Topology and Geometry Seminar

Title: A (gentle) introduction to Poisson geometry, Part 1. 

Abstract: I will give an introduction to Poisson geometry, starting with its role in the dynamics of physical systems and its relationship to symplectic geometry. Through the use of illustrative examples, I will highlight some of the basic features of Poisson manifolds, such as their symplectic leaves. 

Title: A (gentle) introduction to Poisson geometry, Part 2. 

Abstract:  This is the second part of an introduction to Poisson geometry.  In this part, I will discuss the geometry of Poisson manifolds, such as their symplectic leaves, through the use of illustrative examples. I will also briefly discuss the quantization of Poisson manifolds. 

Title: Introduction to deformation quantization

Abstract: A deformation quantization of a manifold is a noncommutative deformation of its algebra of functions; the idea originated in physics, as a way of relating the classical and quantum descriptions of mechanical systems.  At leading order in the deformation parameter, a deformation quantization gives rise to a Poisson structure on the manifold.  A deep theorem of Kontsevich states that this gives an equivalence between Poisson structures and noncommutative deformations; it thus provides a direct link between geometry and noncommutative algebra.  In fact, he gives an explicit formula for the quantization of any Poisson bracket, via integrals of differential forms associated to graphs (a Feynman expansion).  I will give an introduction to deformation quantization, with several examples, leading to a sketch of Kontsevich's proof.

Title:  Dehn Surgery: From Torus Knots to 3-manifold.

Abstract: I will introduce Dehn surgery via torus knots and links, show how 3-manifolds arise from surgeries, note basic surgery equivalences, and sketch homology computations of the resulting manifolds. I will also include surgeries on torus knots that yield lens spaces.

Note: This talk is being held on a Thursday, rather than Tuesday as usual. 

Title: Geometric Approaches for Functional Data

Abstract: Functional data such as time series and images are ubiquitous in applications and are equipped with natural concatenation operations. For machine learning applications, it is often helpful to build structured representations of such data which preserve the underlying algebraic structure, and satisfy universality (allows us to approximate functions) and characteristicness (allows us to characterize probability measures). By viewing such data as paths and surfaces, we discuss how (higher) parallel transport can be used to provide such representations, and discuss their universal and characteristic properties. Based on joint work with Harald Oberhauser (Oxford).

Title: Functor calculus and vector bundle enumerations.

Abstract:  Pick your favorite manifold M and a positive integer r: how many rank r (topological) complex vector bundles are there over M up to isomorphism? While the question is accessible via K-theory in the stable range, unstably such bundles become much harder to compute and to detect.

In this talk, we will demonstrate how the orthogonal/unitary calculus of Weiss — a version of functor calculi — can be applied to study unstable topological vector bundles. We will present several counting results for complex vector bundles over CPn in the metastable range, and, time permitting, introduce an equivariant version of the theory along with some potential applications in equivariant geometry. This talk contains joint work with Hood Chatham and Morgan Opie, and with Prasit Bhattacharya.

Title: Orientation orders and chromatic defects.

Abstract: A ring spectrum (a.k.a. a multiplicative cohomology theory) E is complex-orientable if every complex vector bundle admits a Thom class in E. While many familiar spectra are complex-orientable, notable examples fail to be so — such as real K-theory and topological modular forms. To measure this failure precisely, two competing ideas have been proposed: the complex orientation order and the chromatic defect. A natural question is whether the two notions are related. In this talk, we will present examples that suggest a negative answer. This is joint work with Prasit Bhattacharya and Christian Carrick.

Note: This talk is being held on a Thursday, rather than Tuesday as usual. 

Title: An introduction to Toric Topology

Abstract: In this talk, I will present a crash course in Toric Topology. We will start by discussing basic combinatorial structures (such as polytopes, fans, and simplicial complexes) and their role in studying torus actions. After some examples we will move on to introducing various algebraic objects that arise from the combinatorics of these spaces and pave the way to start discussing polyhedral products if time permits.

Note: This talk is being held on a Thursday, rather than Tuesday as usual. 

Title: An introduction to Toric Topology, part 2

Abstract: We continue last week’s talk by introducing two more combinatorial structures: simplicial and cubical complexes, which we’ll introduce and present some examples and constructions that arise from them. These complexes let us construct a family of Toric spaces we call moment-angle complexes. To study the homotopical features of these spaces we’ll introduce two algebraic objects which arise from the underlying simplicial complex: The Tor algebra and the Stanley-Reisner ring. Both encode the homological features of a MAC; we’ll present this equivalence, a proof sketch and some examples. If time permits, I’ll introduce polyhedral products and talk about some of the current work being done, such as the de-suspension problem and the triviality of the product.

Note: This talk is being held on a Thursday, rather than Tuesday as usual. 

Title: Lie 2-algebras and higher quantization

Abstract: Bundle gerbes on a manifold M provide geometric realizations of degree 3 cohomology classes of M. The space of infinitesimal symmetries of a bundle gerbe naturally carries the structure of a Lie 2-algebra, a deformation of the notion of Lie algebra where the Jacobi identity only holds ‘up to homotopy.’ This Lie 2-algebra of symmetries is related to other Lie 2-algebras associated to a closed differential 3-form, when the 3-form encodes the ‘higher curvature’ of the bundle gerbe—namely, the Poisson Lie 2-algebra of observables, and the Lie 2-algebra of sections of an exact Courant algebroid. This talk reviews the notion of bundle gerbe, and discusses the relations between the aforementioned Lie 2-algebras.

Note: This talk is being held on a Wednesday.

Title: Steenrod algebra

Abstract: The Steenrod algebra acts on ordinary cohomology with coefficients in a finite field, thereby refining the invariant that is cohomology. In this talk, I discuss its formal properties and applications. 

Title: Stacky automorphism groups of holomorphic Poisson manifolds

Time: Wednesday, January 7, 3:30-5:00 pm 

Location: RI 209

Abstract: Lie groupoids are generalizations of Lie groups which allow one to study a broad range of geometric problems in a uniform way. In this talk I will introduce Lie groupoids and Morita equivalences, and explain how these structures combine to form a 2-category. We will then study the automorphism 2-groups, called Picard 2-groups, of objects in this category. After introducing some basic holomorphic Poisson geometry, I will present some results concerning the Picard 2-groups of holomorphic symplectic groupoids integrating holomorphic Poisson manifolds.

Video Recording

Title: An introduction to Toric Topology, part 3

Abstract: Last time, we discussed simplicial complexes and homological properties of their face rings. Every simplicial complex defines a toric space called its moment-angle complex. We'll discuss the construction, examples, its cellular structure, and some important properties. 

Title: Introduction to Algebraic K-theory

Abstract: This is the first of two talks in Algebraic K-theory. I will introduce the lower K-groups (0, 1 and 2) through the study of projective modules and their automorphisms, give a few examples and properties. The exact sequences that arise from the lower K-groups will be presented as a means to motivate the existence of the higher K-groups.

Title: Introduction to Algebraic K-theory

Abstract: This is the second of two talks in Algebraic K-theory. I aim to extend the previously discussed 9-term exact sequence into a long exact sequence by defining the higher K-groups via homotopy theory. I will talk about the construction of the topological space BGL(R)^+, show how its homotopy groups define K_n(R), and confirm that this framework is consistent with the lower K-groups already established.

Title: Identities among relations

Abstract:  A standard way of presenting a group is to choose a collection of generators and then to specify a list of relations that must be satisfied. These are not always optimal: sometimes there are non-trivial identities among the relations. For example, the Hall-Witt identity is an identity between the commutators in a free group and gives the non-abelian analogue of the Jacobi identity. 

In this talk, I will go over different ways of representing these identities: algebraic, pictorial, topological, and homological. We will then take a look at a number of examples in different groups. In particular, I will cover the case of the Steinberg group, which is related to last week’s talk on K-theory. 

The identities among relations are only the first step of an infinite process. Indeed, producing a complete set of identities gives rise to further identities, and this process may continue ad infinitum.  Topologically, this corresponds to a cellular decomposition of the classifying space of a group. In several of the examples that I will mention, these higher syzygies are organized by a recursive sequence of polytopes, such as cubes, simplices, associahedra, and permutahedra. 

Title: The noncommutative geometry of cubes and prisms

Abstract: A map f : K → L between two compact convex sets is affine if it respects convex combinations, meaning that 

f(tx + (1 − t)y) = tf(x) + (1 − t)f(y),

for all x, y ∈ K and 0 ≤ t ≤ 1. The collection of compact convex sets and continuous affine maps forms a category, and the classical representation theorem of Kadison (1951) states that it is dual to the category of Archimedean ordered ∗-vector spaces, where morphisms are unital positive maps. 

A non-commutative (nc) compact convex set is a graded set Knc = Un≤א Kn, where Kn denotes a compact collection of bounded linear operators acting on an n-dimensional Hilbert space, for possibly infinite cardinals n ≤ א. Furthermore, Knc is required to be closed under a non-commutative analogue of convex combinations. The collection of nc compact convex sets also forms a category, where the morphisms are the nc continuous affine maps. Very recently, Davidson and Kennedy (2025) proved that this category is dual to the category of operator systems, consisting of Archimedean matrix ordered ∗-vector spaces and unital completely positive maps. 

Given a compact convex set K ⊆ Cd, a non-commutative realization is defined to be a nc compact convex set Knc whose first level is K1 = K. In this talk, I will describe the non-commutative realizations of three classical geometric objects: cubes, polydiscs, and prisms. I will describe their various geometric and algebraic properties, such as their non-commutative extreme points, and I will apply two classical dilation theorems of Halmos and Mirman to give a complete description of the nc triangular prism in terms of joint unitary dilations. This is joint work with D. Farenick, R. Maleki, and S. Medina Varela.

Title: Constructive logic, computation, and Cartesian Closed Categories.

Abstract: In this talk, I'll introduce the Simply Typed Lambda Calculus (STLC), describing how it connects constructive propositional logic and computation. I then discuss how Cartesian Closed Categories serve as models of the STLC, and how the categorical perspective can be used to prove foundational results about the STLC, such as:

• There are no proofs of falsehood in the base STLC

• Law of the Excluded Middle is not derivable in STLC without additional axioms.

• All closed boolean programs evaluate to "true" or "false" (canonicity)

• The correspondence between datatypes and initial algebras.

Title: Locally Cartesian Closed Categories and Dependent Types

Abstract: Building on last week's talk, I discuss Locally Cartesian Closed Categories (LCCCs) , in which every slice category is Cartesian Closed. I describe how these categories serve model quantifiers in logic, as well as modelling type dependencies in a way not possible with simple types. I discuss how these dependent types are useful for verified programming and computer-checked proofs. To conclude, I briefly describe how different formulations of equality proofs lead to different type theories, such as Extensional Type Theory or Homotopy Type Theory.

Title: Irreducible components of module varieties and some open problems

Abstract: For a finite-dimensional algebra A over an algebraically closed field k, varieties of finite-dimensional A-modules offer a geometric perspective on the homological and categorical properties of A and its modules. These varieties parametrize vector spaces of fixed dimension equipped with an A-module structure. While the connected components of these varieties are known from classical results, a structural understanding of their irreducible components remains largely open.

This talk begins with a brief geometric introduction to classical representation theory, then turns to several modern problems and foundational conjectures in the representation theory of finite-dimensional algebras. I will conclude by presenting recent progress on some of these challenging questions. Some of the new results are based on a series of joint work with Charles Paquette.

No prior background in representation theory is required; familiarity with undergraduate linear algebra and basic category theory will suffice.

Title: Higher Connections 1: Iterated Integrals

Abstract: A connection on a vector bundle is data which tells us how to lift a curve from the base space to the total space. In this talk, we introduce a generalization of them known as "formal connections", and explain how they offer us a way to model the based loop space of a given manifold.

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