Geometry/Topology Seminar
Spring 2012
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday March 29 at 3pm in E308
 Dave Witte Morris, University of Lethbridge, Canada
 Strictly convex norms on amenable groups

Abstract: It is obvious that the usual Euclidean norm is strictly convex, by which we mean that, for all x and all nonzero y, either x + y > x, or x  y > x. We will discuss the existence of such a norm on an abstract (countable) group G. A sufficient condition is the existence of a faithful action of G by orientationpreserving homeomorphisms of the real line. No examples are known to show that this is not a necessary condition, and we will combine some elementary measure theory and dynamics with the theory of orderable groups to show that the condition is indeed necessary if G is amenable. This is joint work with Peter Linnell of Virginia Tech.

 Tuesday April 10 at 3pm in E308
 Mark McLean, MIT
 An upper bound for a certain Lagrangian capacity

Abstract: Every Lagrangian inside a symplectic manifold has a neighbourhood symplectomorphic to an open subset of its cotangent bundle. One can ask, how big is this neighbourhood? One way of bounding the size of this neighbourhood is to find the largest symplectically embedded ball B(r) of radius r so that the intersection of our Lagrangian with B(r) is linear inside B(r) and passing through 0. If our Lagrangian is contained in a Stein manifold then we show that r^2 is bounded above by 8/pi times the displacement energy. We use wrapped Floer cohomology in a new way to prove this. This is joint work with Strom Borman.

 Thursday April 19 at 3pm in E308
 JeanFrancois Lafont, Ohio State
 Isomorphism vs. commensurability for a class of finitely presented groups.

Abstract: I will construct a class of finitely presented groups
within which the isomorphism problem is unsolvable, but the
commensurability problem is solvable. This is based on joint
work with Goulnara Arzhantseva and Ashot Minasyan.

 Monday April 23 at 4pm in Ry251
 Danny Calegari, Caltech
 Namboodiri Lecture I: Topological minimal surface theory and stable commutator length

Abstract: If G is a group and [G,G] is its commutator subgroup, the commutator length of g in [G,G] is the least number of commutators whose product is g, and the stable commutator length is the growth rate of the commutator length of powers of g. Stable commutator length is the algebraic analogue of (relative) 2dimensional GromovThurston norm. We discuss the Rationality and Rigidity theorems for the stable commutator length norm in free groups, and some applications to the construction of surface subgroups and to symplectic rigidity.

 Tuesday April 24 at 4:30pm in E206
 Danny Calegari, Caltech
 Namboodiri Lecture II: Dynamics, integer programming, and surgery

Abstract: The phenomenon of Arnold tongues is a wellknown example of phase locking of coupled nonlinear oscillators. The frequency spectrum of such nonlinear systems obey power laws; such power laws also turn up in integer programming, for example where one considers the Hermite normal form of a random integer matrix. We discuss how stable commutator length in surgery families is parameterized by families of integer programming problems. On the dynamics side, we show how similar power laws arise in the nonlinear ``character varieties'' of the group of homeomorphisms of the circle known as ziggurats.

 Wednesday April 25 at 4pm in Ry251
 Danny Calegari, Caltech
 Namboodiri Lecture III: Statistics, concentration, and compression

Abstract: We discuss the statistical distribution of stable commutator length in various classes of groups, and some applications. For certain classes of groups (e.g. central extensions of lattices in Sp(2n,R)) stable commutator length is distributed like distance to the origin for a random walk in a finite dimensional Euclidean space. For other classes of groups (e.g. hyperbolic groups, braid groups) there is a concentration of values, clustered around some fixed scale Cn/log(n) where the constant C should conjecturally be derived in a simple manner from the (growth) entropy. This concentration should be thought of as a "random" analogue of the phenomenon of Mostow rigidity for hyperbolic manifolds. Finally, the growth rate of stable commutator length is an obstruction to the existence of (nonelementary) homomorphisms to hyperbolic groups, or actions on certain hyperbolic spaces.

 Tuesday May 1 at 3pm in E308
 Frol Zapolsky, Munich
 Geometry of contactomorphism groups and contact rigidity

Abstract: I will use spectral invariants for Legendrians in jet spaces to establish a variety of results. These include (1) the existence of a biinvariant partial order and of a biinvariant integervalued metric on the identity component of the contactomorphism group of the standard contact T^*N \times S^1, (2) contact rigidity, and (3) the existence and multiplicity of orbits of contact flows with Legendrian boundary conditions.

 Thursday May 3 at 3pm in E308
 Sobhan Seyfaddini, Berkeley
 C^0 limits of Hamiltonian paths and spectral invariants

Abstract: After briefly reviewing spectral invariants, I will write down an
estimate, which under certain assumptions, relates the spectral invariants
of a Hamiltonian to the C^0distance of its flow from the identity. I will
also show that unlike the Hofer norm the spectral norm is C^0continuous
on surfaces. Time permitting I will present a few applications to Hofer
geometry.

 Monday May 7 at 4pm in Ry251
 Demetrios Christodoulou, ETH Zurich
 ZygmundCalderon Lecture I: Hyperbolic p.d.e. and Lorentzian geometry

Abstract: In the first part of the lecture I shall begin with a discussion of how EulerLagrange systems of partial differential equations of hyperbolic type arise in classical continuum physics. I shall proceed to discuss the causal structure of spacetime, that is, of the manifold of independent variables, defined by a solution of the system of equations, and how Lorentzian geometry comes into play. In the second part of the lecture I shall focus on what are perhaps the two most important EulerLagrange systems of partial differential equations of hyperbolic type, namely the Euler equations of fluid mechanics, and the Einstein equations of general relativity. The Euler equations govern the motion of a perfect fluid and were first formulated in 1756, but despite the lapse of two and a half centuries, we are still far from adequately understanding the observed phenomena which are supposed to lie within their domain of validity. Among these phenomena are the formation and evolution of shocks in compressible fluids. This shall be the topic of my second and third lecture. The Einstein equations of general relativity, formulated in 1915, govern the geometry of physical spacetime itself, gravitation being the manifestation of spacetime curvature. I shall briefly discuss the global stability of Minkowski spacetime, the trivial solution of the Einstein equations, proved in my joint work with Klainerman (published in 1993), and contrast this with what happens in the case of the Euler equations, where there is a family of trivial solutions, the constant states. A basic notion in general relativity is the concept of a trapped surface. The existence of such a surface in a spacetime implies according to a 1965 theorem of Penrose that the spacetime must come to an end. It also implies that there is a region of the spacetime, called a black hole, which is inaccessible to observation from infinity. The formation of trapped surfaces shall be the topic of my fourth lecture.

 Tuesday May 8 at 3pm in E308
 Yael Karshon, Toronto
 Completely integrable torus actions on complex manifolds with fixed points

Abstract: We show that if a holomorphic ndimensional compact torus action on a compact connected complex manifold of complex dimension n
has a fixed point then the manifold is equivariantly biholomorphic
to a smooth toric variety. This is joint work with Hiroaki Ishida.

 Tuesday May 8 at 4:30pm in E206
 Demetrios Christodoulou, ETH Zurich
 ZygmundCalderon Lecture II: The analysis of shock formation in 3dimensional fluids, Part A

Abstract: In 2007 I published a monograph which treated the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. In this monograph I considered initial data which outside a sphere coincide with the data corresponding to a constant state. Under a suitable restriction on the size of the initial departure from the constant state, I established theorems which gave a complete description of the maximal classical development. In particular, I showed that the boundary of the domain of the maximal classical development has a singular part where the inverse density of the wave fronts vanishes, signalling shock formation. In fact, the theorems which I established gave a complete picture of shock formation in threedimensional fluids. In my lectures I shall give a simplified presentation of these results, assuming from the outset that the initial conditions are irrotational. The basic geometric concept on which the analysis is based is that of the acoustical spacetime manifold. The analysis features the interplay of the original system of equations with another system, the acoustical structure equations, which governs the causal structure of the acoustical manifold. The acoustical geometry degenerates as shocks form, nevertheless things remain smooth relative to a different differential structure, which is what permits a complete analysis of the singular boundary.

 Wednesday May 9 at 4pm in Ry251
 Demetrios Christodoulou, ETH Zurich
 ZygmundCalderon Lecture III: The analysis of shock formation in 3dimensional fluids, Part B

Abstract: (See above)

 Thursday May 10 at 3pm in Ry251
 Demetrios Christodoulou, ETH Zurich
 ZygmundCalderon Lecture IV: The short pulse method and the formation of trapped surfaces in general relativity

Abstract: In 1965 Penrose introduced the fundamental concept of a trapped surface, on the basis of which he proved a theorem which asserts that a spacetime containing such a surface must be incomplete. The presence of a trapped surface implies, moreover, that there is a region of spacetime, the black hole, which is inaccessible to observation from infinity. A major challenge since that time had been to find out how trapped surfaces actually form, by analyzing the dynamics of gravitational collapse. In a monograph published in 2009 I achieved this aim by establishing the formation of trapped surfaces in pure general relativity through the focusing of incoming gravitational waves. The theorems proved therein constitute the first foray into the longtime dynamics of general relativity in the large, that is, when the initial data are no longer confined to a suitable neighborhood of trivial data. The main new method which this work introduces, the short pulse method, applies to general systems of EulerLagrange equations of hyperbolic type, and provides means to tackle problems which have hitherto been inaccessible. The method capitalizes on the assumption that the initial data, although smooth, change abruptly as we cross a certain surface, so there is a small parameter which corresponds to the distance within which the change is effected. A calculus is built in which this small parameter everywhere enters. This calculus is used to demonstrate that when the parameter in question is suitably small we have long time existence independently of the size of the initial data. And in the case of the Einstein equations, when this size is suitably large then trapped surfaces eventually form.

 Thursday May 17 at 3pm in E308
 Ben McReynolds, Purdue
 Fine structure in spectral sets

Abstract: Often the set of lengths of closed geodesics on a closed Riemannian manifold provide us with a discrete subset of the reals. I will take this view and motivate some questions and conjectures on when this set has some additional structure. This talk is suitable for early graduate students. Most of the material is joint with Jean Lafont.

 Thursday May 24 at 3pm in E308
 Tom Church, Stanford
 A stability conjecture for the unstable cohomology of mapping class groups, SL_n(Z), and Aut(F_n)

Abstract: For each of the sequences of groups in the title, the ith rational cohomology is known to be independent of n in a linear range n greater than or equal to Ci.
Furthermore, this "stable cohomology" has been explicitly computed in
each case. In contrast, very little is known about the unstable
cohomology. In this talk I will explain a conjecture on a new kind of
stability in the cohomology of these groups. These conjectures
concern the unstable cohomology, in a range near the "top dimension".
One key ingredient is a version of Poincare duality for these groups
based on the topology of the curve complex and the algebra of modular
symbols. I'll finish by describing the evidence we have for these
conjectures, including some new vanishing theorems for the top
cohomology of M_g and of SL_n(Z). Joint work with Benson Farb and
Andrew Putman.

 Friday June 8 at 3pm in E308
 Robert Penner, Aarhus University/Caltech
 Cell decomposition of the space of stable curves

Abstract:
Fatgraphs index an ideal cell decomposition of open Teichmueller or Riemann moduli space, and an elaboration of them as "partially paired possibly punctured fatgraphs" is shown to likewise index a cell decomposition of augmented Teichmueller or compactified Riemann moduli space. We shall explain these new combinatorial gadgets precisely and discuss aspects of the proof.

 Monday June 11 at 2pm in E308
 Robert Penner, Aarhus University/Caltech
 The geometrical nature of protein hydrogen bonds

Abstract: Recent work has analyzed proteins using combinatorial techniques adapted from the study of moduli spaces. Specifically, a 3d rotation can be associated to each protein hydrogen bond, and these data already embedded in the Protein Data Bank can be analyzed. Nature is economical in exploiting only a small part of the conformational possibilities (hence new constraints for simulation and design) and hydrogen bonds accordingly classified. The geometrically exotic hydrogen bonds have uncanny abilities to predict protein active and functional sites from 3d structure.
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