# 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.

## February 28: TBA

Speaker: Manak Singh

Abstract: TBA

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

Title: TBA

Abstract: TBA

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

Title: TBA

Abstract: TBA

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