### Fall 2020

Wednesday October 7, 12:00-12:50, Zoom

Pierre Py (Strasbourg) 🔗

Title: Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces.

Abstract: A Kodaira fibration is a compact complex surface endowed with a holomorphic submersion onto a Riemann surface, which has connected fibers and is not isotrival. Using a classical construction due to Atiyah and Kodaira, we prove that the mapping class group of a once punctured genus g surface embeds into the mapping class group of a closed surface of larger genus. We also discuss questions related to the existence of CAT(0) metrics on Kodaira fibrations or to the number of distinct fiberings of such complex surfaces. This is based on a joint work with Claudio Llosa Isenrich (arXiv:2001.03694).

Wednesday October 21, 12:00-12:50, Zoom

Matthew Stover (Temple) 🔗

Title: Congruence RFRS towers

Abstract: Ian Agol introduced the concept of a RFRS (residually finite rational solvable) tower, a purely group theoretic condition, in showing certain hyperbolic 3-manifolds virtually fiber over the circle. I will talk about joint work with Agol where we develop general tools one can use to find RFRS towers consisting of congruence subgroups of certain arithmetically defined groups. We will then apply these tools to show that many arithmetic hyperbolic 3-manifolds virtually fiber on a congruence subgroup and to answer a question of Friedl and Vidussi.

Wednesday October 28, 12:00-12:50, Zoom

Jacob Garcia (UCR)

Title: Some aspects of generalized covering spaces theory

Abstract: Covering space theory is a classical tool used to characterize the geometry and topology of spaces. It seeks to separate the main geometric features from certain algebraic properties. For each conjugacy class of a subgroup of the fundamental group, it supplies a corresponding covering of the underlying space and encodes the interplay between algebra and geometry via group actions. The full applicability of this theory is limited to spaces that are, in some sense, locally simple. However, many modern areas of mathematics, such as fractal geometry, deal with spaces of high local complexity. This has stimulated much recent research into generalizing covering space theory by weakening the covering requirement while maintaining most of the classical utility. This talk will focus on the relationships between generalized covering projections, fibrations with unique path lifting, separation properties of the fibers, and continuity of the monodromy. .

Wednesday November 4, 12:00-12:50, Zoom

Corey Bregman (University of Southern Maine) 🔗

Title: TBA

Abstract: TBA

Wednesday November 18, 12:00-12:50, Zoom

Anna Parlak (Warwick) 🔗

Title: TBA

Abstract: TBA

Wednesday November 25, 12:00-12:50, Zoom

Chris Gerig (Harvard) 🔗

Title: TBA

Abstract: TBA

Wednesday December 2, 12:00-12:50, Zoom

Asaf Hadari (University of Hawai'i at Manoa ) 🔗

Title: Mapping class groups in genus at least 3 do not virtually surject to the integers.

Abstract: Mapping class groups of surfaces of genus at least 3 are perfect, but their finite-index subgroups need not be - they can have non-trivial abelianizations. A well-known conjecture of Ivanov states that a finite-index subgroup of a mapping class group in genus at least 3 has finite abelianization. We will discuss a proof of this conjecture, which goes through an equivalent representation-theoretic form of the conjecture due to Putman and Wieland.

### Winter 2020

Wednesday March 11, 12:00-12:50, Skye Building 268

Jonathan Alcaraz (UCR)

Title: Some Group Theory of Surface Bundles

Abstract: Behind every great fiber bundle is an even greater short exact sequence of groups. In this talk, I will discuss the group theory behind surface bundles over surfaces and generalize to certain higher dimensional cases.

Wednesday March 4, 12:00-12:50, Skye Building 268

Sam Nelson (CMC) 🔗

Title: Quandle Module Quivers

Abstract: In this talk we introduce an enhancement of the quandle counting invariant using quandle modules and quandle coloring quivers. The new invariant enhances both previous enhancements and and categorifies the quandle module invariant. This is joint work with Scripps student Karma Istanbouli.

Wednesday February 26, 12:00-12:50, Skye Building 268

Ben Russell

Title: TBA

Abstract: TBA.

Wednesday February 19, 12:00-12:50, Skye Building 268

Stefano Vidussi (UCR)

Title: The invariants of Bieri-Neumann-Strebel VI

Abstract: I will discuss what is known for BNS invariants of group extensions, as well as some open problems.

Wednesday February 12, 12:00-12:50, Skye Building 268

Stefano Vidussi (UCR)

Title: The invariants of Bieri-Neumann-Strebel V

Abstract: I will discuss what is known for BNS invariants of group extensions, as well as some open problems.

Wednesday February 5, 12:00-12:50, Skye Building 268

Stefano Vidussi (UCR)

Title: The invariants of Bieri-Neumann-Strebel IV

Abstract: I will discuss some structural theorem on BNS invariants.

Wednesday January 29, 12:00-12:50, Skye Building 268

Title: On the topology of the space of Riemannian metrics satisfying positive curvature conditions.

Abstract: For a given smooth manifold, one can consider the space of all Riemannian metrics. Much of Riemannian geometry and geometric analysis can be phrased in terms of morphisms to and from this object. While the space of all metrics is itself contractible, in many situations we restrict to metrics satisfying a lower bound on their curvature. When we restrict our space of metrics in this way, the topology can become nontrivial, and it becomes an interesting problem to compute the homotopy and (co)homology of this space. I will give a survey of nontriviality results for the space of metrics with positive scalar curvature or positive Ricci curvature. Concluding with a description of how my own geometric work provides new examples of manifolds for which the space of metrics of positive Ricci curvature has nontrivial topology.

Wednesday January 22, 12:00-12:50, Skye Building 268

Stefano Vidussi (UCR)

Title: The invariants of Bieri-Neumann-Strebel III

Abstract: I will show some explicit computations of the BNS invariants.

Wednesday January 15, 12:00-12:50, Skye Building 268

Stefano Vidussi (UCR)

Title: The invariants of Bieri-Neumann-Strebel II

Abstract: I will discuss some preliminary material and the definition of the BNS invariants.

Wednesday January 8, 12:00-12:50, Skye Building 268

Stefano Vidussi (UCR)

Title: The invariants of Bieri-Neumann-Strebel I

Abstract: I will discuss the historical framework and some of the motivations behind the definition of the BNS invariants.

### Fall 2019

Wednesday December 4, 12:00-12:50, Skye Building 268

Ben Russell (UCR)

Title: A Menagerie of Loop Spaces .

Abstract: The space of based loops of a topological space is a well-studied object. Similar, but less well-understood is the space of free loops. The homotopy quotient of the free loop space by a circle action yields a third space which we will refer to as the "equivariant" loop space. This talk gives an overview of some properties and differences between the pointed loop space and its free counterpart with the goal of determining at least the first Betti number of the equivariant loop space of a specific space.

Wednesday November 27, 12:00-12:50, Skye Building 268

Stefano Vidussi (UCR)

Title: The BNS invariant of the fundamental group of a surface bundle over a surface.

Abstract: We will discuss some new results on the Bieri-Neumann-Strebel invariant of these groups, showing in particular that (with obvious exceptions) they algebraically fiber. As a corollary, we show that for "most" bundles these groups are not coherent.

Wednesday November 20, 12:00-12:50, Skye Building 268

Babak Modami (SUNY) 🔗

Title: Weil-Petersson geometry of the moduli space.

Abstract: The Weil-Petersson metric is a negatively curved Riemannian metric on the moduli space of Riemann surfaces with rich geometry and dynamics. In this talk we focus on the global geometry of the metric and the behavior of its geodesics. We show how deformation theory and combinatorial topology of Riemann surfaces together with various properties of the metric can be employed to construct geodesics with different behaviors. We also compare the behavior of Weil-Petersson geodesics with the thick-thin decomposition of hyperbolic 3-manifolds. This is in part joint with Yair Minsky.

Wednesday November 13, 12:00-12:50, Skye Building 268

Thomas Koberda (UVa) 🔗

Title: Arithmetic groups, thin groups, and commensurators.

Abstract: Arithmetic groups are an important class of lattices in Lie groups which are of interest from a dynamical, geometric, and number theoretic perspective. These groups were characterized among lattices in a purely intrinsically algebraic way, by a famous result of Margulis. I will survey some of the ideas surrounding arithmetic groups and Margulis' theorem, and then move on to a discussion of thin groups. Thin groups are certain discrete subgroups of Lie groups which occur naturally in many contexts in mathematics, from number theory and spectral theory to quantum computing. Thin groups have much less structure than lattices, though they seem to follow some organizational principles analogous to Margulis' theorem. I will survey some recent results in this direction.

Wednesday November 6, 12:00-12:50, Skye Building 268

Jonathan Alcaraz (UCR)

Title: Fibering Rigidity of the Atiyah-Kodaira Bundle.

Abstract: By its construction, the Atiyah-Kodaira Bundle naturally admits two distinct surface bundle structures over surfaces. In 2017, Lei Chen proved that these are in fact the only two bundle structures it admits. In this talk, I will discuss equivalence of bundle structures and a bit on Chen's proof and the machinery she used.

Wednesday October 30, 12:00-12:50, Skye Building 268

Helen Wong (CMC) 🔗

Title: Topological descriptions of proteins.

Abstract: Knotting in proteins was once considered exceedingly rare. However, systematic analyses of solved protein structures over the last two decades have demonstrated the existence of many deeply knotted proteins, and researchers now hypothesize that the knotting presents some functional or evolutionary advantage for those proteins. Unfortunately, little is known about how proteins fold into knotted configurations. In this talk, we approach this problem from a theoretical point of view, using topological techniques. In particular, based on computational and experimental evidence, we propose a new theoretical pathway for proteins to form knots. We then use topological techniques to compare the configurations obtained from the theoretical pathways with known configurations of actual proteins. This is joint work with Erica Flapan and Adam He.

Wednesday October 23, 12:00-12:50, Skye Building 268

Alex Pokorny (UCR)

Title: The Kauffman skein algebra of the torus.

Abstract: A skein module is defined as a set of linear combinations of framed links in a 3-manifold, subject to local skein relations. If the manifold is taken to be a thickened surface, then the skein module carries the structure of an associative algebra. A natural question to ask is: what sort of algebras can arise from this type of construction? In this talk, I will give a presentation of the Kauffman skein algebra of the torus and discuss some of the ideas involved in the proof.

Wednesday October 16, 12:00-12:50, Skye Building 268

Bin Sun (UCR)

Title: Acylindrical hyperbolicity of non-elementary convergence groups.

Abstract: The notion of an acylindrically hyperbolic group was introduced by Osin as a generalization of non-elementary hyperbolic and relative hyperbolic groups. Examples of acylindrically hyperbolic groups can be found in mapping class groups, outer automorphism groups of free groups, 3-manifold groups, etc. Interesting properties of acylindrically hyperbolic groups can be proved by applying techniques such as Monod-Shalom rigidity theory, group theoretic Dehn filling, and small cancellation theory. We showed that non-elementary convergence groups are acylindrically hyperbolic. This result opens the door for applications of the theory of acylindrically hyperbolic groups to non-elementary convergence groups. In addition, we recovered a result of Yang which says a finitely generated group whose Floyd boundary has at least 3 points is acylindrically hyperbolic.

Wednesday October 9, 12:00-12:50, Skye Building 268

Stefano Vidussi (UCR)

Title: Symplectic Calabi-Yau manifolds with a free circle action: a pre-modern approach.

Abstract: I will discuss how to classify SCY 4-manifolds with a free circle action using only results that predate the modern era (i.e. 2011).

### Spring 2019

Thursday June 6, 2:10-3:00, Skye Building 277

Stefano Vidussi (UCR)

Title: Knots and 4-manifolds

Abstract: I will talk about some relations - partly unveiled, partly conjectural - between knots in S3 and a class of simply-connected 4-manifolds. While these relations have been first studied about 20 years ago, interest was recently rekindled as S. Donaldson selected them for the "Solved and Unsolved Problems" article for the EMS Newsletter of March 2019, devoted to Topology.

Thursday May 30, 2:10-3:00, Skye Building 277

Caglar Uyanik (Yale) 🔗

Title: Dynamics on geodesic currents and atoroidal subgroups of Out(FN)

Abstract: Geodesic currents on surfaces are measure theoretic generalizations of closed curves on surfaces and they play an important role in the study of the Teichmüller spaces. I will talk about their analogs in the setting of free groups, and try to illustrate how the dynamics and geometry of the Out(FN) action reflects on the algebraic structure of Out(FN).

Thursday May 23, 2:10-3:00, Skye Building 277

Jonathan Alcaraz

Title: Constructing Atiyah-Kodaira Fibrations

Abstract: To exhibit that signature need not be multiplicative over general fiber-bundles, Atiyah constructed a certain nontrivial surface bundle over a surface. This and a similar construction by Kodaira in the late sixties have been studied extensively including recent results regarding the number of ways these bundles fiber by L. Chen (2017) and N. Salter and B. Tshishiku (2018). In this talk, I will discuss Atiyah's construction in detail and if time permits discuss this result of Chen.

Thursday May 16, 2:10-3:00, Skye Building 277

Ben Russel

Title: The Hausmann-Weinberger Invariant

Abstract: Hausmann and Weinberger defined an invariant of a finitely presented group by minimizing the Euler characteristic over the class of closed orientable 4-manifolds having G as fundamental group. In this talk we present Hausmann and Weinberger's basic inequalities for the invariant as well as results of Kirk and Livingston for free abelian groups. One can modify the invariant by minimizing over a smaller class of 4-manifolds, in particular over closed orientable symplectic 4-manifolds.

Thursday May 2, 2:10-3:00, Skye Building 277

Aaron Mazel-Gee (USC) 🔗

Title: The geometry of the cyclotomic trace.

Abstract: K-theory is a means of probing geometric objects by studying their vector bundles, i.e. parametrized families of vector spaces. Algebraic K-theory, the version applying to varieties and schemes, is a particularly deep and far-reaching invariant, but it is notoriously difficult to compute. The primary means of computing it is through its "cyclotomic trace" map K→TC to another theory called topological cyclic homology. However, despite the enormous computational success of these so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of the cyclotomic trace has remained mysterious. In this talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry. By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices. No prior knowledge of algebraic K-theory or derived algebraic geometry will be assumed. This represents joint work with David Ayala and Nick Rozenblyum.

Thursday April 25, 2:10-3:00, Skye Building 277

Peter Samuelson (UCR)

Title: Jones versus A

Abstract: The A-polynomial of a knot is defined using the SL2-character varieties of the knot complement and of the torus bounding the knot complement. The (colored) Jones polynomials of a knot are defined using the "quantum group" Uq(sl2). In this talk I'll define these and describe a (conjectural) relationship between them called the "AJ conjecture," and give some reasons to believe the conjecture using skein theory.

Thursday April 18, 2:10-3:00, Skye Building 277

Marco Marengon (UCLA) 🔗

Title: Strands algebras and Ozsváth-Szabó's Kauffman states functor

Abstract: Ozsváth and Szabó introduced in 2016 a knot invariant, which they announced to be isomorphic to the usual knot Floer homology. Their construction is reminiscent of bordered Floer homology: for example, their invariant is defined by tensoring bimodules over certain algebras. During the talk I will introduce a more geometric construction, closer in spirit to bordered sutured Floer homology, based on strands on a particular class of generalised arc diagrams. The resulting strands algebras are quasi-isomorphic to the Ozsváth-Szabó algebras, suggesting that Ozsváth and Szabó's theory may be part of a hypothetical generalisation of bordered sutured Floer homology. This is a joint work with Mike Willis and Andy Manion.

Thursday April 11, 2:10-3:00, Skye Building 277

Stefano Vidussi (UCR)

Title: Symplectic Calabi-Yau 4-manifolds.

Abstract: A symplectic manifold is called Calabi-Yau if its canonical class is trivial. I will discuss a few results and conjectures about this class of manifolds in dimension 4.

Thursday Apr 4, 2:10-3:00, Skye Building 277

Francis Bonahon (USC) 🔗

Title: How to multiply matrices? From hyperbolic geometry to quantum topology.

Abstract: We of course know how to multiply matrices when their entries commute with each other. However, extending constructions from classical geometry (such as 2- and 3-dimensional hyperbolic geometry) to quantum topology (such as the Jones polynomial of a knot, or the Kauffman bracket skein algebra of a surface) requires the consistent multiplication of matrices with non-commuting entries. I will explain how to do this in the context of the quantum group Uq(sl2).

### Winter 2019

Thursday Mar 7, 2:10-3:00, Skye Building 277

Wee Liang Gan (UCR)

Title: Homological stability and representation stability.

Abstract: I will give an introduction to and discuss some recent developments in representation stability.

Thursday Feb 28, 2:10-3:00, Skye Building 277

Ben Russell (UCR)

Title: Foliations and the Godbillon-Vey Class.

Abstract: A foliation is a decomposition of a manifold into injectively immersed submanifolds in a way that is locally modeled on the decomposition of Rn by (n-q)-planes. As such foliations reside in the intersection of geometric topology and dynamical systems. In this talk we introduce the basic definitions and machinery of foliations with an eye towards constructing the so-called Godbillon-Vey class of transversely orientable foliation of codimension 1, which Thurston described as measuring the "helical wobble" of the foliation.

Thursday Feb 21, 2:10-3:00, Skye Building 277

Jonathan Alcaraz (UCR)

Title: The Mapping Class Groups of Some Surfaces.

Abstract: In this talk I will define and compute some fun mapping class groups as done in Chapter 2 of "A Primer on Mapping Class Groups" by Benson Farb and Dan Margalit.

Thursday Feb 14, 2:10-3:00, Skye Building 277

Jonathan Alcaraz (UCR)

Title: An Introduction to Surfaces, Fiber Bundles, and Other Words.

Abstract: A first-year graduate student sitting in on a research-level topology seminar will inevitably hear a lot of words they don't understand. If such a graduate student is taking a first-year course in topology, however, some of those words are very approachable. In this talk, I intend to develop intuition for some of this vocabulary using definitions, discussion, examples and exercises.

Thursday Feb 7, 2:10-3:00, Skye Building 277

Lei Chen (Caltech) 🔗

Title: Methods in 2d dynamics and Nielsen realization problem.

Abstract: In this talk, I will discuss several methods in 2-dimensional dynamics including shadowing lemma, Ahlfors trick (describing rigidity of finite order homeomorphisms), Carathéodory's prime end theory, Thurston's stability theorem and Nielsen-Thurston classification theory etc. I will focus on discussing how to use dynamical method to solve different versions of the Nielsen realization problem about realizing subgroups of mapping class groups as a subgroup of homeomorphism groups.

Thursday Jan 31, 2:10-3:00, Skye Building 277

Matt Gibson (UCI)

Title: Symplectic Structures on an Open 4-Manifold with Non-Isomorphic Primitive Cohomology.

Abstract: In this talk we investigate applications of the primitive cohomology on a symplectic manifold, studied by Tseng, Tsai, and Yau. We begin by introducing the necessary background to study these groups and then turn to a famous example of a link complement worked on by McMullen, Taubes, and Vidussi. From this complement, one can construct a symplectic 4-fold fibering over the torus. We show that the dimension of the corresponding primitive cohomology groups can distinguish different fibration structures on this 4-fold, depending on the choice of symplectic form. Time permitting, we briefly introduce the $A_\infty$-structure on the underlying co-chain complex of the primitive cohomology groups and its possible role in the examples discussed. This talk is based on joint work with Li-Sheng Tseng and Stefano Vidussi.

Thursday Jan 24, 2:10-3:00, Skye Building 277

Stefano Vidussi (UCR)

Title: Some Smooth Surfaces with Several Symplectic Structures.

Abstract: Not sure. But the title alliterates beautifully.

Thursday Jan 17, 2:10-3:00, Skye Building 277

Dmitri Gekhtman (Caltech) 🔗

Title: Holomorphic retracts of Teichmüller space.

Abstract: The Teichmüller space of a closed surface carries a natural complex structure, whose analytic properties reflect the topology and geometry of the surface. In this talk, we discuss the problem of classifying the holomorphic retracts of Teichmüller space. Our approach hinges on the analysis of two dynamical flows - one in the moduli space of half-translation surfaces, and the other in the space of bounded holomorphic functions on the polydisk.

Thursday Jan 10, 2:10-3:00, Skye Building 277

Nur Saglam (UCR) 🔗

Title: Constructions of Lefschetz fibrations using cyclic group actions.

Abstract: We construct families of Lefschetz fibrations over S2 using finite order cyclic group actions on product manifolds Σgg for g > 0. We also obtain more families of Lefschetz fibrations by applying rational blow-down operation to these Lefschetz fibrations. This is joint work with Anar Akhmedov.