# Topology and Geometry Seminar

Winter 2024

Topic: Deformation Theory

Time: Wednesdays from 12:30-1:45 PM

Location: CL 251

The topic for our learning seminar this term is deformation theory. We will also have occasional invited research talks over Zoom.

Zoom Link: https://uregina-ca.zoom.us/j/97896109097?pwd=RkI2UkZsMlYyZTBzejhEY1R4RCt4Zz09

## Schedule of talks

## September 13: How does a Lie algebra encode a space?

Speaker: Francis Bischoff

Abstract: The fundamental principle of derived deformation theory is that a formal “space” (i.e. a formal moduli problem) can be encoded by a differential graded Lie algebra. The plan for our learning seminar this term will be to unpack and understand this statement. For the first talk of the seminar, I will give some motivation and outline the big picture. In particular, I will present a very illuminating example of the principle: how to encode the "derived" vanishing locus of a function into a Lie (or L-infinity) algebra using the Taylor expansion.

References: How does a Lie algebra encode a space (Part 1, Part 2)

## September 20: Classical deformation theory of algebras

Speaker: Allen Herman

Abstract: If A is an algebra over a field k, and R is an augmented commutative k-algebra, then an R-deformation of A is (intuitively) an R-algebra B whose underlying k-algebra structure is isomorphic to that of A. Two R-deformations of A are considered equivalent when there is an R-algebra isomorphism between the two that reduces to the expected isomorphism of their underlying k-algebra structures. In this talk we will see that in the cases R = k[[t]] (formal deformations) and R=k[t]/(t^2) (infinitesimal deformations), the equivalence classes of R-deformations of A can be understood using tools from Hochschild cohomology.

Reference: Deformation Theory (Lecture notes)

## September 27: Hochschild cohomology

Speaker: Martin Frankland

Abstract: In Allen's talk, we learned that the deformations of an algebra A can be classified by certain Hochschild cohomology classes. Before we delve into that classification, we must familiarize ourselves with Hochschild cohomology, which is the goal of this talk. We will look at a few definitions and examples.

Reference: Deformation Theory (Lecture notes)

## October 4: Catherine Ray (Universität Münster)

Title: Inverse Galois Theory as Thor's Hammer

Abstract: The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. We can explicitly pin down this squirming action by geometrically modeling it as coming from an action on a moduli space, which we construct using inverse Galois theory and some representation theory (a Hurwitz space). I will show you pretty pictures.

Recording: https://youtu.be/pVMGf6EC61A

## October 18: Ben Williams (University of British Columbia)

Title: Classifying involutions of Azumaya algebras

Abstract: This is joint work with Uriya First. Azumaya algebras are objects that are locally isomorphic to matrix algebras—over a topological space X, they are bundles of matrix algebras. If the base space X is endowed with a self-map of order 2 (which may be trivial) t:X->X, then a t-involution of an Azumaya algebra A over X is a map s: A ->A of order 2 that preserves addition, reverses multiplication, and is compatible with t. These involutions are analogues of transposition or hermitian conjugation of matrix algebras. I will explain a coarse classification of these involutions into types, depending on the base involution t, and produce some exotic examples.

## October 25: A DG Lie algebra controlling the deformation of algebras

Speaker: Martin Frankland

Abstract: We will discuss the link between Hochschild cohomology and deformations of an algebra A. We will see that the Hochschild cochains of A can be endowed with a differential graded (DG) Lie algebra structure. The Maurer-Cartan equation describes the solutions to the deformation problem. In this way, the DG Lie algebra can recover the cohomological classification.

Reference: Deformation Theory (Lecture notes)

## November 1: Elliot Cheung (University of British Columbia)

Title: Towards a discretization of Chern-Simons theory

Abstract: We will describe a discretization of Chern-Simons theory using Whitney forms. Derived moduli spaces are often described using L-infinity algebras and it is interesting to explore how a derived moduli space varies as we modify the 'governing L-infinity algebra' by a homotopy. In this example, we utilize the well-known Dupont homotopy operator to define a discretization of the infinite-dimensional DGLA controlling the moduli problem relevant to Chern-Simons theory. In doing so, we can describe an ( ind-) finite-dimensional model for a derived enhancement of the moduli space of flat connections on an oriented closed 3-manifold M equipped with a triangulation K_M. This derived moduli space has a -1-shifted symplectic structure which also comes with 'geometric quantization data'. This can be used to define a 3-manifold invariant, which can be viewed as a discretization of Witten's Chern-Simons partition function invariant for 3-manifolds.

Recording: https://youtu.be/qRJKkSE2woU

## November 8: Examples from geometry (part 1)

Speaker: Francis Bischoff

Abstract: In this seminar, we have seen that problems in deformation theory are often controlled by differential graded Lie algebras (dgla). For example, in Martin's talk, we saw how the deformations of an associative algebra are controlled by a dgla structure on the Hochschild cochain complex. In this talk and the next, will see an instance of this phenomenon coming from geometry: the theory of flat connections on a smooth manifold.

In part 1, we will cover the background material about connections on smooth manifolds.

## November 15: Examples from geometry (part 2)

Speaker: Francis Bischoff

Abstract: In this seminar, we have seen that problems in deformation theory are often controlled by differential graded Lie algebras (dgla). For example, in Martin's talk, we saw how the deformations of an associative algebra are controlled by a dgla structure on the Hochschild cochain complex. In this talk and the next, will see an instance of this phenomenon coming from geometry: the theory of flat connections on a smooth manifold.

In part 2, we will see that the deformations of flat connections on a smooth manifold are parametrized by the solutions to the Maurer-Cartan equation in a suitable dgla.

## November 22: From DG Lie algebras to deformation functors

Speaker: Martin Frankland

Abstract: So far, we learned that a deformation problem is controlled by a DG Lie algebra, and that solutions to the deformation problem correspond to Maurer-Cartan (MC) elements. However, different MC elements may yield equivalent deformations! The DG Lie algebra gives rise to a group called the gauge group, which acts on the space of MC elements. Two MC elements yield equivalent deformations if and only if they are related by the gauge action. We will illustrate this machinery in the case of deformations of algebras.

## November 29: Rational homotopy theory and a deformation-like moduli problem.

Speaker: Don Stanley

Abstract: We will introduce rational homotopy theory through the CDGA approach and discuss formal and non-formal spaces. We will then move on to a moduli problem related to the homotopy type of complements of polyhedra in closed manifolds.

## December 6: L-infinity algebras

Speaker: Manak Singh

Abstract: What are L-infinity algebras? What is their relevance to deformation theory? How are they an improvement over differential graded Lie algebras? These are the questions I will attempt to answer as we go over the basics of L-infinity algebras.

## January 17: Deformation quantization of Poisson manifolds

Speaker: Matt Alexander

Abstract: Before the winter break we saw a number of key structures of deformation theory: differential graded Lie algebras, the Maurer-Cartan equation and its gauge group, and L-infinity algebras. In this talk, we will see how such structures arise in the deformation theory of Poisson manifolds. We will motivate this theory from the perspective of physics and will see how it relates to the transition between the classical and quantum regimes.

## January 24: What is a formal moduli problem?

Speaker: Martin Frankland

Abstract: In Manak's talk, we learned what a deformation functor is. We will explore this notion more in depth and look at some examples. Deformation functors allow us to make the idea of a "formal moduli problem" more precise. Time permitting, we will discuss Schlessinger's theorem, giving conditions for a deformation functor to be well approximated by a pro-representable one.

## January 31: Deformation functors

Speaker: Martin Frankland

Abstract: In the previous talk, we got comfortable with the category of Artin local rings. In this second part, we will move on to functors of Artin rings, deformation functors, and some examples. Time permitting, we will discuss Schlessinger's theorem, giving conditions for a deformation functor to be well approximated by a pro-representable one.

## February 7: More deformation functors

Speaker: Martin Frankland

Abstract: In the previous talk, we saw a few examples of deformation functors. In this third part, we will discuss some properties of deformation functors and more examples. We will revisit examples coming from DG Lie algebras.

## February 14: DG Lie algebras and coalgebras in rational homotopy theory

Speaker: Martin Frankland

Abstract: A classic theorem of Quillen established an equivalence of homotopy theory between (certain) DG Lie algebras and DG coalgebras. Both are models for (simply-connected) rational homotopy theory. A generalization of Quillen's result by Hinich provided a link with deformation theory.

## March 6: Hari Rau-Murthy (University of Notre Dame)

Location: CL 251 (usual room)

Title: An HKR theorem for factorization homology, and a related project in character theory

Abstract: This talk will primarily be on a generalization of the so-called "HKR theorem". The classical HKR theorem concerns two different ways of generalizing the notion of differential forms on a ring; the theorem asserts they agree. I will give a generalization of this theorem for ring spectra which will allow us to recalculate the factorization homology/higher THH of rational KU.

Time permitting, I will give an advertisement for the ongoing work which motivated the above result (you, yes you! could be my collaborator). I am interested in using K-theory and higher THH to write down maps such as the character of a representation and the Chern character. Written versions of both of these projects are available on my website at hariraumurthy.github.io.

## March 13: Koszul duality of associative algebras

Speaker: Manak Singh

Abstract: A basic aim of homological algebra is to establish invariants for various algebraic objects. This is achieved by constructing resolutions for these objects. These resolutions can be intractable, however for quadratic algebras one obtains a simpler resolution. The aim of Koszul duality theory is to construct explicit quasi-free resolutions for quadratic algebras. We will explore some classic instances of this theory in action.

## March 20: Mahmoud Zeinalian (Lehman College, CUNY)

Title: The Chern character after Toledo-Tong and Green

Abstract: I will review some of the basic machinery used in formulating characteristic classes for coherent sheaves on complex manifolds. The main ideas go back to the fundamental work of Toledo and Tong in the 70s. A natural extension of their ideas leads to defining these invariants for higher stacks. I will showcase some of the main tools and concepts without methodically entering the subject of higher stacks, making the talk appealing to classical differential geometers. This is based on joint works with T. Tradler, M. Miller, C. Glass, and T. Hosgood.

## March 27: The Bar-Cobar construction for dg-algebras

Speaker: Manak Singh

Abstract: The Bar and Cobar functors pair up to form an adjunction between the category of dg-algebras and dg-coalgebras. This adjunction can be used as a tool to provide a quasi-free resolution for any dg-algebra. We work out such a resolution for the case of a quadratic dg-algebra.

## April 3: Olivia Borghi (University of Melbourne)

This talk is at 5 PM on Zoom.

Title: Commutativity in Higher Algebraic Objects

Abstract: A symmetric monoidal category is a category equipped with a monoidal product that is uniquely commutative up to isomorphism. In this way the iterated monoidal product has an action from the symmetric groups. We can generalize this notion by allowing actions from other permutative groups. Examples include braided monoidal categories, coboundary categories and ribbon braided monoidal categories. These generalized commutative monoidal categories find use in the representation theory of quantum groups (coboundary categories) and the study of TQFTs (ribbon braided monoidal categories).

In this talk I will explain we can generalize the definition of symmetric monoidal ∞-category and ∞-operad in the same manner allowing a more generic notion of G-monoidal ∞-category and ∞-G-operad.

Recording: https://youtu.be/y6A5aepWb-4

## April 16: Wojciech Chachólski (KTH Royal Institute of Technology)

This talk is on Tuesday, April 16 at 12:30 – 2:00 pm in Room CL 312

Title: Data, geometry and homology

Abstract: For a successful analysis a suitable representation of data by objects amenable for statistical methods is fundamental. There has been an explosion of applications in which homological representations of data played a significant role. I will present one such representation called stable rank and introduce various novel ways of using it to encode geometry, and then analyse, data. I will provide several illustrative examples of how to use stable ranks to find meaningful results.

Past Seminars: Details on the past seminars can be found at this website.