Time: Tuesdays at 3:05-3:55 pm, unless specified
Location: LOV 231
Fall 2025
08/26: Organizational meeting
09/02: Hyein Choi
Title: Quasi-isometric embeddings of Ramanujan complexes
Abstract: Euclidean buildings (a.k.a. affine buildings and Bruhat-Tits buildings) are considered as a p-adic analogue of symmetric spaces. It is natural to ask how the symmetric space of SL(n,R) differs from the Euclidean building of SL(n,Q_p). We show that there is no quasi-isometric embedding between them. Generalizing this, we distinguish Ramanujan complexes constructed by Lubotzky-Samuels-Vishne as finite quotients of Euclidean buildings of PGL(n,F_p((y))) up to quasi-isometric embeddings. These complexes serve as high dimensional analogues of the optimal expanders, Ramanujan graphs, with fruitful applications in mathematics and computer science.
09/09:
09/16: Miri Son
Title: Classification of SL(n,R)-actions on closed manifolds
Abstract: Recently, Fisher and Melnick classified SL(n,R)-actions on n-dimensional manifolds for n≥3. In this talk, we generalize this result by classifying smooth or real-analytic SL(n,R)-actions on m-dimensional manifolds for 3≤n≤m≤2n-3. This work is motivated by the Zimmer program and is central to it, as Lie group actions restrict to their lattice actions.
This classification relies on the linearization of SL(n,R)-actions when there is a global fixed point. The analytic case was proved by Guillemin—Sternberg and Kushinirenko. We discuss the smooth case which is ongoing joint work with Insung Park.
09/23: Mario Gomez Flores
Title: Ptolemaic spaces and CAT(0)
Abstract: Ptolemy's inequality is a classical result in plane geometry that can be generalized to non-Euclidean geometries and CAT(k) spaces. In particular, CAT(0) spaces satisfy Ptolemy's inequality. I will present a paper dealing with the converse question: when is a Ptolemaic space also CAT(0)?As a result, the authors characterized Euclidean space as the only complete Riemannian manifold whose inversion is also a Riemannian manifold.
09/30: Camilo Arosemena-Serrato
Title: Global rigidity of codimension one actions of higher rank Lie groups
Abstract: In this talk we discuss recent progress on classification of locally free actions of higher rank simple Lie groups G with orbits of codimension one on closed manifolds, under a mild ergodic assumption, namely that there is a P-invariant probability measure that is mixing. We prove that manifolds with such actions are either the simplest possible model, namely, product of G/Gamma and S^1, up to a finite cover; or a fiber bundle over G/Q, for some parabolic subgroup P<Q<G.
We also discuss the existence of elements in the maximal torus of G acting uniformly partially hyperbolically on a fiber of this bundle, and how this implies P is a proper subgroup of Q.
10/07: Ferhat Karabatman
10/14:
10/21: Mujtaba Ali
10/28:
11/04:
11/11: Veterans' Day, holiday at FSU
11/18: Philip Hackney
11/25: Thanksgiving week
12/02:
12/09: Exam week
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Spring 2025
01/14: Organizational meeting
01/21:
01/28: Oishee Banerjee
Title: Functions and line bundles on surface bundles
Abstract: Riemann surfaces are well understood. However, when it comes to families of Riemann surfaces—such as the moduli “space” of Riemann surfaces—and their close relatives, many deep and interesting open questions remain. In this talk, I will discuss one such example: the Hurwitz stack. Along the way, we will see that the notion of stacks arises quite naturally when you want your “spaces” to encode the correct topological information, such as homotopy type. This talk assumes no prior knowledge of stacks.
02/04: Kyounghee Kim
Title: A family of diffeomorphisms on real rational surfaces.
Abstract: We will explore the dynamics of a family of real diffeomorphisms derived from complex rational surface automorphisms.
02/11: Asaf Katz
Title: Rigidity of u-Gibbs states in partially hyperbolic dynamics
Abstract: SRB measures, being physical measures, are of prime importance in partially hyperbolic systems. Their existence is an open problem - in general. Nevertheless, a related, more general class of measures - known as u-Gibbs states, were known to exist by a theorem of Pesin-Sinai. I will explain how one can adapt the factorization technique, pioneered by Eskin-Mirzakhani, to the setting of smooth dynamics and prove that for quantitatively non-integrable systems a (generalized) u-Gibbs state must be an SRB measure. If time permits, I will try to describe some of the key ideas and constructions of the Eskin-Mirzakhani technique.
02/18: Jingyin Huang
Title: Non-positive curved subcomplexes of spherical Deligne complexes
Abstract: The spherical Deligne complex is a simplicial complex introduced in Deligne's work when he studied the K(pi,1) problem for some complex hyperplane arrangement complements. The complex is homotopic to a wedge of spheres, and bear some similarities with spherical buildings, though it is not a building. While the topology of this complex prevent a CAT(0) metric on it, we show that it contains large pieces supporting equivariant non-positive curvature metric. As an application, we deduce new results on the K(pi,1) conjecture for several classes of Artin groups.
02/25: Mario Gomez Flores
Title: A survey of tight spans
Abstract: Given a metric space (X, d), its tight span T(X,d) is a universal metric space that contains an isometric copy of X. Thanks in part to its universal properties, tight spans have many desirable properties, like being a contractible geodesic space. In a 2024 paper (whose preprint was published in 2020), Lim, Mémoli and Okutan found a link between topological data analysis and tight spans. More precisely, they proved that the Vietoris-Rips complex of X is homotopy equivalent to a union of balls around X inside of its tight span. In this talk, I will give a survey of the properties of the tight span, including the aforementioned theorem and some applications to topological data analysis.
03/04:
03/11: Spring break
03/18: Ben Lowe
Title: Minimal submanifolds, higher expanders, and waists of locally symmetric spaces
Abstract: Gromov initiated a program to prove statements of the following form: Suppose we are given two simplicial complexes X and Y, where X is “complicated” and Y is lower dimensional. Then any map f: X-> Y must have at least one ”complicated” fiber. In this talk I will describe various results of this kind for compact locally symmetric spaces, that are proved by bringing new tools into the picture from minimal surface theory and representation theory. Much of the talk will be focused on octonionic hyperbolic manifolds, the case where our approach seems to work best. Time permitting I will also discuss some applications to systolic geometry, global fixed point statements for actions of lattices on contractible CAT(0) simplicial complexes, and/or non-abelian higher expansion and branched cover stability. Based on joint work with Mikolaj Fraczyk.
03/25: Brandon Doherty
Title: The comical model structures
Abstract: We discuss the comical model structures on the category of marked cubical sets, a family of models for (infinity,n)-categories studied in joint work with Kapulkin and Maehara, building on previous work of Campion-Kapulkin-Maehara. In particular, we focus on the Quillen equivalence between the comical model structures and the complicial model structures on marked simplicial sets via the marked triangulation functor and its right adjoint.
04/01: Sam Ballas
Title: Applications of nonstandard analysis in geometry
Abstract: Non standard analysis is a version of analysis where the reals are replaced by the larger non-archimedian field of hyperreals, which contain infinitesimally small elements. In this talk I will introduce the hyperreals and discuss how they give a useful language for describing various degeneration phenomena in geometry.
04/08: Sam Ballas
Title: Applications of nonstandard analysis in geometry
Abstract: Non standard analysis is a version of analysis where the reals are replaced by the larger non-archimedian field of hyperreals, which contain infinitesimally small elements. In this talk I will introduce the hyperreals and discuss how they give a useful language for describing various degeneration phenomena in geometry.
04/15:
04/22:
04/29: Exam week
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Fall 2024
08/27: Organizational meeting
09/03: Thang Nguyen
Title: Rigidity of convex subsets in symmetric spaces
Abstract: Symmetric spaces are manifolds with many rigidity phenomena. In this talk, I will present a phenomenon that Riemannian metric on convex subsets of symmetric spaces is unique once we keep either the same upper bound or lower bound on curvature as of the symmetric metric. Ideas of the work come from classical comparison theorems of Rauch and of Toponogov together with Berger's theorem for Blaschke conjecture. This is from a joint work with C. Connell, M. Islam, and R. Spatzier.
09/10: Sergio Fenley
Title: Existence of quasigeodesic Anosov flows in hyperbolic 3-manifolds
Abstract: A quasigeodesic in a manifold is a curve so that when lifted to the universal cover is uniformly efficient up to a bounded multiplicative and added error in measuring length. A flow is quasigeodesic if all flow lines are quasigeodesics. We prove that an Anosov flow in a closed hyperbolic manifold is quasigeodesic if and only if it is not R-covered. Here R-covered means that the stable 2-dim foliation of the flow, lifts to a foliation in the universal cover whose leaf space is homeomorphic to the real numbers. There are many examples of quasigeodesic Anosov flows in closed hyperbolic 3-manifolds. There are consequences for the continuous extension property of Anosov foliations, and the existence of group invariant Peano curves associated with Anosov flows.
09/17: Sam Ballas
Title: Classical and Exotic Dehn Filling
Abstract: Dehn filling is a topological operation where one starts with a 3-manifold, M, with torus boundary and builds a closed manifold by gluing a solid torus to it along their respective boundaries. There are infinitely many ways to do this gluing and so the construction results in infinitely many distinct manifolds. A great insight of Thurston was that if the interior of M admits a hyperbolic structure then almost all the Dehn fillings on M will admit hyperbolic structures. In this pair of talks I will explain some of the ideas behind Thurston’s construction in the hyperbolic setting and then talk about more recent work with Danciger, Lee, and Marquis where we extend these ideas to construct new examples of closed convex projective 3-manifolds. No background in hyperbolic projective geometry will be assumed, so these talks should hopefully be quite accessible.
09/24: Sam Ballas
Title: Classical and Exotic Dehn Filling (rescheduled)
Abstract: Dehn filling is a topological operation where one starts with a 3-manifold, M, with torus boundary and builds a closed manifold by gluing a solid torus to it along their respective boundaries. There are infinitely many ways to do this gluing and so the construction results in infinitely many distinct manifolds. A great insight of Thurston was that if the interior of M admits a hyperbolic structure then almost all the Dehn fillings on M will admit hyperbolic structures. In this pair of talks I will explain some of the ideas behind Thurston’s construction in the hyperbolic setting and then talk about more recent work with Danciger, Lee, and Marquis where we extend these ideas to construct new examples of closed convex projective 3-manifolds. No background in hyperbolic projective geometry will be assumed, so these talks should hopefully be quite accessible.
10/03 (Thursday) at 2 pm, room LOV 305: Kurt Vinhage (University of Utah)
Title: Entropy in low complexity
Abstract: Entropy is the first and arguably most important invariant of dynamical systems when studying classification problems. We will review two variants of the usual entropy adapted for systems will lower complexity, one of which is an invariant of conjugacy up to controlled time change. We will then consider some systems for which the entropy variant has been computed, and some applications of those computations. Partially joint with M. Cheng, A. Kanigowski, C. Ospina, D. Wei and Y. Zhai.
10/08: Ali Alp Uzman (University of Utah)
Title: : Rigidity of Smooth Actions of R^k
Abstract: To any smooth action of R^k one can associate a hyperspace arrangement given by the kernels of Lyapunov exponents. I will discuss recent work as well as work in progress on actions with some hyperbolicity whose arrangements that are opposites to each other. For the general position case we have global nonuniform and uniform smooth rigidity. For the product (hence symplectic) case we have fragility of rank one factors under generic time changes (joint work with Kurt Vinhage).
10/15: Mario Gomez Flores
Title: Vietoris-Rips complexes of totally split-decomposable spaces
Abstract: Split-metric decompositions are an important tool in the theory of phylogenetics, particularly because of the link between the tight span and the class of totally decomposable spaces, i.e. a generalization of metric trees whose decomposition does not have a “prime” component. The connection with tight spans has been studied at least since the introduction of split-metric decompositions by Bandelt and Dress in 1992 and culminated with the characterization of the polytopal structure of the tight span of a totally decomposable metric by Huber, Koolen, and Moulton in 2018. We use this connection, along with recent results on the Vietoris-Rips complex of the circle and the connection between tight spans and Vietoris-Rips complexes, to characterize the homotopy type of the Vietoris-Rips complex of a large class of totally decomposable spaces.
10/22: Ferhat Karabatman (cancelled)
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10/29:
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11/05: Sam Ballas
Title: Classical and Exotic Dehn Filling
Abstract: Dehn filling is a topological operation where one starts with a 3-manifold, M, with torus boundary and builds a closed manifold by gluing a solid torus to it along their respective boundaries. There are infinitely many ways to do this gluing and so the construction results in infinitely many distinct manifolds. A great insight of Thurston was that if the interior of M admits a hyperbolic structure then almost all the Dehn fillings on M will admit hyperbolic structures. In this pair of talks I will explain some of the ideas behind Thurston’s construction in the hyperbolic setting and then talk about more recent work with Danciger, Lee, and Marquis where we extend these ideas to construct new examples of closed convex projective 3-manifolds. No background in hyperbolic projective geometry will be assumed, so these talks should hopefully be quite accessible.
11/12: Brandon Doherty
Title: Comparison of cubical sets with and without symmetries
Abstract: We discuss an adjoint triple of functors which defines a comparison between categories of cubical sets on which model structures for (infinity,1)-categories have been established, and cubical sets having additional structure which captures the natural symmetry of cubical shapes. This result allows us to prove a variety of results about previously established model structures for (infinity,1)-categories on cubical sets, including their compatibility with the cartesian product, and to establish new model structures for (infinity,1)-categories on symmetric cubical sets. arXiv:2409.13842.
11/19:
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11/26: Thanksgiving week
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12/03: Ferhat Karabatman
Title: Positive Transvections in Symplectic Vector Spaces
Abstract: This talk will provide an introductory overview of symplectic vector spaces, emphasizing their automorphisms, known as symplectomorphisms. We will focus on the elementary generators of these automorphisms, called transvections. A special category of these, called positive transvections—generated by individual vectors—will be examined in detail. The discussion will include their structural properties and applications in mathematical contexts, offering insights into their significance within the study of symplectic vector spaces. This presentation aims to serve as a concise introduction to these fundamental concepts.
12/10: Exam week at FSU
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