N+5th Southern California Topology Colloquium

Saturday, February 13, 2010

The Claremont Colleges Topology Seminar, with funding from Pitzer and Pomona Colleges, is pleased to sponsor the N+5th meeting of the SCTC, to be held at the Claremont Colleges on Saturday, February 13, 2010.


Speaker Time Title and Abstract
Danny Calegari

Cal Tech

11:00-12:00 Title: Scl, sails and surgery

Abstract: Given a group G and an element g \in [G, G], the commutator length of g, denoted cl(g), is the smallest number of commutators in G whose product is g, and the stable commutator length of g is the limit scl(g) := lim_{n \to \infty} cl(g^n)/n. Commutator length in a group extends in a natural way to a pseudo-norm on the real vector space of 1-boundaries (in group homology), and should be thought of as a kind of relative Gromov-Thurston norm. We show that the problem of computing stable commutator length in free products of abelian groups reduces to a (finite dimensional) integer programming problem. Moreover, certain families of elements in such groups (i.e. those obtained by surgery on some element in a bigger group) give rise to families of integer programming problems that are related in explicit ways. In particular one can use this to establish the existence of limit points in the range of scl in such groups, and produce elements whose stable commutator length is congruent to any rational number modulo the integers. This technology relates stable commutator length to the theory of multi-dimensional continued fractions, and Klein polyhedra, and suggests an interesting conjectural picture of scl in free groups. See [1, 2] for background and more details.


[1] D. Calegari, scl, MSJ Memoirs, 20. Mathematical Society of Japan, Tokyo, 2009

[2] D. Calegari, Scl, sails and surgery, preprint arXiv:0907.3541

lunch 12:00-1:30 various eateries are within walking distance
Cameron Gordon

Univ. Texas, Austin

1:30-2:30 Title: Knots with small rational genus

Abstract: If K is a rationally null-homologous knot in a 3-manifold M then the rational genus |K| of K is defined to be (roughly) the infimum of -\chi(S)/2n, over all S and n > 0, where S is an orientable surface in M whose boundary wraps n times homologically around K. If M is a homology sphere then this is essentially the genus of K. One can construct (for example by doing surgery on knots in S^3) knots in 3-manifolds with arbitrarily small rational genus; we show that such knots can be characterized geometrically. More precisely we show that there is a constant C > 0 such that if K in M has 0 < |K| < C then (M,K) belongs to one of a small number of classes: e.g. M is hyperbolic and K is the core of a Margulis tube, M is Seifert fibered and K is a fiber, K lies in a JSJ torus in M, etc. Conversely we show that there are pairs (M,K) in each of these classes with |K| arbitrarily small. This is joint work with Danny Calegari.

Ko Honda


2:45-3:45 Title: Contact structures and categorification

Abstract: In this talk I will introduce the contact category of a compact surface \Sigma. The contact category has enjoys many properties of a triangulated category (such as the octahedral axiom), and we can also construct its Grothendieck group. The braid group naturally acts on a version of the contact category, and categorifies several classical representations of the braid group, such as the Burau and Gassner representations. I will also discuss an exact functor from the contact category to a linear category via the Heegaard Floer homology/sutured Floer homology functor.

Ciprian Manolescu


4:00-5:00 Title: A combinatorial approach to four-manifold invariants

Abstract: Heegaard Floer theory, introduced by Ozsvath and Szabo, is a useful technique in low-dimensional topology: in particular, in four dimensions, it gives rise to invariants that are conjecturally the same as the Seiberg-Witten invariants, and share many of their properties. In this talk, I will describe an algorithm for computing the Heegaard Floer invariants of three- and four-manifolds (modulo 2). The algorithm is based on presenting the manifolds in terms of links in S^3, and then using grid diagrams to represent the links. The talk is based on joint work with P. Ozsvath and D. Thurston.

dinner 5:30 Beer, wine, food, and music!


The talks will be held in Millikan 134 on the Pomona College Campus, located on the North-East corner of 6th Street and College Avenue in Claremont. The lecture hall is on the ground floor at the west end of the building. Parking is free along College Avenue and nearby side streets.

Click here for a nice map of the Claremont Colleges. On the map, Milikan is numbered 8 (but so are other buildings!) Notice on the map, the indications to reach the 210 freeway, or the 10 freeway. To reach Pomona College from the 10 freeway, exit at Indian Hill, go north, turn right (east) on 6th street and then turn left (north) on College. To reach Pomona College from the 210 freeway, if traveling East, exit at Towne Avenue, turn right (south) on Towne, turn left (east) on Foothill Blvd, and turn right (south) onto College Avenue. If traveling West on the 210 freeway, exit at Baseline/Padua, turn right (west) onto Baseline, turn left (south) onto Padua at the first light, turn right (west) onto Foothill Blvd at the third light, turn left (south) onto College Avenue.


There are number of very nice places to eat lunch in the Claremont Village, all within easy walking distance to the lecture hall. Click here for a list of restaurants in Claremont. Note that this list is from the Chamber of Commerce and many of the restaurants on the list are not in the village. Of the ones in the East Village (east side of Indian Hill) we recommend, 42nd Street Bagel, Sacca's Mediterranean, Some Crust Bakery (good sandwiches, but not many places to sit), Pizza 'n' Such (slower), The Press (slower), Walters Restaurant (slower), Heroes (burgers, etc, but slower). In the West Village (west side of Indian Hill), try Le Pain Quotidien, or La Parolaccia osteria Italiana (slower). The places marked "slower" are nice restaurants, but NOT fast food places. If you want a nice meal eat there and allow time for lunch. Click here for a map of the Claremont Village showing the locations of shops and restaurants.

Dinner and Music

Following the talks we will have a buffet dinner at Seaver House located on the North-West corner of Bonita and College Ave. Music will be provided by the Emissaries of Misery.
History of the SCTC.