| Monday September 24 |
Francis
Bonahon (USC): Farey, Ford and other stories. This is an expository talk on the beauty of the geometry and combinatorics of the Ford/Farey circle packing in the upper half-space. It should be accessible to a wide audience. |
| Monday October 1 |
No seminar this week |
| Monday October 8 |
Jian
He (USC): Symplectic field theory of subcritical Stein manifolds We will discuss how to define symplectic invariants for Stein manifolds. Under the assumption that the first Chern clas is zero, the cylindrical contact homology and the rational contact homology algebra of the boundary of a Stein domain is determined. The computation of these invariants also has applications for closed manifolds, in particular a monotone Kahler manifold with a subcritical polarization is uniruled. |
| Monday October 15 |
Nitya
Kitchloo (UC San Diego): Dominant K-theory. I will define an equivariant cohomology theory (called Dominant K-theory), for certain infinite dimensional groups which are amalgams of compact Lie groups. I will further show that the Dominant K-theory of a suitable universal space is related to a natural class of infinite dimensional representations of the group. Among the groups we study are Loop groups, and other Kac-Moody groups. |
| Monday October 22 |
Ko Honda (USC): Contact
structures, Heegaard Floer homology and triangulated categories |
| Friday
October 26 |
Hua
Bai (University of Georgia): Quantum dilogarithm matrices The quantum dilogarithm matrices appear, on one hand as the 6j-symbols associated by Kashaev to tetrahedra in his construction of invariants of links in 3-manifolds, on the other hand as the intertwining operators used in constructing local representations of the quantum Teichm\"uller space of Chekhov-Fock. We will discuss the fundamental functional relations of quantum dilogarithm matrices, such as the pentagon equation related with the 3-2 move, and a symmetry property related with the symmetry group of tetrahedron. |
| Monday October 29 |
Steve Kerckhoff (Stanford
University): CANCELLED. Or more precisely postponed to November 9. |
| Monday November 5 |
Paolo Ghiggini (CalTech): Giroux's
torsion and contact invariants Giroux's torsion was introduced a few years ago by Giroux to distinguish tight contact strucures on toroidal 3-manifolds. I will prove that contact structures with positive torsion have trivial Ozsváth-Szabó contact invariant, and I will show an operation which has the effects of increasing torsion changes the twisted Ozsváth-Szabó contact invariant. |
| Friday
November 9 |
Steve Kerckhoff (Stanford
University): A hyperbolic look at the 24-cell We begin with the (arithmetic) group of reflections in the walls of the 24-cell, realized as a 4-dimensional ideal hyperbolic polyhedron. Removing some faces and using some techniques borrowed from hyperbolic 3-manifolds, we deform it until it becomes a new finite volume, arithmetic reflection group. Lots of pictures. This is joint work with Pete Storm. |
| Monday November 12 |
No seminar this week |
| Monday November 19 |
Jason
Behrstock (Columbia University): Quasi-isometric rigidity of the
mapping class groups This talk will focus on recent work concerning the geometry of the mapping class group. One of the main results I'll discuss is that, up to finite groups, any finitely generated group quasi-isometric to a mapping class group is a mapping class group. This is joint work with Bruce Kleiner, Yair Minsky, and Lee Mosher. |
| Monday November 26 |
Robin Wilson (Cal Poly Pomona):
Almost normal bridge surfaces in knot complements. The analog of a Heegaard surface in the theory of knots and links is a bridge surface that decomposes the knot into two sets of unknotted arcs. Questions about the structure of Heegaard surfaces for 3-manifolds often correspond to questions about bridge surfaces for knots. We will discuss an analog to the result that any strongly irreducible Heegaard surface is isotopic to an almost normal one. In particular, we show that any weakly incompressible bridge surface for a knot is isotopic to an almost normal bridge surface. |
| Friday
November 30 |
François
Guéritaud (Université de Paris-Sud): Canonical
triangulations of Dehn fillings Epstein and Penner showed the canonical or "Delone" decomposition into ideal polyhedra is a complete invariant of cusped hyperbolic 3-manifolds. How does it behave under Dehn filling on some (but not all) of the cusps? We answer this question in what can be considered as the generic case. Joint work with Saul Schleimer (Warwick). |
| ***************************** Spring Semester 07 ***************************** | |
| Monday January 14 |
Thierry Barbot (ENS Lyon): Time
functions in constant curvature spacetimes A Lorentzian manifold is globally hyperbolic if it admits a proper time function, i.e. with compact spacelike fibers. I will present results in the last decade concerning the classification of globally hyperbolic spacetimes of constant curvature, and the existence/uniqueness problem of special time functions (e.g. with constant mean curvature, or constant Gauss curvature) on these spacetimes. I will also discuss the asymptotic properties of the induced metrics on the fibers near the origin of time. |
| Monday January 21 |
No seminar this week (University
holiday), but see Alex Lubotzky's
Whiteman Lectures on Tuesday and Thursday, and Danny Calegari's colloquium on
Wednesday. |
| Monday January 28 |
Linda Keen (Lehman College, CUNY): Palindromes
in two generator groups In a free group on two generators G=<A,B>, every element that can serve as a generator can be written either as a palindrome in A and B or as a product of palindromes in a canonical way. This has geometric implications when G is represented as a group of isometries of hyperbolic three-space. We will prove this and discuss some of these implications. This is based on joint work with Jane Gilman. |
| Monday February 4 |
Double header! 3:30 Emille Davie (UC Santa Barbara): Right-Veering Pseudo-Anosov Mapping Classes Right-veering surface homeomorphisms play an interesting part in 3-dimensional contact topology. We will discuss an algorithm driven by the unreduced Burau representation and an invariant known as the fractional Dehn twist coefficient which allows us to identify a pseudo-Anosov representative of an element of the braid group on 3 strings as right-veering, left-veering or neither. 4:30 Motoo Tange (Osaka University): Knot descriptions derived from lens space surgery on homology spheres There exist many knots in the Poincaré homology sphere which yield lens spaces by positive integral Dehn surgery. Extending these surgeries to variable homology spheres, I will give the knot description in the homology spheres yielding lens spaces. |
| Monday February 11 |
Christelle
Moulin (École Polytechnique Fédérale de Lausanne,
and USC): Simple closed curves on hyperbolic surfaces
(following M. Mirzakhani) |
| Thursday
February 14 |
Michael
Hutchings (UC Berkeley): Foundations of embedded contact
homology The previous day's colloquium will introduce the basic idea of embedded contact homology. This talk will explain some of the details. In particular: What is the mysterious ECH index that makes the whole thing work? Why is the differential really a differential? How can one see that ECH depends only on the contact structure? |
| Monday February 18 |
University holiday |
| Monday February 25 |
Kevin Walker (Microsoft Station Q, UC Santa
Barbara): Tight contact structures and TQFTs There is a well-developed "pictures mod local relations" machinery for constructing topological quantum field theories. This machinery places the Witten-Reshetikhin-Turaev, Turaev-Viro and Dijkgraaf-Witten TQFTs in a uniform context. One can can plug tight contact structures (together with the cut/paste techniques of Honda and others) into this machinery. The result is a new viewpoint on results in contact topology from the last decade, as well as some new, purely topological, questions about contact topology. |
| Monday March 31 |
Richard
Siefring (Stanford University): Intersection theory of
punctured pseudoholomorphic curves Positivity of intersections for pseudoholomorphic curves and the resulting topological controls on intersections and embeddedness of closed pseudoholomorphic curves have been useful for applications of pseudoholomorphic curves to 4-dimensional symplectic topology. In a 4-dimensional symplectic cobordism, a precise description of the asymptotic behavior of punctured pseudoholomorphic curves allows for similar algebraic-topological controls on intersections and embeddedness despite the fact that the intersection number of two curves is no longer a homotopy-invariant quantity. In this talk we will explain these algebraic controls, along with the relevant results about the asymptotic behavior of curves near a puncture. Time permitting, we will also present an application to embedded contact homology. |
| Monday April 7 |
Sergey Tabachnikov (Pennsylvania State
University):
Configuration space of plane polygons, sub-Riemannian geometry and
periodic orbits of inner and outer billiards In the theory of mathematical billiards, the following two problems are distinguished by their importance and difficulty. The first is Brikhoff's conjecture stating that the only integrable plane billiard is the billiard inside an ellipse. The second is Ivrii's conjecture that the set of periodic billiard trajectories is a null set (with respect tot he natural invariant measure). I shall discuss the recent approach to both problems based on ideas of sub-Riemannian geometry. In particular, one can construct inner and outer billiards that possess invariant curves consisting of N-periodic points (such as figures of constant width, for N=2) and to prove that the set of 3-periodic points has zero measure (the case of general period is open). Based on recent work of Yu. Baryshnikov, D. Genin, J. Landsberg, A. Tumanov, V. Zharnitsky and the speaker. |
| Monday April 14 |
Dongping
Zhuang (CalTech): Irrational stable commutator lengths in
finitely presented groups We give examples of finitely presented groups containing elements with irrational(in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse and others. |
| Monday April 21 |
Nathan Geer (Georgia Tech): Modified
quantum dimensions and re-normalized link invariants I will discuss a renormalization of the Reshetikhin-Turaev quantum invariants, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the genuine quantum dimensions. More interestingly I will discuss two examples where the genuine quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras defined by Patureau-Mirand and myself. These invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots. This work is joint with B. Patureau-Mirand and V. Turaev. |