Organizers: Carolyn Abbott (carolynabbott@brandeis.edu), Ruth Charney (charney@brandeis.edu), Kiyoshi Igusa (igusa@brandeis.edu), Danny Ruberman (ruberman@brandeis.edu)

October 5: Michael Sullivan (UMass Amherst) Via Zoom

Title: Displacing Legendrian submanifolds in contact geometry

Abstract: Lagrangian and Legendrian submanifolds of symplectic and contact manifolds are sometimes ``flexible" like smooth topology, and sometimes ``rigid" like differential geometry. Pseudo-holomorphic curves, algebraically packaged into various Floer-theory or Gromov-Witten-theory invariants, have played a (maybe even ``the") main role in proving rigidity results. But if the invariants vanish, does this mean the objects of study are flexible? I will discuss, using the barcodes of a persistence Floer-type homology, how to extract (sometimes optimal) quantitative rigidity results for Legendrian submanifolds, even when the traditional Floer-theory invariants vanish. I plan to give the talk remotely, but in real-time, to keep the pace more accessible. This is joint work with Georgios Dimitroglou Rizell.

October 12: Yajit Jain (Brown University) In person

Title: Topologically Trivial Families of Smooth h-Cobordisms

Abstract: In this talk we will discuss topologically trivial families of smooth h-cobordisms. Using work of Dwyer, Weiss, and Williams, we can assign a K-theoretic invariant to these bundles, the smooth structure characteristic, which is closely related to the higher Franz–Reidemeister torsion invariants studied by Igusa. After describing constructions of these bundles due to Goette and Igusa, we will indicate how one can compute the smooth structure characteristic using Morse theory, and outline a proof of their Rigidity Conjecture. Time permitting, we will also briefly discuss a relationship between these invariants and symplectic topology.

October 19: Daniel Alvarez-Gavela (MIT) In person

Title: The nearby Lagrangian conjecture from the K-theoretic viewpoint.

Abstract: I will discuss two K-theoretic aspects of the nearby Lagrangian conjecture. The first is joint work with M. Abouzaid, S. Courte and T. Kragh and uses a factoring of the Waldhausen derivative to obtain new restrictions on the smooth structure of nearby Lagrangians. The second is joint work in progress with K. Igusa and M. Sullivan and attempts to use a higher Whitehead torsion invariant to obtain new restrictions on the stable isomorphism classes of tube bundles which may be used to generate nearby Lagrangians.

October 26: Zihao Liu (Brandeis) In person

Title: Scaled homology and topological entropy

Abstract: In this talk, I will introduce a scaled homology theory, lc-homology, for metric spaces such that every metric space can be visually regarded as “locally contractible” with this newly-built homology. In addition, after giving a brief introduction of topological entropy, I will discuss how to generalize one of the existing results of entropy conjecture, relaxing the smooth manifold restrictions on the compact metric spaces, by using lc-homology groups. This is joint work with Bingzhe Hou and Kiyoshi Igusa.

November 2: Jacob Russell (Rice University) In person

Title: Searching for geometric finiteness using surface group extensions

Abstract: Farb and Mosher defined convex cocompact subgroups of the mapping class group in analogy with convex cocompact Kleinian groups. These subgroups have since seen immense study, producing surprising applications to the geometry of surface group extension and surface bundles. In particular, Hamenstadt plus Farb and Mosher proved that a subgroup of the mapping class groups is convex cocompact if and only if the corresponding surface group extension is Gromov hyperbolic.

Among Kleinian groups, convex cocompact groups are a special case of the geometrically finite groups. Despite the progress on convex cocompactness, no robust notion of geometric finiteness in the mapping class group has emerged. Durham, Dowdall, Leininger, and Sisto recently proposed that geometric finiteness in MCG(S) might be characterized by the corresponding surface group extension being hierarchically hyperbolic instead of Gromov hyperbolic. We provide evidence in favor of this hypothesis by proving that the surface group extension of the stabilizer of a multicurve is hierarchically hyperbolic.

November 9: Anthony Conway (MIT) In person

Title: Stable diffeomorphism and homotopy equivalence.

Abstract: In this talk, we consider the difference between stable diffeomorphism and homotopy equivalence. Here, two 2n-manifolds are called stably diffeomorphic if they become diffeomorphic after connect summing with enough copies of S^n×S^n. After providing some motivation from surgery theory, we describe families of stably diffeomorphic manifolds that are not pairwise homotopy equivalent. This is based on joint work with Crowley, Powell and Sixt.

November 16: Lorenzo Ruffoni (Tufts University) In person

Title: Strict hyperbolization and special cubulation.

Abstract: Gromov introduced some “hyperbolization” procedures to turn a given polyhedron into a space of non-positive curvature, in a way that preserves some of the topological features of the original polyhedron. For instance, a manifold is turned into a manifold. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. This has been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. We will describe these procedures, and how to construct geometric actions of the resulting groups on CAT(0) cube complexes. As an application, we find that the groups resulting from strict hyperbolization are virtually linear over the integers. This is joint work with J. Lafont.

November 23: Ivan Levcovitz (Tufts University) In person

Title: Coxeter groups with connected Morse boundary

Abstract: The Morse boundary is a quasi-isometry invariant that encodes the possible "hyperbolic" directions of a group. The topology of the Morse boundary can be challenging to understand, even for simple examples. In this talk, I will focus on a basic topological property: connectivity and on a well-studied class of CAT(0) groups: Coxeter groups. I will discuss a criteria that guarantees that the Morse boundary of a Coxeter group is connected. In particular, when we restrict to the right-angled case, we get a full characterization of right-angled Coxeter groups with connected Morse boundary. This is joint work with Matthew Cordes.

November 30: Michah Sageev (Technion) In person

Title: Right angled Coxeter groups acting on CAT(0) cube complexes

Abstract: We will discuss a type of rigidity that one can hope for in the setting of proper, cocompact actions of right angled Coxeter groups acting on CAT(0) cube complexes, and some partial results in this direction. This is joint work with Ivan Levcovitz.

December 7: Cary Malkiewich (Binghamton University) In person

Title: Fixed point theory and the higher characteristic polynomial

Abstract: I'll give a highly revisionist account of classical Nielsen fixed-point theory, putting it in the context of modern trace methods by arguing that its central invariant is most naturally a class in topological Hochschild homology (THH). I'll then describe how this generalizes to periodic points and topological restriction homology (TR), and how these invariants fit together to give a far-reaching generalization of the characteristic polynomial from linear algebra. Much of this is joint work with Ponto, and separately with Campbell, Lind, Ponto, and Zakharevich.