Texas Tech Topology and Geometry Seminar

Online seminar with open participation. Register at the external website to retrieve Zoom credentials.

External speakers on topics close to the previous talks are welcome to contribute a talk, contact the organizers by email for details.

Organizers: Dmitri Pavlov (Lubbock, Texas) and Daniel Grady (Wichita, Kansas).

Departmental website at Texas Tech.

External website at researchseminars.org.

Spring 2025

Joint with Wichita State University, Kansas.

Wednesday at 16:00 UTC (before March 9); 15:00 UTC (after March 9).

February 12

Luuk Stehouwer. Dalhousie University. The Unitary Cobordism Hypothesis. Abstract: The cobordism hypothesis classifies extended topological quantum field theories (TQFTs) in terms of algebraic information in the target category. One of the core principles in quantum field theory - unitarity - says that state spaces are not just vector spaces, but Hilbert spaces. Recently in joint work with many others, we have defined unitarity for extended TQFTs, inspired by Freed and Hopkins. Our main technical tool is a higher-categorical version of dagger categories; categories $C$ equipped with a strict anti-involution $\dagger: C \to C^{op}$ which is the identity on objects. I explain joint work in progress with Theo Johnson-Freyd, Cameron Krulewski and Lukas Müller in which we prove a version of the cobordism hypothesis for unitary TQFTs. The main observation is that the \emph{stably} framed bordism $n$-category is freely generated as a symmetric monoidal dagger $n$-category with unitary duals by a single object: the point.

March 5

Lukas Müller. Perimeter Institute. A Higher Spin Statistics Theorem for Invertible Quantum Field Theories. Abstract: The spin-statistics theorem asserts that in a unitary quantum field theory, the spin of a particle—characterized by its transformation under the central element of the spin group, which corresponds to a 360-degree rotation—determines whether it obeys bosonic or fermionic statistics. This relationship can be formalized mathematically as equivariance for a geometric and algebraic action of the 2-group BZ_2. In my talk, I will present a refinement of these actions, extending from BZ_2 to appropriate actions of the stable orthogonal group O, and demonstrate that every unitary invertible quantum field theory intertwines these O-actions.

March 12

Thomas Rot. Vrije Universiteit Amsterdam. The topology of infinite dimensional spaces and nonlinear proper Fredholm mappings. Abstract: In this talk I will discuss the differential topology of non-linear proper Fredholm mappings. In applications these mappings arise as non-linear PDE problems (of elliptic type). I will discuss work with Lauran Toussaint that relates these mappings to the stable homotopy groups of spheres, and if time permits, I will discuss an ongoing project on defining a new homology theory of singular type for infinite dimensional spaces. This is joint work with Alberto Abbondandolo, Michael Jung and Lauran Toussaint.

March 26

Urs Schreiber. NYU Abu Dhabi. Non-Lagrangian construction of abelian CS/FQH-theory via flux quantization in 2-Cohomotopy. Abstract: After briefly recalling how the analog of Dirac charge quantization in exotic (effective, higher) gauge theories, providing their global topological completion, is encoded in a choice of classifying space 𝒜 whose rationalization reflects the flux Bianchi identities, I explain how the choice 𝒜 ≔ S^2 (“flux quantization in 2-Cohomotopy”) implements effective corrections to ordinary Dirac flux quantization, which over surfaces yields exactly the topological quantum observables of fractional quantum Hall systems, traditionally described by abelian Chern-Simons theory. I close by briefly indicating how this situation is geometrically engineered on probe M5-branes if the M-theory C-field is flux-quantized in 4-Cohomotopy (“Hypothesis H”). This is joint work with Hisham Sati; for more pointers see ncatlab.org/schreiber/show/ISQS25.

April 2

Lory Aintablian. MPIM Bonn. Differentiation of groupoid objects in tangent categories. Abstract: The infinitesimal counterpart of a Lie group(oid) is its Lie algebra(oid). I will show that the differentiation procedure works in any category with an abstract tangent structure in the sense of Rosický, which was later rediscovered by Cockett and Cruttwell. Mainly, I will construct the abstract Lie algebroid of a differentiable groupoid in a cartesian tangent category C with a scalar R-multiplication, where R is a ring object of C. Examples include differentiation of infinite-dimensional Lie groups, elastic diffeological groupoids, etc. This is joint work with Christian Blohmann.

April 9

Sean Sanford. Ohio State University. Manifestly unitary higher Hilbert spaces. Abstract: A key aspect of quantum theory its insistence that states evolve via unitary transformations. In order to understand the symmetries of higher dimensional quantum field theory, we need to develop higher dimensional analogues of unitarity. The language and theory of higher categories has greatly clarified the way we express these higher symmetries, but unfortunately this language imposes a certain dogma seems to be in conflict with various attempts at describing unitarity. In the nLab for example, there is a great debate over whether or not unitary structures on a (higher) category are `evil’; at term which is both dogmatic and technically precise.

Various attempts have been made to force these structures to `play nice’ with one another, to varying degrees of success. In this talk I will present our most recent contribution to these efforts: defining the notion of a 3-Hilbert space. Our work aims to encode a kind of evaluation on spheres of every dimension that plays nicely with duality structures that are imposed by the cobordism hypothesis. I will show how this compatibility is stronger than simply having daggers at all levels, thus differentiating our construction from previous attempts at higher unitarity. If time permits, we will discuss a roadmap for unitarity in any dimension via a unitary version of condensation completion.

April 23

Nivedita. University of Oxford. Bicommutant categories from Conformal Nets. Abstract: Two-dimensional chiral conformal field theories (CFTs) admit three distinct mathematical formulations: vertex operator algebras (VOAs), conformal nets, and Segal (functorial) chiral CFTs. With the broader aim to build fully extended Segal chiral CFTs, we start with the input of a conformal net.
In this talk, we focus on presenting three equivalent constructions of the category of solitons, i.e. the category of solitonic representations of the net, which we propose is what theory (chiral CFT) assigns to a point. Solitonic representations of the net are one of the primary class of examples of bicommutant categories (a categorified analogue of a von Neumann algebras). The Drinfel’d centre of solitonic representations is the representation category of the conformal net which has been studied before, particularly in the context of rational CFTs (finite-index nets). If time permits, we will briefly outline ongoing work on bicommutant category modules (which are the structures assigned by the Segal Chiral CFT at the level of 1-manifolds), hinting towards a categorified analogue of Connes fusion of von Neumann algebra modules.
(Bicommutant categories act on W*-categories analogous to von Neumann algebras acting on Hilbert spaces.)

April 30

Mayuko Yamashita. Perimeter Institute. Topological elliptic genera. Abstract: There is a classical notion of elliptic genera, which assigns Jacobi forms to SU-manifolds. In this talk, I explain my work with Ying-Hsuan Lin (arXiv:2412.02298) to give its homotopy-theoretical refinements and variants, which we call “topological elliptic genera”. The codomain becomes genuinely equivariant twisted Topological Modular Forms. In this talk, I explain the construction and physical idea behind, and discuss an application where we derive an interesting divisibility result of Euler numbers for Sp-manifolds. Also I explain a recent update with Tilman Bauer (in preparation) proving the surjectivity results of topological elliptic genera.

Fall 2025

Lorenzo Riva

Fall 2024

Joint with Wichita State University, Kansas.

Tuesday at 3:30 pm Central Daylight Time (UTC-05; before November 3) or Central Standard Time (UTC-06; after November 3).

September 10, 20:00 UTC.

Daniel Grady (Wichita State University). Deformation classes of invertible field theories and the Freed–Hopkins conjecture. Abstract: In their seminal work, Freed and Hopkins studied the moduli space of topological, reflection positive, invertible, Euclidean field theories, providing a complete classification in terms of certain objects arising in stable homotopy theory. In this work, it was also conjectured that a similar classification holds in the case of nontopological field theories, and this conjecture is already being used in a variety of applications to condensed matter physics. In this talk, I will discuss a recent result which provides an affirmative answer to this conjecture. I will begin by reviewing motivation and background on reflection positive theories. Then I will state the conjecture and sketch of the proof.

November 6 (Wednesday), 16:00 UTC.

Jiaqi Fu (Université Paul Sabatier, Toulouse, France). A filtered Koszul duality of partition Lie algebras(-oids). Abstract: Partition Lie algebras are sophisticated algebraic objects introduced by Brantner–Mathew to control infinitesimal deformations in positive characteristics. This talk will present a Koszul duality between partition Lie algebras and specific complete filtered derived rings. This duality helps to understand the homotopy operations on partition Lie algebras. Additionally, a “many-object” version of this duality connects partition Lie algebroids with infinitesimal derived foliations in the sense of Toën–Vezzosi.

November 19

David Kern (KTH). Categorical spectra and ℤ-categories. Abstract: Categorical spectra, developed by Stefanich, are a directed version of spectra where the suspension of pointed ∞-groupoids is replaced by that of pointed ω-categories. They are very useful for capturing stability phenomena in iterated categorifications, or for defining “∞-vector spaces”. In this talk, I will explain that they can be understood as a weak version of Lessard's ℤ-categories, a kind of category with arrows in all negative as well as positive dimensions, which allows for a more direct study of their structure.

Spring 2024

Joint with Wichita State University, Kansas.

Tuesday at 3:30 pm Central Standard Time (UTC-06; before March 10) or Central Daylight Time (UTC-05; after March 10).

February 13

at 10:00 am UTC-06: Grigorios Giotopoulos. NYU Abu Dhabi. Smooth sets as a convenient setting for Lagrangian field theory. Abstract: In this talk, I will indicate how the sheaf topos of smooth sets serves as a sufficiently powerful and convenient context to host classical (bosonic) Lagrangian field theory. As motivation, I will recall the textbook description of variational Lagrangian field theory, and list desiderata for an ambient category in which this can rigorously be phrased. I will then explain how sheaves over Cartesian spaces naturally satisfy all the desiderata, and furthermore allow to rigorously formalize several more field theoretic concepts. Time permitting, I will indicate how the setting naturally generalizes to include the description of (perturbative) infinitesimal structure, fermionic fields, and (gauge) fields with internal symmetries. This is based on joint work with Hisham Sati.

March 19

Luigi Alfonsi (University of Hertfordshire). Slides. Batalin–Vilkovisky formalism beyond perturbation theory via derived geometry. Abstract: In this talk I will discuss applications of derived differential geometry to study a non-perturbative generalisation of classical Batalin–Vilkovisky (BV-)formalism. First, I will describe the current state of the art of the geometry of perturbative BV-theory. Then, I will introduce a simple model of derived differential geometry, whose geometric objects are formal derived smooth stacks (i.e. stacks on formal derived smooth manifolds), and which is obtained by applying Töen-Vezzosi’s homotopical algebraic geometry to the theory of derived manifolds of Spivak and Carchedi-Steffens. I will show how derived differential geometry is able to capture aspects of non-perturbative BV-theory by means of examples in the cases of scalar field theory and Yang-Mills theory.

March 26

Pelle Steffens (Technische Universität München). Differential geometric PDE moduli spaces: derived enhancements, ellipticity and representability. Abstract: All sorts of algebro-geometric moduli spaces (of stable curves, stable sheaves on a CY 3-folds, flat bundles, Higgs bundles...) are best understood as objects in derived geometry. Derived enhancements of classical moduli spaces give transparent intrinsic meaning to previously ad-hoc structures pertaining to, for instance, enumerative geometry and are indispensable for more advanced constructions, such as categorification of enumerative invariants and (algebraic) deformation quantization of derived symplectic structures. I will outline how to construct such enhancements for moduli spaces in global analysis and mathematical physics, that is, solution spaces of PDEs in the framework of derived C^∞ geometry and discuss the elliptic representability theorem, which guarantees that, for elliptic equations, these derived moduli stacks are bona fide geometric objects (Artin stacks at worst). If time permits some applications to enumerative geometry (symplectic Gromov-Witten and Floer theory) and derived symplectic geometry (the global BV formalism).

April 16

at 10 am UTC-05: Adrian Clough (NYU Abu Dhabi). Homotopical calculi and the smooth Oka principle. Abstract: I will present a new proof of Berwick-Evans, Boavida de Brito, and Pavlov’s theorem that for any smooth manifold A, and any sheaf X on the site of smooth manifolds, the mapping sheaf Hom(A,X) has the correct homotopy type. The talk will focus on the main innovation of this proof, namely the use of test categories to construct homotopical calculi on locally contractible ∞-toposes. With this tool in hand I will explain how a suitable homotopical calculus may be constructed on the ∞-topos of sheaves on the site of smooth manifolds using a new diffeology on the standard simplices due to Kihara. The main theorem follows using a similar argument that for any CW-complex A, and any topological space X the set of continuous maps Hom(A,X) equipped with compact-open topology models the mapping-homotopy-type map(A,X).

April 23

at 10 am UTC-05: Darrick Lee (University of Oxford). Slides. Characterizing paths and surfaces via (higher) holonomy. Abstract: Classical vector valued paths are widespread across pure and applied mathematics: from stochastic processes in probability to time series data in machine learning. Parallel transport of such paths in principal G-bundles have provided an effective method to characterise such paths. In this talk, we provide a brief overview of these results and their applications. We will then discuss recent work on extending this framework to characterizing random and possibly nonsmooth surfaces using surface holonomy. This is based on joint work with Harald Oberhauser.

April 30

Jacob Lebovic (University of Oregon). Iterated K-theory and Functorial Field Theory. Abstract: Using previous work by Bass, Dundas, and Rognes giving a geometric model of the iterated K-theory spectrum K(ku) in terms of bundles of Kapranov-Voevodsky 2-vector spaces, and recent work by Grady and Pavlov providing a rigorous foundation for fully-extended functorial field theories, we construct a model of K(ku) in terms of 2-dimensional functorial field theories valued in KV 2-vector spaces.

Fall 2023

Joint with Wichita State University, Kansas.

Tuesday at 1 pm Central Daylight Time (UTC-05; before November 5) or Central Standard Time (UTC-06; after November 5).

October 3

Rachel Kinard. Sheaves as a Data Structure. Notes. Abstract: Tables, Arrays, and Matrices are useful in data storage and manipulation, employing operations and methods from Numerical Linear Algebra for computer algorithm development. Recent advances in computer hardware and high performance computing invite us to explore more advanced data structures, such as sheaves and the use of sheaf operations for more sophisticated computations. Abstractly, Mathematical Sheaves can be used to track data associated to the open sets of a topological space; practically, sheaves as an advanced data structure provide a framework for the manipulation and optimization of complex systems of interrelated information. Do we ever really get to see a concrete example? I will point to several recent examples of (1) the use of sheaves as a tool for data organization, and (2) the use of sheaves to gain additional information about a system.

October 5

Rachel Kinard. Sheaves as a Data Structure (Part 2). Notes, video. Abstract: We continue our discussion with an example of “Path-Optimization Sheaves” (https://arxiv.org/abs/2012.05974); an alternative approach to classical Dijkstra’s Algorithm, paths from a source vertex to sink vertex in a graph are revealed as Sections of the Path-finding Sheaf.

October 24

Arun Debray. Purdue University. Constructing the Virasoro groups using differential cohomology. Notes. Abstract: The Virasoro groups are a family of central extensions of Diff^+(S^1) by the circle group T. In this talk I will discuss recent work, joint with Yu Leon Liu and Christoph Weis, constructing these groups by beginning with a lift of the first Pontrjagin class to "off-diagonal" differential cohomology, then transgressing it to obtain a central extension. Along the way, I will discuss what the Virasoro extensions are and how to recognize them; a brief introduction to differential cohomology; and lifts of characteristic classes to differential cohomology.

November 7

Severin Bunk. University of Oxford. Smooth higher symmetries groups and the geometry of Deligne cohomology. Abstract: We construct the smooth higher group of symmetries of any higher geometric structure on manifolds. Via a universal property, this classifies equivariant structures on the geometry. We present a general construction of moduli stacks of solutions in higher-geometric field theories and provide a criterion for when two such moduli stacks are equivalent. We then apply this to the study of generalised Ricci solitons, or NSNS supergravity: this theory has two different formulations, originating in higher geometry and generalised geometry, respectively. These formulations produce inequivalent field configurations and inequivalent symmetries. We resolve this discrepancy by showing that their moduli stacks are equivalent. This is joint work with C. Shahbazi.

November 14

Araminta Amabel. Northwestern University. A factorization homology approach to line operators. Abstract: There are several mathematical models for field theories, including the functorial approach of Atiyah–Segal and the factorization algebra approach of Costello–Gwilliam. I'll discuss how to think about line operators in these contexts, and the different strengths of each method. Motivated by work of Freed–Moore–Teleman, I'll explain how to exploit both models to say something about certain gauge theories. This is based on joint work with Owen Gwilliam.

November 21

Joseph Tooby-Smith. Cornell University. Smooth generalized symmetries of quantum field theories. Abstract: In this talk, based on joint work with Ben Gripaios and Oscar Randal-Williams (arXiv:2209.13524 and 2310.16090), we will, with help from the geometric cobordism hypothesis, define and study invertible smooth generalized symmetries of field theories within the framework of higher category theory. We will show the existence of a new type of anomaly that afflicts global symmetries even before trying to gauge, we call these anomalies “smoothness anomalies”. In addition, we will see that d-dimensional QFTs when considered collectively can have d-form symmetries, which goes beyond the (d-1)-form symmetries known to physicists for individual QFTs. We will also touch on aspects of gauging global symmetries in the case of topological quantum field theories.

November 28

Daniel Berwick-Evans. University of Illinois Urbana–Champaign. Twisted equivariant Thom classes in topology and physics. Notes. Abstract: In their seminal work, Mathai and Quillen explained how free fermion theories can be used to construct cocycle representatives of Thom classes in de Rham cohomology. After reviewing this idea, I will describe several avenues of generalization that lead to cocycle representatives of Thom classes in twisted equivariant KR-theory and (conjecturally) in equivariant elliptic cohomology. I will further describe nice properties enjoyed by these cocycle representatives, e.g., compatibility with (twisted) power operations. This is joint work with combinations of Tobi Barthel, Millie Deaton, Meng Guo, Yigal Kamel, Hui Langwen, Kiran Luecke, Alex Pacun, and Nat Stapleton.

Spring 2023

Tuesday at 3:30 pm Central Standard Time (UTC-06) or Central Daylight Time (UTC-05).

Joint with Wichita State University, Kansas.

January 31

Dmitri Pavlov. Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle. Abstract: We prove that the category of diffeological spaces does not admit a model structure transferred via the smooth singular complex functor from simplicial sets, resolving in the negative a conjecture of Christensen and Wu. Embedding diffeological spaces into sheaves of sets (not necessarily concrete) on the site of smooth manifolds, we then prove the existence of a proper combinatorial model structure on such sheaves transferred via the smooth singular complex functor from simplicial sets. We show the resulting model category to be Quillen equivalent to the model category of simplicial sets. We then show that this model structure is cartesian, all smooth manifolds are cofibrant, and establish the existence of model structures on categories of algebras over operads. We use these results to establish analogous model structures on simplicial presheaves on smooth manifolds, as well as presheaves valued in left proper combinatorial model categories, and prove a generalization of the smooth Oka principle established in arXiv:1912.10544. We finish by establishing classification theorems for differential-geometric objects like closed differential forms, principal bundles with connection, and higher bundle gerbes with connection on arbitrary cofibrant diffeological spaces. arXiv:2210.12845.

February 7

Christian Blohmann (Max Planck Institute for Mathematics, Bonn). Elastic diffeological spaces. Abstract: I will introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosický. On elastic spaces there is a natural Cartan calculus, consisting of vector fields and differential forms, together with the Lie bracket, de Rham differential, inner derivative, and Lie derivative, satisfying the usual graded commutation relations. Elastic spaces are closed under arbitrary coproducts, finite products, and retracts. Examples include manifolds with corners and cusps, diffeological groups and diffeological vector spaces with a mild extra condition, mapping spaces between smooth manifolds, and spaces of sections of smooth fiber bundles. arXiv:2301.02583.

February 21

Aaron Mazel-Gee (Caltech). Towards knot homology for 3-manifolds. Abstract: The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin–Turaev using quantum groups. Khovanov homology is a categorification of the Jones polynomial of a knot in R^3, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds. In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided (∞,2)-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.

March 7

Domenico Fiorenza (Sapienza University of Rome). String bordism invariants in dimension 3 from U(1)-valued TQFTs. Slides. Abstract: The third string bordism group is known to be Z/24Z. Using Waldorf's notion of a geometric string structure on a manifold, Bunke–Naumann and Redden have exhibited integral formulas involving the Chern–Weil form representative of the first Pontryagin class and the canonical 3-form of a geometric string structure that realize the isomorphism Bord_3^{String} → Z/24Z (these formulas have been recently rediscovered by Gaiotto–Johnson-Freyd–Witten). In the talk I will show how these formulas naturally emerge when one considers the U(1)-valued 3d TQFTs associated with the classifying stacks of Spin bundles with connection and of String bundles with geometric structure. Joint work with Eugenio Landi (arXiv:2209.12933).

April 4

Emilio Minichiello (CUNY GC). Diffeological Principal Bundles, Čech Cohomology and Principal Infinity Bundles. Abstract: Thanks to a result of Baez and Hoffnung, the category of diffeological spaces is equivalent to the category of concrete sheaves on the site of cartesian spaces. By thinking of diffeological spaces as kinds of sheaves, we can therefore think of diffeological spaces as kinds of infinity sheaves. We do this by using a model category presentation of the infinity category of infinity sheaves on cartesian spaces, and cofibrantly replacing a diffeological space within it. By doing this, we obtain a new generalized cocycle construction for diffeological principal bundles, a new version of Čech cohomology for diffeological spaces that can be compared very directly with two other versions appearing in the literature, which is precisely infinity sheaf cohomology, and we show that the nerve of the category of diffeological principal G-bundles over a diffeological space X for a diffeological group G is weak equivalent to the nerve of the category of G-principal infinity bundles over X. arXiv:2202.11023.

April 11

Robert Fraser (Wichita State). Fourier analysis in Diophantine approximation. Abstract: A real number $x$ is said to be normal if the sequence $a^n x$ is uniformly distributed modulo 1 for every integer $a≥2$. Although Lebesgue-almost all numbers are normal, the problem determining whether specific irrational numbers such as $e$ and $π$ are normal is extremely difficult. However, techniques from Fourier analysis and geometric measure theory can be used to show that certain “thin” subsets of $R$ must contain normal numbers.

April 18

Till Heine (Hamburg). The Dwyer Kan-correspondence and its categorification. Abstract: Extensions of the Dold-Kan correspondence for the duplicial and (para)cyclic index categories were introduced by Dwyer and Kan. Building on the categorification of the Dold-Kan correspondence by Dyckerhoff, we categorify the duplicial case by establishing an equivalence between the $\infty$-category of $2$-duplicial stable $\infty$-categories and the $\infty$-category of connective chain complexes of stable $\infty$-categories with right adjoints. I will further explain the current progress towards a conjectured correspondence between $2$-paracyclic stable $\infty$-categories and connective spherical complexes. Examples of the latter naturally arise from the study of perverse schobers. arXiv:2303.03653.

April 25

Felix Wierstra (Korteweg-de Vries Institute for Mathematics, University of Amsterdam). A recognition principle for iterated suspensions as coalgebras over the little cubes operad. In this talk I will discus a recognition principle for iterated suspensions as coalgebras over the little cubes operad. This is joint work with Oisín Flynn-Connolly and José Moreno-Fernádez. arXiv:2210.00839.

Fall 2022

November 8

Alastair Hamilton. The BV formalism and Matrix Models. In this talk we will discuss how ideas from noncommutative geometry put forward by Kontsevich can be used as a framework to study matrix models. We will explore the case of the Gaussian Unitary Ensemble from this perspective and determine some of its large N behavior.

November 15

Alastair Hamilton. The BV formalism and Matrix Models II.

November 29

Santosh Kandel. (California State University, Sacramento.) Two-dimensional perturbative scalar QFT and Atiyah–Segal gluing. We will discuss perturbative path integral quantization of scalar field theory with a polynomial interaction potential on two-dimensional compact Riemannian manifolds with boundary. The perturbative partition function is defined in terms of configuration space integrals on the surface. Moreover, partition functions can be organized into a functor, in the sense of Atiyah-Segal axiomatics, from the Riemannian cobordism category to the category of Hilbert spaces. A crucial role in the result is played by non-trivial behavior of tadpoles (short loops) under gluing. This is based on joint work with Pavel Mnev and Konstantin Wernli. Reference: arXiv:1912.11202. Handwritten notes.

December 6

Alastair Hamilton. The BV formalism and Matrix Models III.

Spring 2022

February 1

Anna Cepek (University of Oregon). The combinatorics of configuration spaces of R^n. Abstract: We approach manifold topology by examining configurations of finite subsets of manifolds within the homotopy-theoretic context of ∞-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration spaces in the case of Euclidean space in terms of the category Θ_n.

February 15

Daniel Brügmann (MPIM Bonn). Vertex Algebras and Factorization Algebras. Abstract: Vertex algebras and factorization algebras are two approaches to chiral conformal field theory. Costello and Gwilliam describe how every holomorphic factorization algebra on the plane of complex numbers satisfying certain assumptions gives rise to a Z-graded vertex algebra. They construct some models of chiral conformal theory as factorization algebras. We attach a factorization algebra to every Z-graded vertex algebra.

March 8

Ryan Grady (Department of Mathematical Sciences, Montana State University). Bordism, Genera, and QFT. Abstract: Bordism is an equivalence relation on manifolds and has been a powerful tool in algebraic topology for the last 60 years. A genus is a $\mathbb{Q}$-valued function on equivalence classes of manifolds (actually it's a ring homomorphism); genera have played an important role in 4-manifold topology, index theory, and homotopy theory. In this talk, I will recall the cobordism ring and discuss generators in terms of explicit manifolds. Next, I will introduce some examples of genera and then recall a general construction due to Hirzebruch. Finally, I will present an interpretation of genera through the lens of quantum field theory and share some recent computations.

March 22

Alexander Schenkel (School of Mathematical Sciences, University of Nottingham). An AQFT perspective on quantum gauge theories. Abstract: Algebraic quantum field theory (AQFT) is a time-honoured axiomatic approach to describe and study QFTs on Lorentzian manifolds. In this talk I will try to summarize some of our main insights and results about “levelling up” traditional AQFT to the higher categorical world, which leads to a refined framework that I believe is suitable for quantum gauge theories. I will focus on both the underlying higher algebraic structures, i.e., the question “What does an ∞-AQFT assign? And why?”, and on concrete examples (mostly toy-models) that one can construct through methods from derived algebraic geometry. This talk is based on a long-term research program with Marco Benini.

April 12

Robin Koytcheff (Department of Mathematics, University of Louisiana at Lafayette). Configuration space integrals for spaces of knots, links, and braids. Abstract: Configuration space integrals and graph complexes have been used to study a number of problems in topology. We will discuss the application to cohomology of spaces of embeddings, which directly generalizes their application to Vassiliev knot invariants. In joint work with Komendarczyk and Volić, we showed that such integrals describe the whole cohomology of spaces of 1-dimensional pure braids in any Euclidean space of dimension at least 4. We related them to Chen’s iterated integrals and showed that the inclusion of 1-dimensional pure braids into 1-dimensional long links induces a surjection in cohomology. This motivated further work of ours on the relationship between higher-dimensional pure braids and string links. In work in progress, I have been using these integrals to relate classes of long links to classes of long knots in various dimensions.

April 19

John Huerta (Department of Mathematics, Instituto Superior Técnico, University of Lisbon). Splitting supermanifolds. Abstract: We give a gentle introduction to supermanifolds and the splitting problem. Supermanifolds are a mildly noncommutative geometry where the coordinate functions either commute or anticommute. We recount Batchelor's theorem: every supermanifold in the smooth category is noncanonically isomorphic to a vector bundle. Such an isomorphism is called a splitting. Splittings may not exist at all in the holomorphic category, which turns out to be deeply significant to string theory in a way we will sketch. We close with basic examples of splittings and nonsplittings.

April 26

Severin Bunk (Mathematical Institute, University of Oxford) Higher symmetries of gerbes. Abstract: Gerbes are geometric objects describing the third integer cohomology group of a manifold and the B-field in string theory. Like line bundles, they admit connections and gauge symmetries. In contrast to line bundles, however, there are now isomorphisms between gauge symmetries: the gauge group of a gerbe is a smooth 2-group. Starting from a hands-on example, I will explain gerbes and some of their properties. The main topic of this talk will then be the study of symmetries of gerbes on a manifold with G-action, and how these symmetries assemble into smooth 2-group extensions of G. In the last part, I will survey how this construction can be used to provide a new smooth model for the String group, via a theory of ∞-categorical principal bundles and group extensions.

Fall 2021

September 7

Daniel Grady. The geometric cobordism hypothesis I. Video recording. Abstract: This series of talks will survey two joint papers with Dmitri Pavlov, which together prove a geometric enhancement of the cobordism hypothesis. As a special case, taking homotopy invariant geometric structures, our work provides the first rigorous proof of the topological cobordism hypothesis of Baez--Dolan, after Lurie's 2009 sketch. In this first talk, I will begin with motivation from quantum field theory and string theory. Then I will survey the basic mathematical structures that appear in the statement of the cobordism hypothesis and end with the statement of the main theorem. References: Extended field theories are local and have classifying spaces.

September 14

Daniel Grady. The geometric cobordism hypothesis II. Video recording. Abstract: In this talk, I will begin by introducing geometric structures on bordisms. I will then motivate and introduce the fully extended bordism category. Finally, I will end with the statement of the geometric cobordism hypothesis.

September 21

Daniel Grady. The geometric cobordism hypothesis III. Video recording. Abstract: In this talk, I will state the main theorem and begin introducing the background needed to understand the statement. I will begin by briefly reviewing the theory of simplicial presheaves and their localizations.

October 5

Daniel Grady. The geometric cobordism hypothesis IV.

October 19

Daniel Grady. The geometric cobordism hypothesis V.

November 2

Razvan Gelca. Chern–Simons theory I.

November 9

Razvan Gelca. Chern–Simons theory II.

November 16

Stephen Peña. Geometric factorization homology.

Spring 2021

January 26

No seminar.

February 2

No seminar.

February 9

No seminar.

February 16

Nilan Manoj Chathuranga.

February 23

No seminar.

March 2

Stephen Peña. Factorization Algebras I.

March 9

Stephen Peña. Factorization Algebras II.

March 16

No seminar.

March 23

Stephen Peña. Factorization Algebras III.

March 30

No seminar.

April 6

Stephen Peña. Factorization Algebras IV.

April 13

Stephen Peña. Factorization Algebras V.

April 20

Stephen Peña. Factorization Algebras VI.

April 27

Stephen Peña. Factorization Algebras VII.

May 4

Stephen Peña. Factorization Algebras VIII.

Fall 2020

August 25

James Francese. Fibrations and Special Structures for Categorified Computability Theory. Abstract: In this talk we will examine a result (due to Robin Cockett and Richard Garner) identifying Cockett and Stephen Lack's notion of restriction category as a certain class of 2-categories weakly enriched over a particular base, constructed via Day convolution in a weak double category. We then suggest some examples from differential geometry, functional and complex analysis, and computability theory which call for a generalization of the restriction category concept, and a strategy for extending Cockett and Garner's structure theory result for ordinary restriction categories to this new setting.

September 1

Nilan Manoj Chathuranga. Inverse semigroups and etale groupoids I.

September 8

James Francese. Overview and Motivation for Special Structures in Categorified Computability Theory. Abstract: In this talk, we provide a general overview of a few concepts in computability theory which have motivated our study of restriction categories and related structures: partial combinatory algebras, their so-called functional completeness, and their ability to generate categories known as realizability toposes, with structures known as triposes as an intriguing intermediate step in one account of how realizability toposes may be constructed. We then introduce Turing categories as a certain categorification of partial combinatory algebras based on restriction categories, and illustrate how some of previous concepts translate into this new setting.

September 15

Daniel Grady. The homotopy type of the cobordism category I. Abstract: This talk is the first in a series which reviews the seminal work of Soren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss. The main result is a refinement of the Pontryagin-Thom equivalence to a space level equivalence, going between the classifying space of the cobordism category and a certain spectrum. The authors use this equivalence to prove a conjecture by David Mumford about the cohomology of the mapping class group of Riemann surfaces.
In this first talk, I will begin motivation and review the classical Pontryagin-Thom construction. I will then describe a categorification of the collapse map, which is claimed to induce an equivalence at the level of classifying spaces. Various bordism categories will be introduced, all of which will be equivalent. If there is time, I will introduce the Madsen-Tillmann spectrum.

September 22

Daniel Grady. The homotopy type of the cobordism category II. Abstract: In the second installment of this series, I will introduce the topological bordism category and discuss its classifying space. Then I will review some basics of sheaf theory and discuss the geometric realization (or shape) of sheaves on manifolds. A sheaf variant of the bordism category will be introduced and it will be shown (next time) that its geometric realization is precisely the classifying space of the bordism category.

September 29

Daniel Grady. The homotopy type of the cobordism category III. Abstract: In this talk, I will introduce the Madsen–Tillmann spectrum and discuss its connection with the Thom spectrum. I will prove the first of the two main theorems that imply Mumford's conjecture.

October 6

Nilan Manoj Chathuranga. Complete Inverse Semigroups – Etale Localic Groupoids Correspondence. Abstract: In this talk, I will discuss an equivalence between the category of etale groupoids internal to locales and a certain subcategory of inverse semigroups. This generalizes the well-known equivalence of pseudogroups and effective etale Lie groupoids, as well as the correspondence between etale groupoids and quantales, due to Pedro Resende and Lawson–Lenz.

October 13

No seminar.

October 20

No seminar.

October 27

No seminar (winter storm).

November 3

Dmitri Pavlov. Gelfand-type duality for commutative von Neumann algebras. Abstract: We show that the following five categories are equivalent: (1) the opposite category of commutative von Neumann algebras; (2) compact strictly localizable enhanced measurable spaces; (3) measurable locales; (4) hyperstonean locales; (5) hyperstonean spaces. This result can be seen as a measure-theoretic counterpart of the Gelfand duality between commutative unital C*-algebras and compact Hausdorff topological spaces.

November 10

No seminar.

November 17

No seminar.

November 24

Rachel Harris. Algorithms for Skein Manipulation and Automation of Skein Computations. Abstract: Skein manipulations prove to be computationally intensive due to the exponential nature of skein relations. Resolving each crossing in a knot diagram produces 2 new knot diagrams; knot diagrams with over 5 crossings become increasingly difficult to work with. In this talk, I will introduce a method for automating these computations and discuss how this method was implemented as a Python program. I will illustrate the use of the program in several known examples, demonstrating how examples obtained through several months of work can be can now be obtained in less than 5 minutes. This program will be used to generate a library of examples for testing various conjectures in Chern-Simons theory.

Spring 2020 (Tuesdays, 3:30–5)

January 21

Alastair Hamilton. Introduction to the Batalin-Vilkovisky Formalism. Abstract: I will discuss some of the basic ideas and geometry of the Batalin-Vilkovisky formalism as well as its connection to Chern-Simons theory.

January 28

Daniel Grady. Lifting M-theory fields to cohomotopy via obstruction theory. Abstract: We show that the Postnikov tower for the 4-sphere gives rise to obstruction classes which correctly recover various quantization conditions and anomaly cancellations on the M-theory fields. This further adds weight to the hypothesis that the M-theory fields take values in cohomotopy, rather than cohomology.

February 4

No seminar (snow storm).

February 11

Alastair Hamilton. Introduction to the Batalin-Vilkovisky Formalism, Part II. Abstract: I will discuss some of the basic ideas and geometry of the Batalin-Vilkovisky formalism as well as its connection to Chern-Simons theory.

February 18

Alastair Hamilton. Noncommutative Geometry in the BV-formalism. Abstract: I will begin to talk about noncommutative analogues of the objects that were introduced in previous lectures.

February 25

Alastair Hamilton. BV-formalism IV.

March 3

Alastair Hamilton. BV-formalism V.

Fall 2019 (Tuesdays 3:30–5)

August 27

Dmitri Pavlov. What is a geometric cohomology theory?

September 3

Daniel Grady. An introduction to differential cohomology.

September 10

Daniel Grady. The Atiyah–Hirzebruch spectral sequences for differential cohomology theories I.

September 17

Daniel Grady. The Atiyah–Hirzebruch spectral sequences for differential cohomology theories II.

September 24

Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence I.

October 1

Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence II.

October 8

Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence III.

October 15

Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence IV.

October 22

Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence V.

October 29

James Francese. Manifolds of Many Holomorphies: Almost-Clifford Moduli Problems with Discussion of Higher Spectral Sequences. Abstract: In this talk we will formulate the existence of almost-Clifford structures on smooth manifolds of appropriate dimension in terms of a Kuranishi–Kodaira–Spencer theory, obtaining local structural equations analogous to Cauchy–Riemann conditions. Globally, the satisfaction of these structural equations have obstructions detected precisely by (higher) prolongations of the corresponding G-structures, governed by differential-graded Lie algebras. These obstructions can be compared to the well-understood almost-complex and almost-quaternionic cases (classical Kodaira–Spencer vs. twistor theory). We present also the obstruction for the second complex Clifford algebra, known as the bicomplex numbers, describing an existence result for integrable almost-bicomplex structures, and two compatible double-complexes of differential forms (one elliptic, one non-elliptic) which has its own cohomology and notion of spectral sequence. We draw attention to the previously unobserved similarities between this formalism and work on the “generalized geometry” of Hitchin, Gualtieri, Cavalcanti, and others, suggesting applications of the bicomplex differential geometry to problems in T-duality. If time allows we may suggest a related spectral sequence for almost-quaternionic geometry based on the work of Widdows.

November 5

James Francese. Manifolds of Many Holomorphies: Almost-Clifford Moduli Problems with Discussion of Higher Spectral Sequences II.

November 12

Daniel Grady. Natural operations on differential cohomology.

November 19

James Francese. Manifolds of Many Holomorphies: Almost-Clifford Moduli Problems with Discussion of Higher Spectral Sequences III.

November 26

Rachel Harris. Excision of Skein Categories and Factorization Homology (after Juliet Cooke). Part I.

December 3

Rachel Harris. Excision of Skein Categories and Factorization Homology (after Juliet Cooke). Part II.

Spring 2019 (Geometry Seminar, Wednesdays 4–5)

January 16

No seminar.

January 23

Cezar Lupu. The dilogarithm function in geometry and number theory (Part III).

January 30

Cezar Lupu. The dilogarithm function in geometry and number theory (Part IV).

February 6

Cezar Lupu. The dilogarithm function in geometry and number theory (Part V).

February 13

Cezar Lupu. The dilogarithm function in geometry and number theory (Part VI).

February 20

Cezar Lupu. The dilogarithm function in geometry and number theory (Part VII).

February 27

Cezar Lupu. The dilogarithm function in geometry and number theory (Part VIII).

March 6

Vlad Matei (University of California, Irvine). Point counting and cohomology. Abstract: I will explain how most of the arithmetic statistics questions over functions can be approached from a geometrical viewpoint. The main tool involved is the Grothendieck Lefschetz trace formula and twisted versions of it, which allows to translate between point counts and studying the geometry of the underlying variety that parametrizes the objects we want to count.

March 11

(Monday at 3 p.m.) Peter Ulrickson (Catholic University of America). Supersymmetric Euclidean Field Theories and K-theory. Abstract: A functorial quantum field theory is a symmetric monoidal functor from a category of bordisms to a category of vector spaces. I will present some aspects of Stolz and Teichner's approach to relating functorial quantum field theories and cohomology theories. Specifically, I will sketch the case of 1-dimensional supersymmetric Euclidean field theories and topological K-theory.

March 20

No seminar.

March 27

Alastair Hamilton. Ribbon graph decomposition of the moduli space of Riemann surfaces. Abstract: In this talk I will describe a decomposition of the moduli space of Riemann surfaces into orbi-cells. We will need this background for subsequent talks.

April 3

Alastair Hamilton. Ribbon graph decomposition of the moduli space of Riemann surfaces. Part II. Abstract: This time around, I will discuss orbi-cell decompositions of compactifications of the moduli space. We will need this background for subsequent talks.

April 10

Alastair Hamilton. Algebraic model for homology of the moduli space of Riemann surfaces. Abstract: In this talk, I will describe a theorem due to Kontsevich that recovers the homology of the one-point compactification of the moduli space of Riemann surfaces in terms of the homology of a certain infinite-dimensional Lie algebra. Time permitting, I will discuss more refined compactifications of the moduli space.

April 17

Charles Frohman. Invariants of Geometric structures of three-manifolds derived from the Kauffman bracket. Abstract: I will recap the representation theory of the Kauffman bracket skein algebra of a surface and show how it can be used to derive invariants depending on the hyperbolic of a three-manifold.

April 24

Alastair Hamilton. An isomorphism between the graph complex and the Chevalley–Eilenberg complex of a differential graded Lie algebra.

May 1

Adrian Zahariuc (University of California, Davis). A Riemann-Hurwitz-Plucker formula. Abstract: The classical Riemann-Hurwitz and Plucker formulas give the number of ramification points of maps between (algebraic) curves, and of maps from a curve to a projective space respectively. They overlap in the case of a map from a curve to the projective line, where they give the same formula. I will state a common generalization of these two formulas, obtained in joint work with Brian Osserman.

May 8

Alastair Hamilton. Moduli spaces of Riemann surfaces. IV.

Fall 2018 Schedule (Geometry Seminar, Wednesdays 4–5)

August 29

No seminar.

September 5

Dmitri Pavlov. From whence do differential forms come? Handwritten notes. Abstract: From Newton to Cartan, infinitesimal quantities were productively used in analysis and differential geometry. This language was cast aside completely by the mid 20th century in favor of ugly limit-style arguments. Yet at the same time André Weil and Alexander Grothendieck came up with a very simple and elegant way to formalize infinitesimals: nilpotent elements in rings of functions. In this expository talk I will explain an extremely elegant formulation of differential forms in this language: the de Rham complex is isomorphic (and not merely quasi-isomorphic) to the infinitesimal smooth singular cochain complex with real coefficients. The talk will be accessible to graduate students, no prior knowledge of differential forms will be assumed or required.

September 12

Continuation of the previous talk.

September 19

Razvan Gelca. The volume conjecture. Abstract: The volume conjecture is a difficult question in Chern-Simons theory. It belongs to the realm of semiclassical analysis, and it establishes a bridge between Witten's and Thurston's theories.

September 26

Razvan Gelca. The volume conjecture II.

October 3

Dmitri Pavlov. A very gentle introduction to derived smooth manifolds. Abstract: The intersection of two transversal submanifolds of a smooth manifold is again a smooth manifold. What happens when we intersect nontransversal submanifolds? In this talk I will give a very slow and gentle introduction (accessible to graduate students) to the subject of derived differential geometry, which provides an answer to this question. It has numerous applications in theoretical physics, including the famous BV-BRST formalism.

October 10

Razvan Gelca. The action of the Kauffman bracket skein algebra of the torus on the skein module of the complement of the 3-twist knot. Abstract: I will present a computation that Hongwei Wang and myself have done using skeins in knot complements. This is a laborious endeavor, and the aim of the talk is to make my collaborators aware of the difficulties of this computation.

October 17

Josh Padgett. Numerical integration techniques on manifolds and their Hopf algebraic structure. Abstract: Lie group integrators are a class of numerical integration methods which approximate the solution to differential equations which preserve the underlying geometric structure of the true solution. In this talk, we consider a commutative graded Hopf algebraic structure arising in the order theory and backward error analysis of such Lie group methods. We will consider recursive and direct formulae for the coproduct and antipode, while emphasizing the connection to the Hopf algebra of classical Butcher theory and the Hopf algebra structure of the shuffle algebra. The talk will provide the necessary background to make it accessible to graduate students.

October 24

Josh Padgett. Continuation of the previous talk.

October 31

No seminar.

November 7

Cezar Lupu. Clausen function and a dilogarithmic integral arising in quantum field theory. Abstract: In this talk, I shall bring into perspective a transcendental function which connects many branches of mathematics (analysis, number theory, algebra) and physics (quantum field theory). This function is called nowadays the Clausen integral. This was introduced for the first time by Thomas Clausen in 1832 and it is intimately connected with the polylogarithm, polygamma function and ultimately with the Riemann zeta function. Next, we state some basic properties of this function and we give a new proof of the Clausen acceleration formula (based on a joint work with Derek Orr). We also make some important remarks why this acceleration formula is important in physics and towards the end of the talk, I shall discuss about a dilogarithmic integral from quantum field theory.

November 14

Cezar Lupu. Continuation of the previous talk.

November 21

Thanksgiving Break.

November 28

Cezar Lupu. The dilogarithm function in geometry and number theory (Part I). Abstract: In this first part of a series of talk, we introduce and explore basic properties of another special function called the dilogarithm. First defined by Euler, the dilogarithm function is one of the simplest non-elementary functions, but also one of the strangest. It was also studied by mathematicians such as Abel, Lobachevsky, Kummer, and Ramanujan among others. In recent years, it has become much better known due to its connections with hyperbolic geometry, algebraic K-theory and mathematical physics.

December 5

Cezar Lupu. Continuation of the previous talk.