Past Topology Seminar

23 November 2020
15:45
Luciana Bonatto
Abstract

Operads are tools to encode operations satisfying algebro-homotopic relations. They have proved to be extremely useful tools, for instance for detecting spaces that are iterated loop spaces. However, in many natural examples, composition of operations is only associative up to homotopy and operads are too strict to captured these phenomena. This leads to the notion of infinity operads. While they are a well-established tool, there are few examples of infinity operads in the literature that are not the nerve of an actual operad. I will introduce new topological operad of bracketed trees that can be used to identify and construct natural examples of infinity operads. The key example for this talk will be the normalised cacti model for genus 0 surfaces.

Glueing surfaces along their boundaries defines composition laws that have been used to construct topological field theories and to compute the homology of the moduli space of Riemann surfaces. Normalised cacti are a graphical model for the moduli space of genus 0 oriented surfaces. They are endowed with a composition that corresponds to glueing surfaces along their boundaries, but this composition is not associative. By using the operad of bracketed trees, I will show that this operation is associative up to all higher homotopies and hence that normalised cacti form an infinity operad.

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16 November 2020
15:45
Abstract

We study a group theoretic analog of Dehn fillings of 3-manifolds and derive a spectral sequence to compute the cohomology of Dehn fillings of hyperbolically embedded subgroups. As applications, we generalize the results of Dahmani-Guirardel-Osin and Hull on SQ-universality and common quotients of acylindrically hyperbolic groups by adding cohomological finiteness conditions. This is a joint work with Nansen Petrosyan.

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9 November 2020
15:45
Abstract

Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of locally finite affine A_n buildings. From this data, we will show how to construct new fiber functors on the category of tilting modules for SL(n+1) in characteristic p (related to order of the projective geometry) using the web calculus of Cautis, Kamnitzer, Morrison and Brundan, Entova-Aizenbud, Etingof, Ostrik.

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2 November 2020
15:45
Abstract

The main result is the existence of smooth, properly embedded 3-discs in S¹ × D³ that are not smoothly isotopic to {1} × D³. We describe a 2-variable Laurent polynomial invariant of 3-discs in S¹ × D³. This allows us to show that, when taken up to isotopy, such 3-discs form an abelian group of infinite rank. Joint work with David Gabai.

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26 October 2020
15:45
Abstract

I will discuss recent progress on the study of homological duality properties of complex algebraic manifolds, with a view towards the projective Singer-Hopf conjecture. (Joint work with Y. Liu and B. Wang.)

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19 October 2020
16:00
Michael Freedman
Abstract

In Joint work with Modj Shokrian-Zini we study (numerically) our proposal that interacting physics can arise from single particle quantum Mechanics through spontaneous symmetry breaking SSB. The staring point is the claim the difference between single and many particle physics amounts to the probability distribution on the space of Hamiltonians. Hamiltonians for interacting systems seem to know about some local, say qubit, structure, on the Hilbert space, whereas typical QM systems need not have such internal structure. I will discuss how the former might arise from the latter in a toy model. This story is intended as a “prequel” to the decades old reductionist story in which low energy standard model physics is supposed to arise from something quite different at high energy. We ask the question: Can interacting physics itself can arise from something simpler.

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12 October 2020
15:45
Vlad Marković
Abstract

I will discuss connections between ambient geometry of Moduli spaces and Teichmuller dynamics. This includes the recent resolution of the Siu's conjecture about convexity of Teichmuller spaces, and the (conjectural) topological description of the Caratheodory metric on Moduli spaces of Riemann surfaces.

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22 June 2020
15:45
Yair Minsky
Abstract

There is a well-known correspondence between Weil-Petersson geodesic loops in the moduli space of a surface S and hyperbolic 3-manifolds fibering over the circle with fibre S. Much is unknown, however, about the detailed relationship between geometric features of the loops and those of the 3-manifolds.

In work with Leininger-Souto-Taylor we study the relation between WP length and 3-manifold volume, when the length (suitably normalized) is bounded and the fiber topology is unbounded. We obtain a WP analogue of a theorem proved by Farb-Leininger-Margalit for the Teichmuller metric. In work with Modami, we fix the fiber topology and study connections between the thick-thin decomposition of a geodesic loop and that of the corresponding 3-manifold. While these decompositions are often in direct correspondence, we exhibit examples where the correspondence breaks down. This leaves the full conjectural picture somewhat mysterious, and raises many questions. 

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15 June 2020
15:45
Severin Bunk
Abstract

In this talk I will present results from an ongoing joint research  program with Konrad Waldorf. Its main goal is to understand the  relation between gerbes on a manifold M and open-closed smooth field  theories on M. Gerbes can be viewed as categorified line bundles, and  we will see how gerbes with connections on M and their sections give  rise to smooth open-closed field theories on M. If time permits, we  will see that the field theories arising in this way have several characteristic properties, such as invariance under thin homotopies,  and that they carry positive reflection structures. From a physical  perspective, ourconstruction formalises the WZW amplitude as part of  a smooth bordism-type field theory.

8 June 2020
15:45
Abstract

We study the countable set of rates of growth of a hyperbolic 
group with respect to all its finite generating sets. We prove that the 
set is well-ordered, and that every real number can be the rate of growth 
of at most finitely many generating sets up to automorphism of the group.

We prove that the ordinal of the set of rates of growth is at least $ω^ω$, 
and in case the group is a limit group (e.g., free and surface groups), it 
is $ω^ω$.

We further study the rates of growth of all the finitely generated 
subgroups of a hyperbolic group with respect to all their finite 
generating sets. This set is proved to be well-ordered as well, and every 
real number can be the rate of growth of at most finitely many isomorphism 
classes of finite generating sets of subgroups of a given hyperbolic 
group. Finally, we strengthen our results to include rates of growth of 
all the finite generating sets of all the subsemigroups of a hyperbolic 
group.

Joint work with Koji Fujiwara.

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