Past Topology Seminar

15 March 2021
15:45
Marc Lackenby
Abstract

I will outline a new algorithm for unknot recognition that runs in quasi-polynomial time. The input is a diagram of a knot with n crossings, and the running time is n^{O(log n)}. The algorithm uses hierarchies, normal surfaces and Heegaard splittings.

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8 March 2021
15:45
Abstract

In rational conformal field theory, the sewing and factorization properties are probably the most important properties that conformal blocks satisfy. For special examples such as Weiss-Zumino-Witten models and minimal models, these two properties were proved decades ago (assuming the genus is ≤1 for the sewing theorem). But for general (strongly) rational vertex operator algebras (VOAs), their proofs were finished only very recently. In this talk, I will first motivate the definition of conformal blocks and VOAs using Segal's picture of CFT. I will then explain the importance of Sewing and Factorization Theorem in the construction of full rational conformal field theory.

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1 March 2021
15:45
Sam Shepherd
Abstract

Studying quasi-isometries between groups is a major theme in geometric group theory. Of particular interest are the situations where the existence of a quasi-isometry between two groups implies that the groups are equivalent in a stronger algebraic sense, such as being commensurable. I will survey some results of this type, and then talk about recent work with Daniel Woodhouse where we prove quasi-isometric rigidity for certain graphs of virtually free groups, which include "generic" cyclic HNN extensions of free groups.

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22 February 2021
15:45
Akhil Matthew
Abstract

I will give an overview of the interactions between chromatic homotopy theory and the algebraic K-theory of ring spectra, especially around the subject of Ausoni-Rognes's principle of "chromatic redshift," and some of the recent advances in this field.

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15 February 2021
15:45
John Greenlees

Further Information: 

joint work with G.Stevenson and D.Benson

Abstract

For an ordinary commutative Noetherian ring R we would define the singularity category to be the quotient of the (derived category of) finitely generated modules modulo the (derived category of) fg projective modules [``the bounded derived category modulo compact objects’’]. For a ring spectrum like C^*(BG) (coefficients in a field of characteristic p) it is easy to define the module category and the compact objects, but finitely generated objects need a new definition. The talk will describe the definition and show that the singularity category is trivial exactly when G is p-nilpotent. We will go on to describe the singularity category for groups with cyclic Sylow p-subgroup.

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8 February 2021
15:45
Anna Parlak
Abstract

Veering triangulations are a special class of ideal triangulations with a rather mysterious combinatorial definition. Their importance follows from a deep connection with pseudo-Anosov flows on 3-manifolds. Recently Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. During the talk I will discuss how and why it is connected to the Alexander polynomial of the underlying manifold.  

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1 February 2021
15:45
Davide Spriano
Abstract

Hierarchically Hyperbolic Groups (HHGs) were introduced by Behrstock—Hagen—Sisto to provide a common framework to study several groups of interest in geometric group theory, and have been an object of great interest in the area ever since. The goal of the talk is to provide an introduction to the theory of HHGs and discuss the advantages of the unified approach that they provide. If time permits, we will conclude with applications to growth and asymptotic cones of groups.

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25 January 2021
15:45
Dawid Kielak
Abstract

I will introduce the Friedl-Tillmann polytope for one-relator groups, and then discuss how it can be generalised to the Friedl-Lück polytope, how it connects to the Thurston polytope, and how we can view it as a convenient source of intuition and ideas.

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18 January 2021
15:45
Oscar Randal-Williams
Abstract

I will discuss joint work with S. Galatius and A. Kupers in which we investigate the homology of general linear groups over a ring $A$ by considering the collection of all their classifying spaces as a graded $E_\infty$-algebra. I will first explain diverse results that we obtained in this investigation, which can be understood without reference to $E_\infty$-algebras but which seem unrelated to each other: I will then explain how the point of view of cellular $E_\infty$-algebras unites them.

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30 November 2020
15:45
Kasia Jankiewicz
Abstract

Artin groups are a family of groups generalizing braid groups. The Tits conjecture, which was proved by Crisp-Paris, states that squares of the standard generators generate an obvious right-angled Artin subgroup. In a joint work with Kevin Schreve, we consider a larger collection of elements, and conjecture that their sufficiently large powers generate an obvious right-angled Artin subgroup. In the case of the braid group, regarded as a mapping class group of a punctured disc, these elements correspond to Dehn twist around the loops enclosing multiple consecutive punctures. This alleged right-angled Artin group is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for some classes of Artin groups. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.

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