# Seminars in geometry/topology

The Topology Seminar Spring 2020 will be on Mondays 13:15 - 14:15 (or 14:00 - 15:00) in room 734, Sentralbygg 2.

## ???, 13:15 - 14:00, room 734, Sentralbygg 2

Markus Szymik (NTNU): A$_\infty$algebras and morphisms

Abstract:

This is the first half of a double feature on Homotopy Probability Theory after Drummond-Cole, Park, and Terilla.
In this half, I will explain the notions of A$_\infty$algebras and morphisms as needed to state the main result.

## ???, 14:15 - 15:00, room 734, Sentralbygg 2

Kurusch Ebrahimi-Fard (NTNU): Homotopy Probability Theory

Abstract:

This is the second half of the double feature on Homotopy Probability Theory after Drummond-Cole, Park, and Terilla.

## January 23, 10:15 - 11:15, room 734, Sentralbygg 2

Kathryn Hess (EPFL): Equivariant flavors of (topological) Hochschild homology

Abstract: Relative topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this talk I will present tools for computing relative THH and explain how to apply them to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum. I will describe in particular an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild of Green functors, first introduced by Blumberg, Gerhardt, Hill, and Lawson. (Joint work with Katharine Adamyk, Teena Gerhardt, Inbar Klang, and Hana Kong.)

## January 23, 14:15 - 15:15, room 734, Sentralbygg 2

Jose Perea (Michigan State University): Topological dimensionality reduction

Abstract: When dealing with complex high-dimensional data sets, several machine learning tasks rely on having appropriate low-dimensional representations. These reductions are often phrased in terms of preserving statistical or metric information and may fail to recover important topological features. We will describe in this talk several schemes to take advantage of the underlying topology of a data set, in order to produce informative low-dimensional coordinates.

## February 10, 13:15 - 14:15, room 734, Sentralbygg 2

Joachim Kock (Universitat Autònoma de Barcelona): Objective Combinatorics and Decomposition Spaces I

Abstract:

Coalgebras and Hopf algebras in combinatorics arise from the ability to decompose combinatorial structures. The theory of decomposition spaces (also called 2-Segal spaces) serves among other things to give an objective approach to combinatorial Hopf algebras. 'Objective' here means working with the combinatorial objects themselves, instead of with vector spaces derived from them. Compared to the classical theory of incidence coalgebras of posets, there are three generalisations involved: the passage from vector spaces to slice categories; the passage from sets to groupoids (and infinity-groupoids); the simplicial viewpoint which prompts the passage from categories to decomposition spaces.

The first lecture, after some introductory stuff, will outline the main points of objective linear algebra, where slice categories replace vector spaces, and spans and linear functors replace linear maps. One advantage of this abstraction is the ability to deal with infinite quantities. (While one should generally avoid infinite numbers, there is no reason to be afraid of an infinite set.) This leads to an neat interpretation of the duality between vector spaces and pro-finite-dimensional vector spaces, and a combinatorial interpretation of continuity.

## February 17, 13:15 - 14:15, room 734, Sentralbygg 2

Joachim Kock (Universitat Autònoma de Barcelona): Objective Combinatorics and Decomposition Spaces II

Abstract:

The second lecture will start with incidence coalgebras of posets, then generalise to Möbius categories, and see how these constructions look objectively. Then we shall look at some combinatorial Hopf algebras that cannot arise directly in this way. Analysing this from a simplicial viewpoint leads to the notion of decomposition spaces, which are simplicial objects subject to some conditions tailor-made to make the incidence coalgebra construction work. For the theory to work smoothly, it is necessary to upgrade from sets to groupoids and use homotopy methods. This prompts the further upgrade from groupoids to infinity-groupoids, and leads to surprising connections with other areas of mathematics (the theory of higher Segal spaces).

References: for the viewpoint of combinatorics taken here, see various papers of Gálvez-Kock-Tonks, starting perhaps with "Decomposition spaces in combinatorics". For an ampler perspective, see Dyckerhoff-Kapranov, "Higher Segal Spaces" (Springer Lecture Notes).