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: 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.
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.