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 Speaker Series 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: Let P be a polygon with n sides and let S and T be triangulations of P using no extra vertices. I will define the notion of a "sequence of signed flips" taking S to T. The Signed Flip Conjecture (SFC) says that such a sequence exists for every n, S and T. I will give the easy argument, modulo Whitney's first theorem in graph theory (1931), that the SFC implies the 4 Color Theorem (4CT). But searching for a short proof of the 4CT by proving the SFC might be futile if the SFC is false. I will then give the elegant proof by Gravier and Payan that the 4CT implies the SFC. Thus the 4CT and SFC are equivalent, and trying to prove the 4CT through the SFC might be worthwhile.TIME : 1:15-2:15pm; PLACE : LN2205
Abstract: I'll talk about the current state of efforts to understand non-resolvable ANR homology manifolds.
Abstract: We will discuss amenability of the topological full group of a minimal Cantor system. Together with the results of H. Matui this provides examples of finitely generated simple amenable groups. Joint with N. Monod.
Abstract: In this talk we will discuss the recent joint work with Zheng Huang on the dynamic stability of the surface area preserving mean curvature flow in Euclidean space. More precisely, we show that the flow exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).Robert Bieri is giving a colloquium talk at 4:30pm.
Abstract: For a finitely generated group, the conjugacy problem asks if there is an algorithm which determines whether a pair of elements are conjugate or not. This is one of Max Dehn's well-known decision problems and has been widely studied over the past century. Related to this problem is the question of understanding, for a pair of conjugate elements, the family of conjugators between them. We focus in particular on estimating the minimal length of such an element. This problem naturally extends beyond the reach of the conjugacy problem itself and can be asked of any group which admits a left-invariant metric. We will discuss a result for pairs of hyperbolic elements in a semisimple Lie group. The issue of extending results from here to their lattices, such as SL(n,Z), remains unsolved, with questions raised of a number theoretic nature.
Abstract: The talk will begin with the definition and a brief overview of real rational knots in the projective space. We will then discuss rational knots in the 3-sphere, including methods to construct and classify them.
Abstract: Let G split as a finite graph of free abelian groups with cyclic edge groups. We characterize when G acts freely on a CAT(0) cube complex. We show that if G acts properly and semi-simply on a CAT(0) space then G acts freely on a CAT(0) cube complex.TIME : 9:40-10:40am; PLACE : LN2201
Abstract: I will discuss some relations between the horospherical limit set of a finite volume nonpositively curved manifold and the toplogy of its end. These relations are especially strong in the case of a locally symmetric manifold or a product of visibility manifolds. The above terms will be defined (or at least illustrated) in the talk. This is joint work with Dave Witte Morris.
Abstract: We show that all of the higher-dimensional Brin-Thompson groups nV are 2-generated, and outline work in progress towards giving small presentations of these finitely-presented, infinite simple groups. Along the way, we will discuss various approaches to building small presentations for groups in general, and groups in the extended Thompson family in particular. This is joint work with Martyn Quick.
Abstract: This talk will discuss the problem of how to decide when one 4-dimensional symplectic ellipsoid embeds into another. I will try to explain the connection of this question with resolving singularities in complex surface theory, and with invariants coming from gauge theories. The talk will be aimed at a fairly general audience and will not assume any knowledge of symplectic geometry.
Abstract: I will talk about material that will prepare for Justin Moore's talk a week later about the amenability of Thompson's group F.
Abstract: I will describe some results bounding the isometry groups of Riemannian metrics on aspherical manifolds and of the lifted metrics on their universal covers. The general theme is that topological properties of an aspherical manifold often restrict the symmetries of an arbitrary complete Riemannian metric on that manifold. I will illustrate this by explaining why on a finite volume irreducible locally symmetric manifold, no metric has more symmetry than the locally symmetric metric. Possibly, I will also discuss why moduli space is a minimal orbifold.
Abstract: In the 1970's, Graeme Segal proved that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding continuous mapping space through a range of dimensions increasing with degree. I will address if a similar result holds when other almost complex structures are put on projective space. For CP^2, I prove that the inclusion map from the space of J-holomorphic maps to the space of continuous maps induces a homology surjection through a range of dimension tending to infinity with degree. The proof involves comparing the scanning map of topological chiral homology (Salvatore, Lurie, Andrade) with gluing of J-holomorphic curves (Floer, McDuff-Salamon, Sikorav).
Abstract: There are several notions of complexity for three-manifolds. One such definition was introduced by Matveev in the 90's. For a three-manifold M, with or without boundary, his complexity of M is a nonnegative integer c(M). Two properties of c(M) are: (A) For any n there are only finitely many closed, irreducible orientable three-manifolds with c(M) less than n. (B) c(M) is additive with respect to connected sums: c(M#N)=c(M)+c(N). While there are many combinatorial measures of complexity which satisfy (A), the natural ones to try, such as the minimum number of vertices or tetrahedra in a simplicial triangulation of M, badly fail (B). My goal is to see how c(M) can help understand face enumeration of simplicial triangulations of M. I will also explore the possibility that there might be simple combinatorial invariants which satisfy (A) and (B) for 3-manifolds, and which have natural extensions to PL-triangulations in higher dimensions which still satisfy (A). The talk will assume no specialized knowledge of 3-manifolds.TIME : 1:15-2:15pm; PLACE : LN2205
Abstract: Differential forms give geometric representatives for cocycles of real cohomology classes via de Rham cohomology. Differential cohomology theories are refinements of topological cohomology theories whose cocycles give geometric models for these classes. We will discuss a few examples of these theories and, in particular, how differential K-theory is the natural setting for a refinement of the Atiyah-Singer Index theorem.
Abstract: It follows from work of Crowley-Loeh (d>3) and Barge-Ghys (d=2) that in all degrees distinct from d=3, the l^1-seminorm and the manifold semi-norm coincide on homology of degree d. We show that when d=3, the two semi-norms are bi-Lipschitz to each other, with an explicitly computable constant. This was joint work with Christophe Pittet (Univ. Marseille).
Abstract: In 2006, Alves and Ontaneda gave a formula for the Whitehead group of a 3-dimensional crystallographic group G in terms of the Whitehead groups of the virtually infinite cyclic subgroups of G . In this talk, I will present a general splitting formula for the lower algebraic K-theory of all 3-dimensional crystallographic groups, which generalizes the one for the Whitehead group obtained by Alves and Ontaneda. This joint work with Dan Farley.
Abstract: Let G(n,p) be the Bernoulli random graph on n vertices, i.e., G(n,p) is the probability space of all graphs on n vertices where each edge is inserted with uniform probability p. Let X(n, p) be the associated random flag complex (or clique complex) obtained by filling in a simplex for each complete subgraph. Usually p will be a function of n. Write f << g to mean f = o(g). A famous result of Erdos-Renyi states that if p << (log n)/n, then, with high probability (abbreviated w.h.p.), X(n, p) is not connected, while if (log n)/n << p, it is connected w.h.p. My collaborator, Matt Kahle, has generalized this by showing that in other ranges of p, the reduced cohomology (with rational coefficients) of X(n, p) is concentrated in a single degree w.h.p. It turns out that the degree where the cohomology is concentrated is the greatest integer < d/2, where d is the dimension of X(n,p). One can also associate to a graph and a sequence of groups indexed by the vertex set of the graph, a new group called the "graph product." For example, when each group is cyclic of order 2, the graph product is a right-angled Coxeter group. One can compute the cohomology of such a graph product of groups, with coefficients in its group algebra, from the cohomology of the flag complex determined by the graph. So, the notion of a random graph leads to the notion of a random graph product of groups. In previous work I calculated the cohomology of graph products in terms of the cohomology of the associated flag complex. For example, it follows that, with group algebra coefficients, a random graph product of finite groups .w.h.p. has cohomology concentrated in a single degree, i.e., is a rational duality group.
Abstract: TBA