Geometry and Topology

Please contact Saul Schleimer or Robert Kropholler if you would like to speak or to suggest a speaker. A picture of a cat that is made out of cubes

We will also attempt to maintain an up-to-date listing at researchseminars.org.

The seminar will be hybrid, and will be run weekly. The talk is in B3.02 Zeeman Building on Thursdays, starting at 14:05. We will open and close the Zoom session on the hour. All of the talks will be streamed at this link.

Click on a title to view the abstract!

Upcoming Seminars

15 May 2025 at 13:30 in MB0.07 (Part of the Tropical Geometry and its Applications meeting)
Speaker: Hannah Markwig (University of Tübingen)
Title: Tropical Curve Counting

Abstract: In enumerative geometry, we fix geometric objects and conditions and count how many objects satisfy the conditions. For example, there are 2 plane conics passing through 4 points and tangent to a given line. Tropical geometry can be viewed as a degenerate version of algebraic geometry and has proved to be a successful tool for enumerative problems. We review tropical curve counting problems. In particular, we show how tropical methods can be applied for quadratically enriched counts, which can be viewed as generalizations that allow results over any ground field.

29 May 2025 at 13:30 in B3.02
Speaker: Daniele Turchetti (University of Durham)
Title: Non-Archimedean limits of families of Schottky groups

Abstract: Non-Archimedean analytic geometry provides a powerful framework for understanding families of geometric objects defined over the complex numbers. This perspective proves particularly fruitful when studying degenerations, compactifications, and limiting behaviors that are otherwise elusive in the Archimedean setting.

This talk aims to provide a gentle introduction to these techniques, with a particular focus on the case of Schottky spaces and moduli of curves, where recent work has highlighted deep connections between classical and non-Archimedean geometries. The final aim is to present joint work with Jérôme Poineau, in which we construct Schottky spaces over rings of integers of number fields and its connections with the outer space in geometric group theory. Finally, I will explain how these ideas have been used by Dang–Mehmeti and Li–Matheus–Pan–Tao to analyze the limiting behavior of Schottky groups and their associated spectra.

Past Seminars

08 May 2025 at 13:30 in B3.02
Speaker: Anna Felikson (University of Durham)
Title: Hyperbolic geometry of friezes

Abstract: Frieze patterns were introduced by Coxeter in the 1970s who, with Conway, established a correspondence between frieze patterns and triangulated polygons. It turned out later that this object is very rich in connections with different fields in mathematics. We use hyperbolic geometry to provide a classification of positive integral friezes on marked bordered surfaces. This is a joint work with Pavel Tumarkin.

20 March 2025 at 13:30 in TBC
Speaker: Macarena Arenas (University of Cambridge)
Title: Taut smoothings and shortest geodesics

Abstract: In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between a smoothing, which is a type of surgery on a curve that resolves a self-intersection, and k-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay.

06 March 2025 at 13:30 in B3.02
Speaker: Ismael Morales (University of Oxford)
Title: Fixed points, splittings and division rings

Abstract: Let G be a free group of rank N, let f be an automorphism of G and let Fix(f) be the corresponding subgroup of fixed points. Bestvina and Handel showed that the rank of Fix(f) is at most N, for which they developed the theory of train track maps on free groups. Different arguments were provided later on by Sela, Paulin and Gaboriau-Levitt-Lustig. In this talk, we present a new proof which involves the Linnell division ring of G. We also discuss how our approach relates to previous ones and how it gives new insight into variations of the problem.

20 February 2025 at 13:30 in B3.02
Speaker: Rachael Boyd (University of Glasgow)
Title: TBA

Abstract: TBA

05 December 2024 at 13:30 in B3.02
Speaker: Peter Patzt (University of Oklahoma)
Title: Unstable cohomology of SL_n Z and Hopf algebras

Abstract: The cohomology of SL_n Z has many connections to geometry and number theory and is largely unknown. In this talk, I will give a survey about what is known about it. In particular, I will include newly found unstable classes which come from a Hopf algebra structure. This talk is on joint work with Avner Ash and Jeremy Miller.

28 November 2024 at 13:30 in B3.02
Speaker: Simon Machado (ETH Zurich)
Title: Approximate lattices: structure and beyond

Abstract: Approximate lattices are aperiodic generalisations of lattices in locally compact groups. Yves Meyer first introduced them in abelian groups before studying them as mathematical models for quasi-crystals. Since then, their structure has been thoroughly investigated in both abelian and non-abelian settings. The primary motivation behind this research was to extend Meyer’s foundational theorem to non-abelian locally compact groups. This generalisation has now been established, and I will discuss the resulting structure theory. I will highlight certain concepts, including a notion of cohomology that lies between group cohomology and bounded cohomology, which plays a significant role in their study. Additionally, I will formulate open problems and conjectures related to approximate lattices.

21 November 2024 at 13:30 in B3.02
Speaker: Harry Petyt (University of Oxford)
Title: Obstructions to cubulation

Abstract: One can get a lot of information about a group by getting it to act geometrically on a CAT(0) cube complex. When this is possible there is a standard framework for trying to find the action, known as Sageev's construction. On the other hand, whilst most groups will not admit such actions, there is a real lack of ways to actually rule out the possibility that they exist. This talk will discuss joint work with Zach Munro, where we give a geometric obstruction to the possibility of cubulating groups.

14 November 2024 at 13:30 in B3.02
Speaker: Mikhail Hlushchanka (University of Amsterdam)
Title: Canonical decomposition of rational maps

Abstract: There are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics (which include celebrated results by Bill Thurston). The goal of this talk is to introduce a powerful decomposition of rational maps based on the topological structure of their Julia sets. Namely, we will discuss the following result: every postcritically-finite rational map with non-empty Fatou set can be canonically decomposed into crochet maps (these have very "thinly connected" Julia sets”) and Sierpinski carpet maps (these have very "heavily connected" Julia sets). If time permits, I will discuss applications of this result in various aspects of geometric group theory. Based on a joint work with Dima Dudko and Dierk Schleicher.

07 November 2024 at 13:30 in B3.02
Speaker: Sami Douba (IHES)
Title: Zariski closures of linear reflection groups

Abstract: We show that linear reflection groups in the sense of Vinberg are often Zariski dense in PGL(n). Among the applications are examples of low-dimensional closed hyperbolic manifolds whose fundamental groups virtually embed as Zariski-dense subgroups of SL(n,Z), as well as some one-ended Zariski-dense subgroups of SL(n,Z) that are finitely generated but infinitely presented, for all sufficiently large n. This is joint work with Jacques Audibert, Gye-Seon Lee, and Ludovic Marquis.

31 October 2024 at 13:30 in B3.02
Speaker: Stefanie Zbinden (Heriot-Watt University)
Title: Morse directions in classical small cancellation groups

Abstract: Morse geodesics are geodesics that capture the hyperbolic-like features of not necessarily hyperbolic spaces. They were studied in order to generalize proofs about hyperbolic groups. However, it quickly became clear that having a Morse geodesic is not enough to exclude various types of pathological behaviours, which makes many genearlizations impossible. Luckily, it turns out that having slightly stronger assumptions on the group, such as having a WPD element or being "Morse-local-to-global" makes certain pathologies impossible. In this talk, we explore how those stronger assumptions relate to each other in the case of small cancellation groups.

17 October 2024 at 13:30 in B3.02
Speaker: Mireille Soergel (MPIM Leipzig)
Title: Dyer groups: Coxeter groups, right-angled Artin groups and more...

Abstract: Dyer groups are a family encompassing both Coxeter groups and right-angled Artin groups. Among many common properties, these two families admit the same solution to the word problem. Each of these two classes of groups also have natural piecewise Euclidean CAT(0) spaces associated to them. In this talk I will introduce Dyer groups, give some of their properties.

10 October 2024 at 13:30 in B3.02
Speaker: Shaked Bader (University of Oxford)
Title: Hyperbolic subgroups of type FP_2(Ring)

Abstract: In 1996 Gersten proved that if G is a word hyperbolic group of cohomological dimension 2 and H is a subgroup of type FP_2, then H is hyperbolic as well. In this talk, I will present a joint work with Robert Kropholler and Vlad Vankov generalising this result to show that the same is true if G is only assumed to have cohomological dimension 2 over some ring R and H is of type FP_2(R).

03 October 2024 at 13:30 in B3.02
Speaker: Saul Schleimer (University of Warwick)
Title: Solving the word problem in the mapping class group in quasi-linear time

Abstract: Mapping class groups of surfaces are of fundamental importance in dynamics, geometric group theory, and low-dimensional topology. The word problem for groups in general, the definition of the mapping class group, its finite generation by twists, and the solution to its word problem were all set out by Dehn [1911, 1922, 1938]. Some of this material was rediscovered by Lickorish [1960's] and then by Thurston [1970-80's] -- they gave important applications of the mapping class group to the topology and geometry of three-manifolds. In the past fifty years, various mathematicians (including Penner, Mosher, Hamidi-Tehrani, Dylan Thurston, Dynnikov) have given solutions to the word problem in the mapping class group, using a variety of techniques. All of these algorithms are quadratic-time. We give an algorithm requiring only O(n log^3(n)) time. We do this by combining Dynnikov's approach to curves on surfaces, M\"oller's version of the half-GCD algorithm, and a delicate error analysis in interval arithmetic. This is joint work with Mark Bell.