Spring and Summer 2020
January 28 -
Christoforos Neofytidis (OSU)
Manifolds with no Ruelle-Sullivan cohomology classes
We exploit notions of geometric and algebraic topology as well as of geometric group theory, such as Gromov's simplicial volume, Steenrod-Thom's realisability of (co)homology classes by manifolds and outer automorphism groups, to show that the cohomology rings of certain classes of product manifolds do not contain Ruelle-Sullivan classes. We obtain consequences with respect to a long-standing problem in dynamics, namely classes of manifolds that do not support Anosov diffeomorphisms.
February 4-
Michelle Chu (UIC)
Virtual properties of arithmetic hyperbolic 3-manifolds
We will introduce arithmetic methods to construct hyperbolic manifolds and discuss how arithmeticity can be used in the study of virtual properties of hyperbolic 3 manifolds.
February 6 -
Rob Kropholler (Tufts University)
Incoherence and virtual fibering of free-by-free groups.
The class of free-by-free groups comes as a natural extension of the class of free-by-cyclic groups. We are interested in what properties this class has in common with free-by-cyclic groups. We are particularly interested in the properties of coherence and algebraic fibering. I will discuss results of this type for related classes of groups. I will then talk about recent work with Genevieve Walsh on fibering and incoherence for free-by-free and related groups.
Jun Choi (Purdue University)
Rationality of torus bundle group
Gromov's polynomial growth theorem states that a finitely generated group has a polynomial volume growth if and only if it is virtually-nilpotent. One may ask under what condition the volume of virtually nilpotent groups is a polynomial, or more generally, a rational function. It turns out the rationality depends very much on the structure of the group as well as the choice of the generators. In the first half of the talk, I will give a brief survey on the rational volume growth and the established strategy. In the second half, we will see that the torus bundle group has rational growth using different approach. This work is joint with Turbo Ho and Mark Pengitore.
February 13 - Rescheduled for April 2
Burns Healy (University of Wisconsin - Milwaukee)
Model Spaces for Relatively Hyperbolic Pairs
Relatively hyperbolic pairs are groups with preferred peripheral subgroups and are meant to generalize the behavior of non-uniform lattices in rank one symmetric spaces of noncompact type. While in geometric actions hyperbolic spaces are well-defined up to quasi-isometry, cusp-uniform actions by relatively hyperbolic pairs require two choices to be made in order to determine a QI-class of hyperbolic space. We examine the symmetric space case for the motivation behind the choice of a preferred type of space and prove these model spaces exist and are uniquely determined. In doing this we examine different kinds of horospherical geometry, prove internal geometry conditions sufficient for uniform perfection of the boundary of a hyperbolic space when acted on cusp-uniformly, note the connection between space quasi-isometries and boundary quasi-symmetries, and demonstrate existence of some classes of cusp-uniform actions on non-model spaces.
February 18-
Solenn Estier (University of Geneva)
Multidegree and Sp(1)-principal bundles over the seven-sphere
The determination of the possible degrees of maps between closed oriented manifolds of same dimension is a well-studied question. In this talk, we focus on a refinement called the multidegree of a map, in the specific case of sphere bundles over spheres. Following work by Lafont and Neofytidis, Wang and Kennedy, we restrict to principal S^3-bundles over spheres, and will present a work in progress, aiming to apply homotopical methods to provide necessary and sufficient conditions to the existence of maps with given multidegree. We use the fact that many spaces under scrutiny admit an H-space structure, along with a simple CW-structure, to extend maps using the so-called Puppe sequence of a cofibration.
February 20-
Denis Gorodkov (Steklov Mathematical Institute)
Combinatorial formulas for the first rational Pontryagin class and applications
The combinatorial computation of rational Pontryagin classes of a manifold given its triangulation is a classical problem of algebraic topology arising in the 1970’s. Works on the subject include the praised Gabrielov-Gelfand-Losik 1975 article, several articles by Gelfand and MacPherson and many more. In 2004 Prof. Alexander Gaifullin explicitly described the set of all local formulas for the first Pontryagin class in a computable way. We will give a general overview of the subject, present some recent results and demonstrate a nice application proving that a special simplicial complex is the vertex-minimal triangulation of the quaternionic projective plane.
March 19- 2:00 pm Eastern Time
Talk will be delivered remotely via a Zoom meeting.
Join the Zoom meeting at https://osu.zoom.us/j/549470831
To participate, you may need to download and install the freely available basic version of Zoom https://zoom.us/download
Matthew Zaremsky (University at Albany)
Quasi-isometric embeddings into simple groups
It is a classical fact that every finitely generated group embeds as a subgroup of a finitely generated simple group. In the 90's Bridson proved that if one relaxes "simple" to "no proper finite index subgroups" then such an embedding can be done in a quasi-isometric way. In joint work with Jim Belk, we prove that this is true even keeping the word "simple": every finitely generated group quasi-isometrically embeds as a subgroup of a finitely generated simple group. The simple groups we construct are "twisted" variants of Brin-Thompson groups. Certain of these twisted Brin-Thompson groups also provide examples of groups with interesting finiteness properties, and using them we can produce the second known family of simple groups with arbitrary finiteness properties (the first being due to Skipper-Witzel-Z).
March 26- 3:00 pm Eastern Time
Talk will be delivered remotely via a Zoom meeting.
Join the Zoom meeting at https://osu.zoom.us/j/697493218
To participate, you may need to download and install the freely available basic version of Zoom https://zoom.us/download
Chris Connell (Indiana University, Bloomington)
Entropy rigidity for RCD-spaces
We present extensions of the entropy rigidity results of Ledrappier-Wang and Besson-Courtois-Gallot to the class of RCD metric spaces. These are the metric-measure spaces with a weak notion of "Ricci curvature" being bounded from below. Most importantly, they are the natural class of metric spaces containing measured-Gromov-Hausdorff limits of manifolds with Ricci curvature bounded below. For this reason, they have been intensely studied over the past decade by Lott, Villani, Gigli, Mondino and many others. From our results we will be able to also derive new corollaries about manifolds. This is joint work with Xianzhe Dai, Jesus Nunez-Zimbron, Raquel Perales, Pablo Suarez-Serrato and Guofang Wei.
March 24-Postponed to next academic year
Tian-Jun Li (University of Minnesota)
April 2 -
Talk will be delivered remotely via a Zoom meeting.
Join the Zoom meeting at https://osu.zoom.us/j/991191369
To participate, you may need to download and install the freely available basic version of Zoom https://zoom.us/download
Burns Healy (University of Wisconsin - Milwaukee)
Model Spaces for Relatively Hyperbolic Pairs
Relatively hyperbolic pairs are groups with preferred peripheral subgroups and are meant to generalize the behavior of non-uniform lattices in rank one symmetric spaces of noncompact type. While in geometric actions hyperbolic spaces are well-defined up to quasi-isometry, cusp-uniform actions by relatively hyperbolic pairs require two choices to be made in order to determine a QI-class of hyperbolic space. We examine the symmetric space case for the motivation behind the choice of a preferred type of space and prove these model spaces exist and are uniquely determined. In doing this we examine different kinds of horospherical geometry, prove internal geometry conditions sufficient for uniform perfection of the boundary of a hyperbolic space when acted on cusp-uniformly, note the connection between space quasi-isometries and boundary quasi-symmetries, and demonstrate existence of some classes of cusp-uniform actions on non-model spaces.
April 16 - 3pm Eastern Tim
Talk will be delivered remotely via a Zoom meeting.
Join the Zoom meeting at https://osu.zoom.us/j/781557151
To participate, you may need to download and install the freely available basic version of Zoom https://zoom.us/download
Merlin Incerti-Medici (University of Zurich)
Circumcenter extension maps for Hadamard manifolds
Given a CAT(-1) space, we can associate to it a boundary at infinity and a cross ratio on said boundary. There is a series of results that tell us that, for sufficiently nice CAT(-1) spaces, the boundary together with the cross ratio uniquely determines the interior space. The proof of these results can be understood in terms of the construction of a circumcenter extension map. It turns out that this relationship between boundary and interior space generalizes nicely to a large class of CAT(0) spaces. In this talk, we will survey some known results, explain the construction of the circumcenter extension and show that the boundary roughly determines the interior space for a large class of manifolds.
June 16 - 3pm Eastern Time
Jim Belk (University of St Andrews)
Talk will be delivered remotely via a Zoom meeting.
Join the Zoom meeting at https://osu.zoom.us/j/92136148494
On Finitely Presented Groups that Contain Q
It is a consequence of Higman's embedding theorem that the additive group Q of rational numbers can be embedded into a finitely presented group. Though Higman's proof is constructive, the resulting group presentation would be very large and ungainly. In 1999, Martin Bridson and Pierre de la Harpe asked for an explicit and "natural" example of a finitely presented group that contains an embedded copy of Q. In this talk, we describe some solutions to this problem related to Thompson's groups F, T, and V, including a new simple group of type F infinity that contains Q. This is joint work with James Hyde and Francesco Matucci.
July 14- 2pm Eastern Time
Johnny Nicholson (University College London)
Talk will be delivered remotely via a Zoom meeting.
Join the Zoom meeting at https://osu.zoom.us/j/97619118082
Projective modules and the homotopy classification of CW-complexes
A basic question in the homotopy classification of CW-complexes is to ask for which finitely presented groups $G$ does $X \vee S^2 \simeq Y \vee S^2$ imply $X \simeq Y$, where $X$ and Y$ are finite 2-complexes with fundamental group $G$. Despite early interest by Cockroft-Swan and Dyer-Sieradski, it wasn’t until 1976 that examples of non-cancellation were found by Dunwoody and Metzler. This led Browning to complete the classification in the finite abelian case. In recent years, applications to Wall’s D2 problem and the classification of manifolds have sparked renewed interest in this problem. In this talk, we will show how the case where $G$ has periodic cohomology can largely be reduced to a question about projective $\mathbb{Z} G$ modules. We then resolve this by generalising results of Swan from the 1980s.