UIUC Dept. of Mathematics Seminar Calendar

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Tuesday, October 25, 2011

Ergodic Theory
11:00 am in 347 Altgeld Hall, Tuesday, October 25, 2011

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Submitted by jathreya.

Florin Boca (Department of Mathematics, University of Illinois)
Lost on the modular surface ?
Abstract: Abstract: I will discuss some results and calculations concerning the limiting pair correlation of angles between geodesic rays [z,gz], g in PSL(2,Z), when z is a fixed point on the modular surface H/SL(2,Z). This is joint work with Alexandru Popa and Alexandru Zaharescu.

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, October 25, 2011

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Submitted by sabok.

Jay Williams (Rutgers)
Universal countable Borel quasi-orders
Abstract: Descriptive set theory gives us a framework for analyzing the relative complexity of quasi-orders (i.e. reflexive transitive relations) arising in many areas of mathematics, such as Turing reducibility of sets of natural numbers or embeddability of countable groups, using the notion of a Borel reduction. I will discuss a special class of quasi-orders, the countable Borel quasi-orders. There is a universal countable Borel quasi-order. After giving a couple of examples, I will discuss the Borel complexity of embeddability of finitely-generated groups. I will also discuss the more general case of embeddability of countable groups.

Geometry Seminar
2:00 pm in 243 Altgeld Hall, Tuesday, October 25, 2011

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Submitted by sba.

Spencer Dowdall (Department of Mathematics, University of Illinois)
Teichmuller distances via the the combinatorics of curves on surfaces
Abstract: Given a fixed topological surface S, one may consider the Teichmuller space of Riemann surface structures on S. This space has a rich and well-studied geometry (the Teichmuller metric) defined in terms of quasi-conformal mappings. One may also consider the Curve Complex of S, which is a combinatorial object consisting of all simple closed curves on S packaged into a nice geometric complex. It turns out these two seemingly unrelated objects have a lot in common. By simply choosing a short curve on a given Riemann surface, one obtains a natural map from Teichmuller space to the curve complex that captures a lot of geometric information. However, this map is blind to the vast "thin parts" of Teichmuller space, which is where a lot of the interesting and subtle geometry lies. In this talk I will describe how the geometry of these thin parts can also be understood via subsurface projections, giving a complete combinatorial interpretation of the Teichmuller distance. This will be primarily an expository talk about the work of Masur--Minsky and Rafi. If time permits, I'll indicate some recent results building on their work.

Algebra, Geometry and Combinatoric
2:00 pm in 345 Altgeld Hall, Tuesday, October 25, 2011

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Submitted by darayon2.

Ben Wyser (University of Georgia)
K-orbits on G/B, Richardson varieties, and a positive rule for (p,q)-Schubert constants
Abstract: For G a complex, reductive algebraic group, the fixed point subgroup of an involution of G is typically denoted K, and is referred to as a symmetric subgroup. K acts on the flag variety G/B (by left translations) with finitely many orbits. The geometry of such orbits and their closures is important in the infinite-dimensional representation theory of real forms of G.
One interesting example of a symmetric pair is (G,K) =(GL(p+q), GL(p) x GL(q)). Restricting attention to this example, I will discuss a recent result which establishes that a number of the K-orbit closures in this case coincide with certain Richardson varieties. When combined with a theorem of M. Brion on expressing the class of such an orbit closure in the Schubert basis, this observation implies a positive (in fact, multiplicity-free) rule for certain Schubert structure constants c_{u,v}^w --- those for which u,v form what I refer to as a "(p,q)-pair".

Analysis Seminar
2:00 pm in 241 Altgeld Hall, Tuesday, October 25, 2011

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Submitted by aimo.

Chris Kottke (Brown University)
Callias' Index Theorem and Magnetic Monopoles
Abstract: I will discuss two different extensions of the index theorem originally due to C. Callias and later generalized by N. Anghel and others, concerning operators on a complete Riemannian manifold of the form $D + i\Phi$, where $D$ is a Dirac operator and $\Phi$ is a family of self-adjoint invertible matrices. The first result is a pseudodifferential version of this index theorem, in the spirit of the $K$-theoretic proof of the Atiyah-Singer index theorem, for an appropriate class of pseudodifferential operators on asymptotically conic (asymptotically locally Euclidean) spaces. The second result is an extension to the case where $\Phi$ has constant rank nullspace bundle at infinity. Finally, I will show how these results can be applied to compute the dimension of the moduli space of magnetic monopoles on asymptotically conic manifolds.

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, October 25, 2011

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Submitted by west.

Bill Kinnersley (UIUC Math)
Chain games on posets
Abstract: In the chain game on a poset $P$, two players called "Maker" and "Breaker" take turns choosing elements of $P$. Maker wins by collecting all elements of some chain of prescribed size; Breaker wins by preventing this. We show that on a product of chains, Maker can build a chain of size $k - \lfloor{r/2}\rfloor$, where $k$ is the maximum size of a chain in the product and $r$ is the maximum size of a factor chain. Moreover, Breaker can prevent Maker from building any longer chain.

In the ordered chain game, we additionally require that Maker choose the elements of his chain in order. We study this game on the poset consisting of the bottom $k$ levels of the product of $d$ arbitrarily long chains. When $d = 2$, we show that Maker can build a chain of size $\lceil{2k/3}\rceil$ but cannot guarantee a longer chain. We further show that, when $d \ge 14$, Maker can build a chain hitting all levels; this result employs a winning strategy for Conway's Angel-Devil game. (Joint work with Dan Cranston, Kevin Milans, Greg Puleo, and Douglas West.)