UIUC Dept. of Mathematics Seminar Calendar

     January 2012          February 2012            March 2012     
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Tuesday, February 14, 2012

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, February 14, 2012

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Submitted by jathreya.

Albert Fisher (University of Sao Paolo)
Invariant measures for adic transformations on Bratteli diagrams
Abstract: Vershik's adic transformations are a class of combinatorially defined maps that can be used to topologically and measure-theoretically model a wide variety of dynamical systems, including substitution dynamical systems, cutting-and-stacking constructions and interval exchange transformations. They act on the path space of a Bratteli diagram, defined by a sequence of nonnegative integer matrices and so generalizing subshifts of finite type to nonstationary combinatorics. Let us say a matrix sequence is primitive if for all $k$ there exists $n>k$ such that $M_k... M_n> 0$. Primitivity implies minimality (that every orbit is dense) for adic transformations, but in the nonstationary case, unique ergodicity (that there is a unique unvariant probability measure) does not always follow. In recent work we classify the invariant Borel measures which are finite positive on the path space of some sub-Bratteli diagram, for the bounded alphabet but not necessarily primitive case. This includes some interesting (and naturally occuring) measures which are infinite on every open subset. Our results extend theorems of Bezuglyi, Kwiatkowski, Medynets and Solomyak. This is joint work with Marina Talet, Universite de Provence.

Topology Seminar
11:00 am in 243 Altgeld Hall, Tuesday, February 14, 2012

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Thomas Kragh (MIT)
Stable homotopy types and orientations in Hamiltonian Floer theory
Abstract: I will start by outlining the basic ideas in Morse theory and Conley index theory. Then I will describe Hamiltonian Floer homology using infinite dimensional Morse homology. I will then describe the ideas of finite dimensional approximations, and discuss existence and uniqueness.

For cotangent bundles these finite dimensional approximations exists canonically - but are not natural. I will explain this in more detail for a nearby Lagrangian, and describe how this lead to new insights into the coherent orientations in Floer homology. If time permits I will talk about some generalizations of the finite dimensional approximations and relations to complex periodic cobordism.

Geometry Seminr
2:00 pm in 243 Altgeld Hall, Tuesday, February 14, 2012

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Joseph Rosenblatt (UIUC )
Distribution of parameters in optimal approximations
Abstract: Given a compact $d$-dimensional manifold with boundary, the computation of its $d$-dimensional volume can be carried out by approximating it by simpler manifolds for which the volume calculation is easy. This is the classical approach that is often used to define the volume of the manifold. The choice of optimal approximations for this purpose raises some interesting geometrical issues. Here is a very simple case. Take an increasing smooth function $f$ on $[0,1]$ and choose a partition $P_n$ of $[0,1]$ containing $n$ points for which the lower Riemann sum gives the best approximation of $\int_0^1 f(s)\, ds$ among all possible partitions containing just $n$ points. How are the points of $P_n$ distributed? For example, if $P_n =(x_k:k=1,\dots,n)$, and we take the discrete measure $\mu_n = \frac 1n\sum_{k=1}^n \delta_{x_k}$, does the sequence $(\mu_n)$ converge weakly as $n\to \infty$? If so, what is its limit?

Algebraic Geometry
3:00 pm in 243 Altgeld Hall, Tuesday, February 14, 2012

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Submitted by schenck.

Wenbo Niu [email] (Purdue)
Asymptotic Regularity of Ideal Sheaves
Abstract: Let $I$ be an ideal sheaf on $\mathbb{P}^n$ . Associated to $I$ there are three elementary invariants: the invariant $s$ which measures the positivity of $I$, the minimal number $d$ such that $I(d)$ is generated by its global sections, and the Castelnuovo-Mumford regularity reg $I$. In general one has $s \leq d\leq \mbox{reg }I$. If we consider the asymptotic behavior of the regularity of $I$, that is the regularity of $I^p$ when $p$ is sufficiently large, then we could have a clear picture involving these invariants. We will talk about two main theorems in this direction. The first one is the asymptotic regularity of $I$ is bounded by linear functions as $sp\leq \mbox{reg }I^p\leq sp+e$, where $e$ is a constant. The second one is that if $s=d$, i.e., $s$ reaches its maximal value, then for $p$ large enough reg $I^p=dp+e$ for some positive constant $e$.

Mathematics Colloquium - Special Lecture 2011-12
4:00 pm in 245 Altgeld Hall, Tuesday, February 14, 2012

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Submitted by kapovich.

P. Di Francesco (Institut de Physique Theorique, CEA Saclay and Mathematical Sciences Research Institute, Berkeley, CA)
Discrete Integrable Systems and Cluster Algebras
Abstract: Recursive systems arising from integrable quantum spin chains, such as Q,T and Y-systems display remarkable combinatorial properties. These are actually part of a more general mathematical structure called Cluster Algebras, introduced by Fomin and Zelevinsky around 2000, and which has found a host of mathematical applications so far, ranging from the theory of total positivity, Teichm�ller space geometry, to the representation theory of quantum groups. A cluster algebra is a sort of dynamical system describing the mutation of a vector of data along the edges of an infinite tree, with rules guaranteeing that only Laurent polynomials of the initial data are generated. A longstanding conjecture of Fomin and Zelevinsky states that these have non-negative integer coefficients. In this talk, we will describe the very simple example of discrete integrable systems, and use their exact solutions in terms of paths on graphs or networks to explain this positive Laurent phenomenon. Non-commutative extensions will also be discussed.