Caltech Geometry and Topology Seminar Fall 2003 

Time: Fri 3pm (Except as noted)

Place: Sloan 257 (Except as noted)

Date Speaker Organization Title Abstract
3 October

 (Joint Analysis/ Geometry and Topology seminar)

 Sa'ar Hersonsky Ben-Gurion University / Caltech Boundary value problems and tiling by squares  
10 October Daniel Groves Caltech Dehn functions and mapping tori of free group automorphisms

Dehn functions are the most natural measure of the complexity of the word problem for finitely presented groups. I will define, and motivate the Dehn function, concentrating particularly on the linear and quadratic cases.

I will then discuss free group automorphisms, and particularly mapping tori. I will talk about why they have a quadratic or linear Dehn function, why I think this is interesting, and where to go from here.
31 October Hee Oh Caltech Lattice action on a symmetric space and its boundary Let $X$ be a Riemannian symmetric space of non-compact type and $X(\infty)$ its geometric boundary. For a lattice $\Gamma$ in the isometry group of $X$, we discuss the distribution of $\Gamma$ orbits on $X\times X(\infty)$. This is a joint work with Alexander Gorodnik.
7 November 4pm (note time change) Marty Scharlemann UCSB Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting An updated proof of a 1933 theorem of Goeritz, exhibiting a finite set of generators for the group of automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. The group is analyzed via its action on a certain connected 2-complex.
11 November 10:30am Manfred Einsiedler University of Washington Invariant measures and the set of exceptions to Littlewood's conjecture In a recent joint work with A. Katok and E. Lindenstrauss we classified the measures on SL(k,R)/ SL(k, Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. I will highlight a few steps of this proof and sketch in greater detail the connection to Littlewood's conjecture, and how to the deduce that the set of exceptions has Hausdorff dimension zero.
14 November Jim Hoste Pitzer College Trace fields, A-polynomials, and commensurability classes of 2-bridge knots Two-bridge knots have particularly nice fundamental groups that allow computations of various hyperbolic invariants to be undertaken. Especially for twist knots, and a few other infinite families of 2-bridge knots, the degree of the cusp and trace fields, and recursive formulas for the A-polynomials can be obtained. A criteria of Calegari and Dunfield regarding commensurability with fibered knots can also be applied in special cases.
20 November 3pm-5pm Sloan 151 Nicolas Monod University of Chicago Orbit equivalence rigidity Geometric group theory leads naturally to the notion of Measure Equivalence. This concept generalizes the classical Orbit Equivalence studied in ergodic theory. I will introduce these topics and present new superrigidity statements. This illustrates a new approach initiated in collaboration with Y. Shalom.
21 November Kasra Rafi UCSB Continued fraction expansion for foliations on surfaces The sequence of simple closed curves that are the best approximating a foliation on a Riemann surface is in many ways analogous to the continued fraction expansion of a real number. We study geodesics in Teichmüller space and foliations on a surface using the combinatorics of this sequence.
1 December 3pm-5pm Jason Behrestock SUNY Stony Brook Asymptotic geometry of the mapping class group and Teichmüller space I will discuss some recent progress towards understanding the asymptotic geometry of mapping class groups. Using techniques involving the complex of curves, the first part of my talk will explain some properties of maps from the mapping class group to the curve complex of a subsurface X, which are closely related to the map taking a curve in S to its intersection with X. The second half will describe how the topology of the asymptotic cone of the mapping class group is encoded by these maps and discuss how my description of the asymptotic cone computes the geometric rank of the mapping class group. I will also compute the geometric rank of Teichmüller space with the Weil-Petersson metric.
5 December Paul Norbury University of Melbourne Closed geodesics on incomplete surfaces Motivated by the question of when a Heegaard surface is isotopic to a minimal surface, we consider the problem of finding embedded closed geodesics on the two-sphere equipped with an incomplete metric.
22 December, 2pm Alexander Gorodnik University of Michigan Oppenheim conjecture and generalizations I discuss some old and new results related to the Oppenheim conjecture on values of irrational quadratic forms at integer points. This conjecture was proved by Margulis using the methods of topological dynamics on homogeneous spaces of Lie group. I give a gentle introduction to the ideas of the proof, derive a generalization of the Oppenheim conjecture for pairs consisting of linear and quadratic forms, and discuss open problems in the area. I also talk about the joint work with Nevo and Weiss on asymptotic distribution of values of quadratic forms on integer frames.
Back to main seminar page.