Geometry and Topology Seminar 2012-2013
Anne Thomas (Sydney)
Divergence in right-angled Coxeter groups
Abstract: The divergence of a pair of geodesic rays emanating from a point is a measure of how quickly they are moving away from each other. In Euclidean space divergence is linear, while in hyperbolic space divergence is exponential. Gersten used this idea to define a quasi-isometry invariant for groups, also called divergence, which has been investigated for classes of groups including fundamental groups of 3-manifolds, mapping class groups and right-angled Artin groups. I will discuss joint work with Pallavi Dani on divergence in right-angled Coxeter groups (RACGs). We characterise 2-dimensional RACGs with quadratic divergence, and prove that for every positive integer d, there is a RACG with divergence polynomial of degree d.
Liviu Nicolaescu (Notre Dame)
Random Morse functions and spectral geometry
Abstract: I will discuss the distribution of critical values of a smooth random function on a compact m-dimensional Riemann manifold (M,g) described as a random superposition of eigenfunctions of the Laplacian. The notion of randomness that we use has a naturally built in small parameter $\varepsilon$, and we show that as $\varepsilon\to 0$ the distribution of critical values closely resemble the distribution of eigenvalues of certain random symmetric $(m+1)\times (m+1)$-matrices of the type introduced by E. Wigner in quantum mechanics. Additionally, I will explain how to recover the metric $g$ from statistical properties of the Hessians of the above random function.
Chris Hruska (UW Milwaukee)
Local topology of boundaries and isolated flats
Abstract: Swarup proved that every one-ended word hyperbolic group has a locally connected Gromov boundary. However for CAT(0) groups, non-locally connected boundaries are easy to construct. For instance the boundary of F_2 x Z is the suspension of a Cantor set.
In joint work with Kim Ruane, we have studied boundaries of CAT(0) spaces with isolated flats. If G acts properly, cocompactly on such a space X, we give a necessary and sufficient condition on G such that the boundary of X is locally connected. As a corollary, we deduce that such a group G is semistable at infinity.
Eriko Hironaka (FSU)
Small dilatation pseudo-Anosov mapping classes
The theory of fibered faces implies that pseudo-Anosov mapping classes with bounded normalized dilatation can be partitioned into a finite number of families with related dynamics. In this talk we discuss the problem of finding concrete description of the members of these families. One conjectural way generalizes a well-known sequence defined by Penner in '91. However, so far no known examples of this type come close to the smallest known accumulation point of normalized dilatations. In this talk we describe a different construction that uses mixed-sign Coxeter systems. A deformation of the simplest pseudo-Anosov braid monodromy can be obtained in this way, and hence this model does realize the smallest known accumulation point.
Yu-Shen Lin (Harvard)
Open Gromov-Witten Invariants on K3 surfaces and Wall-Crossing
Strominger-Yau-Zaslow conjecture suggests that the Ricci-flat metric on Calabi-Yau manifolds might be related to holomorphic discs. In this talk, I will define a new open Gromov-Witten invariants on elliptic K3 surfaces trying to explain this conjecture. The new invariant satisfies certain wall-crossing formula and multiple cover formula. I will also establish a tropical-holomorphic correspondence. Moreover, this invariant is expected to be equivalent to the generalized Donaldson-Thomas invariants in the hyperK\"ahler metric constructed by Gaiotto-Moore-Neitzke. If time allowed, I will talk about the connection with disks counting on Calabi-Yau 3-folds.
Sylvain Cappell (NYU)
Manuel Gonzalez Villa (Heidelberg)
The monodromy conjecture for plane meromorphic germs
Joint work with Ann Lemahieu (Lille). A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function, introduced by Gussein-Zade, Luengo and Melle, invite to consider the notion of topological zeta function for meromorphic germs and the corresponding monodromy conjecture. We try to motive these notions and discuss the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.
Emmy Murphy (MIT)
Exact Lagrangian immersions with few transverse self intersections
This talk will focus on the following question: supposing a smooth manifold immerses into C^n as an exact Lagrangian, what is the minimal number of transverse self-intersections necessary? Finding lower bounds on the number of intersections of two embedded Lagrangians is a central problem in symplectic topology which has seen much success; in contrast bounding the number of self-intersections of an exact Lagrangian immersion requires more advanced tools and the known results are far less general. We show that no Arnold-type lower bound exists for exact Lagrangian immersions by constructing examples with surprisingly few self-intersections. For example, we show that any three-manifold immerses as an exact Lagrangian in C^3 with a single transverse self-intersection. We also apply Lagrangian surgery to these immersions to give some interesting new examples of Lagrangian embeddings. (This is joint work of the speaker with T. Ekholm, Y. Eliashberg, and I. Smith.)
Yuan-qi Wang (UCSB)
Bessel Functions, Heat Kernel and the Conical Kahler-Ricci Flow
Inspired by Donaldson's program, we introduce the Kahler Ricci flow with conical singularities. The main part of this talk is to show that the conical Kahler Ricci flow exists for short time and for long time in a proper space. These existence results are highly related to heat kernel and Bessel functions. We will also discuss some easy applications of the conical Kahler Ricci flow in conical Kahler geometry.
Yong-Geun Oh (Wisconsin)
Analysis of contact instantons and contact homology
In this talk, we explain the analysis of the following system of (degenerate) elliptic equation $$ \overline \partial^\pi w = 0, \, d(w^*\lambda \circ j) = 0 $$ associated for each given contact triad $(Q,\lambda,J)$ on a contact manifold $(Q,\xi)$. (Such an equation was first introduced by Hofer.) We directly work with this equation on the contact manifold without involving the symplectization process. We explain the basic analytical ingredients towards the construction of moduli space of solutions, which we call contact instantons. I will indicate how one can define contact homology type invariants using such a moduli space, which is still in progress. The talk is partially based on the joint work with Rui Wang.
|September 21||Owen Sizemore (Wisconsin)||local|
|September 28||Mireille Boutin (Purdue)||Mari Beffa|
|October 5||Ben Schmidt (Michigan State)||Dymarz|
|October 12||Ian Biringer (Boston College)||Dymarz|
|October 19||Peng Gao (Simons Center for Geometry and Physics)||Wang|
|October 26||Jo Nelson (Wisconsin)||local|
|November 2||Jennifer Taback (Bowdoin)||Dymarz|
|November 9||Jenny Wilson (Chicago)||Ellenberg|
|November 16||Jonah Gaster (UIC)||Kent|
|November 30||Shinpei Baba (Caltech)||Kent|
|December 7||Kathryn Mann (Chicago)||Kent|
Owen Sizemore (Wisconsin)
Operator Algebra Techniques in Measureable Group Theory
Measurable group theory is the study of groups via their actions on measure spaces. While the classification for amenable groups was essentially complete by the early 1980's, progress for nonamenable groups has been slow to emerge. The last 15 years has seen a surge in the classification of ergodic actions of nonamenable groups, with methods coming from diverse areas. We will survey these new results, as well as, give an introduction to the operator algebra techniques that have been used.
Mireille Boutin (Purdue)
The Pascal Triangle of a discrete Image: definition, properties, and application to object segmentation
We define the Pascal Triangle of a discrete (gray scale) image as a pyramidal ar- rangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal Triangle. We study the induced action of some common group actions, such as translation, rotations, and reflections, and we propose simple tests for equivalence and self- equivalence for these group actions. The motivating application of this work is the problem of recognizing ”shapes” on images, for example characters, digits or simple graphics. Application to the MERGE project, in which we developed a fast method for segmenting hazardous material signs on a cellular phone, will be also discussed.
This is joint work with my graduate students Shanshan Huang and Andrew Haddad.
Ben Schmidt (Michigan State)
Three manifolds of constant vector curvature.
A Riemannian manifold M is said to have extremal curvature K if all sectional curvatures are bounded above by K or if all sectional curvatures are bounded below by K. A manifold with extremal curvature K has constant vector curvature K if every tangent vector to M belongs to a tangent plane of curvature K. For surfaces, having constant vector curvature is equivalent to having constant curvature. In dimension three, the eight Thurston geometries all have constant vector curvature. In this talk, I will discuss the classification of closed three manifolds with constant vector curvature. Based on joint work with Jon Wolfson.
Ian Biringer (Boston College)
Growth of Betti numbers and a probabilistic take on Gromov Hausdorff convergence
We will describe an asymptotic relationship between the volume and the Betti numbers of certain locally symmetric spaces. The proof uses an exciting new tool: a synthesis of Gromov-Hausdorff convergence of Riemannian manifolds and Benjamini-Schramm convergence from graph theory.
Peng Gao (Simons Center for Geometry and Physics)
string theory partition functions and geodesic spectrum
String theory partition functions often have nice modular properties, which is well understood within the context of representation theory of (supersymmetric extensions) of Virasoro algebra. However, many questions of physical importance are preferrably addressed when string theory is formulated in terms of non-linear sigma model on a Riemann surface with a Riemannian manifold as target space. Traditionally, physicists have studied such sigma models within the realm of perturbation theory, overlooking a large class of very natural critical points of the path integral, namely, closed geodesics on the target space Riemannian manifold. We propose how to take into account the effect of these critical points on the path integral, and initiate its study on Ricci flat targe spaces, such as the K3 surface.
Jo Nelson (Wisconsin)
Cylindrical contact homology as a well-defined homology theory? Part I
In this talk I will define all the concepts in the title, starting with what a contact manifold is. I will also explain how the heuristic arguments sketched in the literature since 1999 fail to define a homology theory and provide a foundation for a well-defined cylindrical contact homology, while still providing an invariant of the contact structure. A later talk will provide us with a large class of examples under which one can compute a well-defined version of cylindrical contact homology via a new approach the speaker developed for her thesis that is distinct and completely independent of previous specialized attempts.
Jennifer Taback (Bowdoin)
The geometry of twisted conjugacy classes in Diestel-Leader groups
The problem of computing the Reidemsieter number R(f) of a group automorphism f, that is, the number of f-twisted conjugacy classes, is related to questions in Lefschetz-Nielsen fixed point theory. We say a group has property R-infinity if every group automorphism has infinitely many twisted conjugacy classes. This property has been studied by Fel’shtyn, Gonzalves, Wong, Lustig, Levitt and others, and has applications outside of topology.
Twisted conjugacy classes in lamplighter groups are well understood both geometrically and algebraically. In particular the lamplighter group L_n does not have property R-infinity iff (n,6)=1. In this talk I will extend these results to Diestel-Leader groups with a surprisingly different conclusion. The family of Diestel-Leader groups provides a natural geometric generalization of the lamplighter groups. I will define these groups, as well as Diestel-Leader graphs and describe how these results include a computation of the automorphism group of this family. This is joint work with Melanie Stein and Peter Wong.
Jenny Wilson (Chicago)
FI-modules for Weyl groups
Earlier this year, Church, Ellenberg, and Farb developed a new framework for studying sequences of representations of the symmetric groups, using a concept they call an FI--module. I will give an overview of this theory, and describe how it generalizes to sequences of representations of the classical Weyl groups in Type B/C and D. The theory of FI--modules has provided a wealth of new results by numerous authors working in algebra, geometry, and topology. I will outline some of these results, including applications to configurations spaces and groups related to the braid group.
Jonah Gaster (UIC)
A Non-Injective Skinning Map with a Critical Point
Following Thurston, certain classes of 3-manifolds yield holomorphic maps on the Teichmuller spaces of their boundary components. Inspired by numerical evidence of Kent and Dumas, we present a negative result about the regularity of such maps. Namely, we construct a path of deformations of the hyperbolic structure on a genus-2 handlebody, with two rank-1 cusps. The presence of some extra symmetry yields information about the convex core, which is used to conclude some inequalities involving the extremal length of a certain symmetric curve family. The existence of a critical point for the associated skinning map follows.
Shinpei Baba (Caltech)
Grafting and complex projective structures
A complex projective structure is a certain geometric structure on a (real) surface, and it corresponds a representation from the fundamental group of the base surface into PSL(2,C). We discuss about a certain surgery operation, called a 2π–grafting, which produces a different projective structure, preserving its holonomy representation. This surgery is closely related to three-dimensional hyperbolic geometry.
Kathryn Mann (Chicago)
The group structure of diffeomorphism groups
Abstract: What is the relationship between manifolds and the structure of their diffeomorphism groups? On the positive side, a remarkable theorem of Filipkiewicz says that the group structure determines the manifold: if Diff(M) and Diff(N) are isomorphic, then M and N are diffeomorphic. On the negative side, we know little else. Could the group Diff(M) act by diffeomorphisms on M in nonstandard ways? Does the "size" of Diff(M) say anything about the complexity of M? Ghys asked if M and N are manifolds, and the group of compactly supported diffeomorphisms of N injects into the group of compactly supported diffeomorphisms of M, can the dimension of M be less than dim(N)? We'll discuss these and other questions, and answer these in the (already quite rich) case of dim(M)=1.