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.
Some seminar speakers will also give a colloquium talk at 4:30 pm on the same day as the seminar talk. This seminar is partly funded as one of Dean's Speaker Series in Harpur College (College of Arts and Sciences) at Binghamton University.
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Abstract: We use the inverse mean curvature flow to prove a geometric inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic n-space. This inequality is similar to the Alexandrov-Fenchel inequality for hypersurfaces in Euclidean space. As an application, we will prove the Penrose inequality for asymptotically hyperbolic graphs. This is joint work with Levi Lopes de Lima.
Abstract: We shall discuss the enumerative problem of counting the number of complex curves (in complex projective space of dimension 2) which pass through the requisite number of generic points and has a prescribed singularity at one point. Our exposition will be from a topological point of view via the Euler class. This is joint work with Ritwik Mukherjee.
Abstract: In his book ``Partial Differential Relations", Gromov gives the definition for an intrinsic isometry between metric spaces as a generalization of the definition of an isometry between Riemannian manifolds. In 2010, Petrunin proved that a compact metric space admits an intrinsic isometry into n-dimensional Euclidean space if and only if it is a pro-Euclidean space of rank at most n, and that either of these assumptions implies that the Lebesgue covering dimension of X is at most n. He also shows that this is not true, in general, for path isometries, thus motivating the definition of an intrinsic isometry. One of the main results of my thesis allows for a partial generalization of Petrunin's result to intrinsic isometric embeddings. In this talk we will define all of the necessary terminology, and sketch proofs of the various results.
Abstract: Fake projective plane was first introduced by David Mumford. It has the smallest possible Euler number among all smooth surfaces of general type. The main purpose of the talk is to explain the joint work of Gopal Prasad and myself on classification of fake projective planes. Some observations related to exotic four manifolds will be mentioned. Related work of Klingler, myself, Prasad and Cartwright-Steger will also be discussed.
Abstract: I will discuss a result which shows that every right-angled Artin group can be embedded in a planar braid group. I will then give some examples of closed hyperbolic 4-manifold groups which were previously not known to embed in braid groups, as well as some applications to the algebra of right-angled Artin groups. Finally, I will discuss some applications to decision problems for finitely presented subgroups of braid groups, showing that there exists a finitely presented subgroup of a braid group with an unsolvable conjugacy problem and an unsolvable membership problem, as well as the fact that the isomorphism problem for finitely presented subgroups of braid groups is unsolvable. This work is joint with Sang-hyun Kim.
Abstract: If M is a 3-manifold that admits the Sol geometry, then either M is a torus bundle over the circle where the gluing map is hyperbolic, or M is a sapphire (also known as torus semi-bundle) with the said geometry. Given such a manifold, we study the structure of its cohomology ring.
Abstract: Given a smooth manifold, we get a natural symplectic structure on its cotangent bundle. In certain cases, we can obtain invariants of the smooth structure of the base from symplectic invariants of the cotangent bundle. I will give a short introduction to exotic smooth structures on some open 4-manifolds and show that their cotangent bundles are symplectically equivalent in a natural way.
Abstract: We investigate a set of groups {SBC_n}. Each group SBC_n sits very naturally in the full group of automorphisms of {0,1, ... , n-1}^Z, the full shift on n letters, and is somehow a very natural object. Still, the structure of each group SBC_n, at least initially, was quite a mystery. These groups' elements are describable as finite transducers, and so the groups SBC_n are linked strongly to the rational group R introduced by Grigorchuk, Nekrashevych, and Suschanski. Furthermore, the group SBC_n corresponds precisely to the outer automorphism group of the generalized Higman-Thompson group G_{n,1} = V_n. In this talk, we exploit a connection with De Bruijn graphs to begin to explore the structure of the groups SBC_n. Amongst other results, we will show that for m \neq n, SBC_m is not isomorphic to SBC_n, and we will also show that for n>2, the groups SBC_n are infinite, locally finite groups (and hence are not finitely generated). This talk features work from two separate projects involving collaborators Y. Maissel, A. Navas, and P. Cameron.
Abstract: Character varieties on unitary groups were perhaps the first understood examples of a vast and rich theory with diverse flavors. In this talk I will describe a concrete, and in a certain way, minimal example arising from consideration of the fundamental group of a punctured sphere, its classical complex-analytic interpretation (in terms of monodromies of Fuchsian systems), and a relation between its geometric properties (in the form of a K\"{a}hler structure) and a suitable interpretation of the WZNW models in conformal field theory.
Joint with Combinatorics seminar
Abstract: The Arf-Kervaire invariant problem arose from Kervaire-Milnor's classification of exotic spheres in the early 1960s. Browder's theorem of 1969 raised the stakes by connecting it with a deep question in stable homotopy theory. In 2009 Mike Hill, Mike Hopkins and I proved a theorem that solves all but one case of it. The talk will outline the history and background of the problem and give a brief idea of how we solved it.The talk will take place in S2 140 from 3:15-4:15 p.m. There is a reception at Chenango Room from 4:30 p.m.
Abstract: Let M be an oriented, noncompact, complete, Riemannian manifold. Gromov proved that if the sectional curvature of M negative and bounded, and if the volume of M is finite, then M is homeomorphic to the interior of a compact manifold \overline{M} with boundary B. I will discuss the question what manifold B can be. I will prove that when M has dimension 4, each boundary component of \overline{M} is aspherical.
Abstract: Let M_{g,n} be the moduli space of genus g Riemann surfaces with n marked points. Given a non negative integer i, we want to understand how the i-th rational cohomology group of M_{g,n} changes as the parameter n increases. It turns out that the symmetric group S_n acts on it and the sequence of S_n-representations ``stabilizes'' in a certain sense once n is large enough. In this talk I will explain the behavior of this and other examples via the language of representation stability. Moreover, I will introduce the notion of a finitely generated FI-module and show our sequence of interest has this underlying structure which explains the stability phenomena mentioned above. As a consequence we obtain that, for n large enough with respect to i, the i-th Betti number of M_{g,n} is a polynomial in n of degree at most 2i.
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