal dilatation and it descends to the cotangent bundle of the moduli space under the action of the mapping class group. It is well-known that the Lyapunov spectrum of this flow is determined by $g$ numbers $$1 = \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_g \geq 0.$$ The Kontsevich-Zorich conjecture, proven by Forni and Avila-Viana, showed that generically all the inequalities are strict with respect to the canonical absolutely continuous measures. However, Forni found an example of a measure on the genus three moduli space, and Forni-Matheus found a measure in genus four, with completely degenerate spectrum, i.e. $$1 = \lambda_1 > \lambda_2 = \cdots = \lambda_g = 0.$$ We prove that these are the only such measures in genus three and four. Furthermore, there are no such measures for $g=2$ and $g \geq 13$. Finally, if there are no square-tiled surfaces in genus five that determine a measure with completely degenerate spectrum, then there are no examples for $g \geq 5$.

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, February 21, 2012

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Submitted by ssolecki.

Ward Henson (UIUC)
Uncountably categorical Banach space structures
Abstract: Model theory is applied to (unit balls of) Banach spaces (and structures based on them) using the $[0,1]$-valued continuous version of first order logic. A theory $T$ of such structures is said to be $\kappa$-categorical if $T$ has a unique model of density $\kappa$. Work of Ben Yaacov and Shelah-Usvyatsov shows that Morley's Theorem holds in this context: if $T$ has a countable signature and is $\kappa$-categorical for some uncountable $\kappa$, then $T$ is $\kappa$-categorical for all uncountable $\kappa$. Known examples of uncountably categorical such structures are closely related to Hilbert space. After the speaker called attention to this phenomenon, Shelah and Usvyatsov investigated it and proved a remarkable result: if $M$ is a nonseparable Banach space structure (with countable signature) whose theory is uncountably categorical, then $M$ is prime over a Morley sequence that is an orthonormal Hilbert basis of length equal to the density of $M$. There is a wide gap between this result and verified examples of uncountably categorical Banach spaces, which leads to the question: can a stronger such result be proved, in which the connection to Hilbert space structure is clearly expressed in the geometric language of functional analysis? The main part of this talk will focus on some new examples of uncountably categorical Banach spaces that the speaker has studied. This is based on joint work with Yves Raynaud.

Differential Geometry
2:00 pm in 243 Altgeld Hall, Tuesday, February 21, 2012

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Submitted by palbin.

Fr�d�ric Rochon (Universit� du Qu�bec � Montr�al)
Asymptotics of the Ricci flow on quasi projective varieties
Abstract: PLEASE NOTE THE TIME CHANGE TO 2 PM! After a brief review on the Kahler-Ricci flow, we will consider Kahler metrics on quasiprojective varieties with a certain asymptotic behavior at infinity and show that this behavior is preserved as the metrics evolve according to the Kahler-Ricci flow. A key ingredient in our approach will be to compactify quasiprojective varieties by manifolds with corners. Time permitting, we will also discuss what happens to the limiting Kahler-Einstein metric when the flow converges. This is a joint work with Zhou Zhang.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, February 21, 2012

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Submitted by west.

Jozef Skokan (London School of Economics)
Monochromatic cycles in 2-edge-colored graphs
Abstract: The Ramsey number $r(H)$ of a graph $H$ is the smallest integer $n$ such that every 2-coloring of the edges of the complete graph on $n$ vertices contains a monochromatic copy of $H$. Schelp conjectured that if $H$ is a path or a cycle and $G$ is any graph on $r(H)$ vertices with minimum degree larger than $3r(H)/4$, then every 2-coloring of the edges of $G$ contains a monochromatic copy of $H$. In this talk, we shall discuss the ideas in our proof of the conjecture for long paths and cycles. (Joint work with B. Bollobas, F. Benevides, T. Luczak, A. Scott, and M. White.)