Geometry and Topology Seminar
![[Hopf fibration]](hopf.gif)
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Department of Mathematical Sciences
Geometry and Topology Seminar |
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Unless stated otherwise, the seminar takes place Thursdays at 2:50–3:50 pm in LN-2205 followed by refreshments served from 4:00–4:25 pm in the Anderson Memorial Reading Room.
Directions to the department. LN-2205 is on the same floor as the department offices. Stop at the department offices at LN-2200 and ask directions.
Some seminar speakers will also give a colloquium talk at 4:30 on the same day as the seminar talk. Titles of the colloquium talks are also given below where they apply.
This seminar is partly funded as one of Dean's Workshops in Harpur College (College of Arts and Sciences) at Binghamton University.
To receive announcements of seminar talks by email, please join the seminar's mailing list.
To subscribe to an on-line calendar with the seminar schedule, please choose a format: iCal or xml.
Abstract: Newton introduced divided differences to define interpolation polynomials fitting data points. More recently, Demazure and Bernstein-Gelfand-Gelfand have used divided difference operators when studying the cohomology of flag varieties. These operators may be defined more generally on Borel's equivariant cohomology. I shall begin with a brief introduction to equivariant cohomology, discuss the construction of the divided difference operators, and show how they can be used computationally. This talk is based on joint work with Reyer Sjamaar.
Abstract: The topological space obtained by removing a set of hyperplanes from a finite dimensional complex vector space has many interesting features. For instance, every such space is minimal, in the sense that it has the homotopy type of a CW complex with as many cells in every dimension as there are generators of the corresponding homology group. The central question about complements of hyperplane arrangements is to study to what extent the topology of the complement is determined by the combinatorics of the pattern of intersections of the hyperplanes.
The goal of the talk is to introduce some basics of the theory of hyperplane arrangements and to show that the above-mentioned minimality property can be deduced from purely combinatorial considerations (at least in the case where the hyperplanes are defined by real linear forms).
Abstract: We discuss a recent project with Kim Ruane and Genevieve Walsh, in which we prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.
Abstract: Thompson's group F is the group of all PL dyadic increasing homeomorphisms of the closed unit interval. This fascinating (finitely presented) group has relevance in a number of areas of mathematics, and has been widely studied in recent years. I will briefly introduce F and describe some of its known properties. Then I will discuss the following Theorem: For each n≥0 there is a subgroup of F of type Fn which is not of type Fn+1. (The properties "type Fn" are the topological finiteness properties of groups: a group has type F1 if it is finitely generated, has type F2 if it is finitely presented, etc.) The proof involves the Bieri-Neumann-Strebel-Renz invariants of groups; we have determined these for F, and have proved a general product formula for these invariants. I will explain how the combination of these two ideas yields the desired subgroups of F. This is joint work with Robert Bieri and Dessislava Kochloukova.
Abstract: A convex real projective structure on a manifold gives a (non-Riemannian) metric which in turn assigns a length to every element of the fundamental group. We show that two distinct structures assign the same length to every element iff the two structures are dual, and that a structure is self dual iff it is a hyperbolic structure. This result was also (mostly) proved by Inkang Kim.
March 12, 4:30–5:30 pmAbstract: The Farrell-Jones conjecture predicts that the K-theory of a group Γ can be built up from the K-theory of virtually cyclic subgroups of Γ. Suppose G is a semisimple, simple real Lie group or rank one. We prove the Farrell-Jones conjecture when Γ is a lattice in G, i.e., Γ is a discrete subgroup of G and G/Γ has finite invariant volume. There is an analogous result for L-theory. There is a large overlap with already existing results of Farrell–Jones and Berkove–Farrell–Juan-Pineda–Pearson, but new groups are covered and the result holds for higher K-theory and arbitrary coefficient rings.
Abstract: Locally finite Kac-Moody groups generalize finite groups of Lie-type in a natural way. While the latter are finite groups, the former are in general not, but have interesting finiteness properties (such as being finitely generated in general, or finitely presented if they are of two-spherical type). An important tool in the study of both classes of groups are so-called Tits buildings. In the present talk, we introduce the notion of a (generalized) unitary form of a Kac-Moody group, and present a result on finite generation of these groups.
Abstract: The Charney-Davis conjecture is an easily stated linear inequality on the face numbers of simplicial spheres that are also flag complexes, that is, complexes for which every minimal non-face is a two element set. While the conjecture is reasonably well known, it is only known to hold in a handful of rather narrow cases, and the proofs are often very complicated. I will discuss some known results and give an easy proof of a relatively broad case.
Abstract: In a well-known paper from 1992, D. Wright showed that certain contractible open manifolds admit no nontrivial proper free group actions and thus do not occur as universal covers of any other manifolds. That paper also includes interesting results that apply to non-manifolds. We will discuss an alternative approach to Wright's work which leads to a new proof of his manifold theorem and a strengthening of his non-manifold results. This talk will cover joint work with Ross Geoghegan.
March 26, 4:30–5:30 pm in LH-9 (Colloquium)Abstract: In this talk, we will discuss the (α,K)-differential Harnack inequality for the positive solutions of the linear heat equation on general complete Riemannian manifolds with Ricci curvature bounded from below, i.e., Rij>-Kgij with some real constant K, and prove that the corresponding (α,K)-entropy formula is monotone non-increasing. As applications, we obtain the pointwise Harnack inequality for the positive solutions of the linear heat equation. We also prove the sharp differential Harnack inequality for the positive heat kernel of the linear heat equation on general complete Riemannian manifolds, and obtain a sharp upper bound of the Nash's entropy for heat kernel. This is the joint work with Dr. Junfang Li in UAB.
Abstract: While it is not known whether Thompson's group F is amenable, I will establish a lower bound on the cardinality of Følner sets. In particular, I will demonstrate the following: There is a constant C>1 such that if A is a C -4n-Følner set in F, then A contains at least H(n) elements, where H(0)=0 and H(n+1)=2H(n).
Abstract: I will discuss progress on a conjecture by Adem and Ruan relating the twisted K-theory and the Chen-Ruan cohomology of an effective orbifold.
April 23, 4:30–5:30 pm (Colloquium)Abstract: The group Out(FN) is a (more difficult to study) cousin of the mapping class group and the Culler-Vogtmann Outer space is a free group analog of the Teichmuller space of a hyperbolic surface. There is a companion space to the Outer space, the space Curr(FN) of geodesic currents on the free group FN. A geodesic current is a measure-theoretic analog of the notion of a closed curve (or of a conjugacy class). We will describe a natural Bonahon-type “geometric intersection form” on the product of the Outer space and the space of currents, and its continuous extension to the closure of the Outer space obtained in our recent work with Lustig. We will discuss various aplications of the intersection form, including to the study of curve complex analogs in the free group context and the construction of analogs of free purely pseudo-Anosov subgroups in Out(FN). The talk is based mostly on the joint work with Martin Lustig.
Abstract: The study of finitely generated groups obtained by various kinds random constructions has been an actively developing topic in Geometric Group Theory in recent years. A particularly important application of such probabilistic methods is a result of Gromov proving the existence of a finitely prsented group that does not admit a uniform embedding into a Hilbert space. We will discuss a number of results related to algebraic properties of “generic” groups. In particular, we show that random one-relator groups and “generic” finitely presented quotients of the modular group enjoy a strong Mostow-type isomorphism rigidity property and are essentially algebraically incompressible. Applications include counting isomorphism types for several classes of groups given by generators and relators. The talk is based mostly on joint work with Paul Schupp.
Abstract:
Whitehead's algorithm takes a collection of (cyclic) words in a free
group and reduces them to minimal length. Results of Berge and
Zieschang relate the tools used in Whitehead's algorithm to the
realizability problem for Heegaard splittings. (This is the question
of which finite group presentations are presentations for fundamental
groups of 3-manifolds of a special form which comes from a Heegaard
diagram.) It turns out these ideas are also applicable to a recent
question of Gordon and Wilton about whether every presentation with a
single relator is “virtually geometric” — this is
related in turn to the question of which hyperbolic groups contain
surface subgroups. I'll discuss some of these ideas and answer Gordon
and Wilton's question.
Abstract: Consider a robot, Bob, that has to get from its start position S to a certain target point T. At all times, Bob knows its location via a global positioning system. We examine the situation when Bob has only tactile sensors and no visual sensors. In particular, Bob does not know the locations of any obstacles until it touches them. We discuss some known algorithms for mobile robot navigation in 2 dimensions. Further, we present complexity estimates for higher dimensional environments, and describe some new algorithms in this setting.
Abstract: I will present results about the quasi-isometry invariance of the existence and location of certain infinite cyclic subgroups and their commensurizers in one-ended finitely presented groups. An application of this is the quasi-isometry invariance of certain vertex groups of the Scott-Swarup JSJ decomposition for groups.
Abstract: For a group G that splits as an amalgamation of A and B over a common subgroup C, there is an associated Waldhausen Nil-group, measuring the "failure" of Mayer-Vietoris for algebraic K-theory. Assume that (1) the amalgamation is acylindrical, and (2) the groups A, B, and G satisfy the Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen Nil-group splits as a direct sum of Nil-groups associated to certain (explicitly describable) infinite virtually cyclic subgroups of G. This was joint work with Ivonne Ortiz.
Abstract: I will examine the enumerative properties of three simplicial complexes associated to matroids: the independence complex of a matroid, the broken circuit complex of a matroid, and the order complex of a geometric lattice.
Abstract: We define the Hochschild homology groups of a group ring ZG relative to a family of subgroups F of G. These groups are the homology groups of a space which can be described as a homotopy colimit, or as a configuration space, or, in the case F is the family of finite subgroups of G, as a space constructed from stratum preserving paths. This is joint work with David Rosenthal.
Abstract: We study faithful unitary representations of Thompson's group F in L2(Rn). The amenability question of F is translated within this framework in terms of weak containments with respect to the left regular representation. We prove that the extension of each representation to the universal group algebra has non-trivial kernel. Our main result says that the direct sum of all representations weakly contains the left regular representation. We discuss the relevance of this result to the amenability problem of F: for example, the lack of projections of the reduced C*-algebra generated by F should transfer to the C*-algebras generated by the above representations. Moreover, this result could allow for other C*-algebraic invariants to be tested.
Abstract: The problem of Clifford-Klein forms is to determine for which pairs of Lie groups (H, J), with J a closed subgroup of H, there is a discrete subgroup Γ of H so that the quotient J\H/Γ is a compact manifold. Methods from topology, geometry, dynamics, and representation theory have been used to approach this problem and many cases are still open. For example, the basic test case of SLn-k(R)\SLn(R) is not fully solved – nonexistence has been proved for k>2 and for k=1 and n odd, but the other cases are open. In this talk I'll very briefly survey the general problem and then present a result on compact forms of SLn-2(R)\SLn(R) which gives an algebraic characterization of any possible compact form and reduces the problem to an algebraic one about subgroups of SLn(R). The proof relies on techniques from three important parts of rigidity theory for group actions, cocycle superrigidity, Ratner's theorems for unipotent flows and measure rigidity.
Abstract: Algebraic objects in general do not have good homotopy properties: a space homotopy equivalent to e.g. a topological group usually will have no group structure itself. On the other hand, several classes of objects naturally occurring in algebraic topology (loop spaces, Eilenberg-MacLane spaces, spectra) exhibit properties closely resembling those known from algebra. These two facts inspired an intensive study which showed that in special cases algebraic structures can be described in homotopy meaningful terms. This research led to the development of operads, PROPS, Gamma-spaces, etc. The talk will present an overview of these results. It will also explain how they all admit a common generalization. As it turns out, there is a very broad class of algebraic objects which have their interesting homotopy theoretical analogs.
Abstract: I will show that the directed group of automorphism of a rooted tree is amenable if and only if the valency is bounded. Depending on time, I will discuss relations with questions about word growth of groups.
Abstract: First I will explain the definition of stringy cohomology ring of a global quotient orbifold [X/G] where X is a symplectic or almost complex manifold with an action of a finite group G. Its relation to Chen-Ruan orbifold cohomology ring and Gromov-Witten invariants will be also explained. Then I will show the computational result of the stringy cohomology of symmetric product of an orbifold and its application to Ruan's crepant resolution conjecture.
November 20, 4:30–5:30 pm (Colloquium)Abstract: I'll describe joint with with Greg Arone on the chain rule for Goodwillie's calculus of homotopy functors. For functors of stable model categories such as spectra, this chain rule takes a simple form that mirrors the chain rule for higher derivatives in everyday calculus. In the unstable case, this formula must be adjusted to take into account the operad formed by the derivatives of the identity functor.
Abstract: Given a closed negatively curved manifold M, we study the statistical asymptotic penetration behaviour of geodesic lines of M in small neighborhoods of a point q in M and we prove a Khinchine type theorem for the spiraling of geodesic lines around it.
Abstract: The so-called additivity theorem is one of the fundamental results in algebraic K-theory. I will present a shortcut through Waldhausen's proof of this theorem, using only a minor strengthening of Quillen's theorem A instead of theorem B. Although unavoidably technical, this talk will describe the result and focus on the simplicial techniques involved in its proof.