UIUC Dept. of Mathematics Seminar Calendar

    September 2009          October 2009          November 2009    
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Tuesday, October 20, 2009

Topology seminar
11:00 am in 241 Altgeld Hall, Tuesday, October 20, 2009

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Submitted by bertg.

Julie Bergner (UC Riverside)
Algebraic theories and (\infty, n)-categories
Abstract: One approach to the comparison of simplicial monoids and Segal monoids, or diagrams of simplicial sets which look like simplicial monoids but only up to homotopy, makes use of the algebraic theory of monoids. One can then construct more complicated algebraic theories in order to extend this comparison to more general simplicial categories and Segal categories with a given fixed object set. This approach becomes extremely useful in the most general comparison of simplicial categories and Segal categories, two of the models for (\infty, 1)-categories. Thus, it is expected that having algebraic theories corresponding to n-categories with a fixed set of objects would be helpful in comparing analogous models for (\infty, n)-categories. In this talk we'll look at work in progress in this direction.

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, October 20, 2009

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Submitted by w-henson.

Yevgeniy Gordon (EIU)
What does G. Birkhoff's Ergodic Theorem mean for very big finite spaces? (part II)
Abstract: The trivial proof of the ergodic theorem for a finite set X and a permutation s of X shows that for an arbitrary real-valued function f on X, the sequence of ergodic means A_n(f,s) stabilizes for large n. We show that if X is a very big finite set and f(x) is much less than the size of X for almost all x, then A_n(f,s) stabilizes for a significantly long segment of big numbers n that are, however, much less than the size of X. This statement has a natural rigorous formulation in terms of nonstandard analysis, which is, in fact, equivalent to the ergodic theorem for infinite probability spaces. Its standard formulation in terms of sequences of finite probability spaces is complicated. We discuss also a new notion of approximation of dynamical systems by finite dynamical systems, whose definition in terms of nonstandard analysis is much easier and much more natural than in classical terms. This is joint work with Lev Glebsky and Ward Henson.

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, October 20, 2009

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Submitted by jarouse.

Bruce Reznick (UIUC Math)
Sums of two cubes
Abstract: This will include a discussion of some very old formulas of Viete (1591) and Euler-Binet on equal sums of two cubes of rational numbers, polynomials and rational functions over C. A simple necessary and sufficient condition will be given for a polynomial to be a sum of two cubes of polynomials; for example, xy(x^4-y^4) has six essentially different representations as a sum of two cubes of quadratic forms and x^6 + y^6 has four such representations. There are some analogues to the addition of points on elliptic curves. The speaker will present a lot of questions which he cannot answer.

Probability Seminar
2:00 pm in 347 Altgeld Hall, Tuesday, October 20, 2009

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Submitted by pankikim.

Panki Kim [email] (Seoul National University, UIUC)
The boundary Haranck principle of the independent sum of Brownian motion and symmetric stable process.
Abstract: In this talk, we consider the family of pseudo differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$. We establish a uniform boundary Harnack principle with explicit boundary decay rate for nonnegative functions which are harmonic with respect to $\Delta +b = \Delta^{\alpha/2}$ (or equivalently, the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with constant multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, October 20, 2009

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Submitted by west.

Paul Wenger (UIUC Math)
Uniquely H-saturated graphs
Abstract: For a fixed graph H, a graph G is uniquely H-saturated if G does not contain H as a subgraph, but the addition to G of any missing edge completes exactly one copy of H in G. We examine uniquely H-saturated graphs when H is a path or a small cycle. In particular, we prove that there are only nine uniquely C₄-saturated graphs, all of which have at most nine vertices. Doing so includes an algebraic proof using eigenvalues of graphs that there is only one such graph with girth 5. This is joint work with Joshua Cooper, Timothy LeSaulnier and Douglas B. West.