Topology and Geometry

Time and place: Tuesday, 3:30PM, Math 011

Fall 2023

Joint with Texas Tech University

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

Abstract: 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.

Abstract: Sheaves as a Data Structure (Part 2). Notes, video. Abstract: We continue our discussion with an example of “Path-Optimization Sheaves” (; 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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

Abstract: 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 Texas Tech University

Abstract: 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.

Abstract: 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.

Abstract: 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.

Abstract: 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).

Abstract: 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.

Abstract: 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.

Abstract: 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.

Abstract: 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.

Spring 2022 Schedule

Abstact: 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.

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.

Fall 2021 Schedule

Spring 2021 Schedule

Abstract: In this talk I will lay the foundation for building classical field theory models (and eventually quantum models) in both pAQFT and the Factorization Algebra formalism.

Fall 2020 Schedule 

Spring 2020 Schedule

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. 

 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. 

Fall 2019 Schedule

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. 

Quantum Homotopy Seminar

Time and place: Thursday, 3:30pm, Math 011

Fall 2021 Schedule

Abstract: I will discuss how to give a precise definition of a functorial field theory, formalizing a variety of ideas due to Segal, Atiyah, Kontsevich, Freed, Lawrence, Stolz,  Teichner, Hopkins, Lurie, and many others. This will provide motivation for subsequent talks, which provide details for ingredients used in the definition. The following topics will be examined: The notion of a smooth symmetric monoidal (∞,n)-category. The smooth bordism category as a smooth symmetric monoidal (∞,n)-category. Examples of target categories: spans, cospans, E_n-algebras

Spring 2021 Schedule

Abstract:  We review what are arguably the four most important unifying ideas in geometry: (1) The duality between algebras and spaces. (2) Sheaves. (3) Stacks. (4) Derived stacks.

Fall 2020 Schedule

Abstract: I will give an introduction to the language of simplicial presheaves, which lies at the foundation of modern differential and algebraic geometry. In particular, I will explain sheaf cohomology in this language. 

Abstract: In this talk, I will survey three convenient categories for studying the homotopy theory of spaces equipped with the action of a group. I will present a theorem of Elmendorf, which shows that all three variants are equivalent.

Abstract: In this talk I will discuss the basics of higher topos theory with an emphasis on the theory's applications to geometry. Particular emphasis will be placed on diffeological spaces and sheaf toposes. 

Abstract: In this talk I will begin by finishing the discussion of a theorem relating infinity toposes and infinity stacks which started last week. After this, I will give basic results on over-infinity-toposes and bundles over fixed elements. I will end with a discussion on truncated objects and a correspondence theorem between groupoids internal to an infinity topos and infinity stacks. 

Abstract: In this talk, I will discuss Section 2.2 from the recent paper “Proper Orbifold Cohomology” by Sati and Schreiber in which the concept of groups and group actions are formulated for infinity-toposes. Externally, these structures are known as grouplike E_n-algebras, but can be constructed internally in a more natural way. I will define groups, group actions, principal bundles, and fiber bundles. 

Abstract: Refining the fundamental ∞-groupoid functor Π: Top → ∞Grpd to the context of topological ∞-groupoids Sh∞(Top), we introduce an abstract shape operation ∫: Sh∞(Top) → ∞Grpd which exists in many ∞-toposes, in particular those known as cohesive, where this shape operation has particular left and right adjoints (respectively sharp # and flat ♭), and preserves finite products.

We illustrate the use of these adjoints again in the exemplary context of topological ∞-groupoids/topological stacks, in particular to define the “points-to-pieces” transformation. In the axiomatic setting of ∞-toposes, we explain how these operations specify (co)reflective subuniverses, and provide geometric interpretations of this fact. The shape and flat (co)modalities preserve group objects and their deloopings, as well as group object homotopy-quotients, which results in a formulation of differential cohomology internal to any cohesive ∞-topos. For example, given objects X, A in a cohesive ∞-topos, we explain how a morphism X →♭A represents a A-local system on X, i.e., a cocycle in (nonabelian) cohomology with A-coefficients. 

Abstract: Refining the previous shape operation to possess the infinitesimal property that the “points-to-pieces” transformation ♭X → ∫X is an equivalence of ∞-groupoids, we explain how this condition axiomatizes certain infinitesimal behavior in a cohesive ∞-topos. However, it is also not enough for differential geometry. We explain that this equivalence holds, in particular, when there is a universal internal notion of “tangent space” for objects X, computed by a universal object of contractible infinitesimal shape. This is the richer setting of differential cohesion, where all the cohesion modalities factor through a sub-∞-topos of infinitesimal shapes. This extends the setting of fundamental path ∞-groupoids and differential cohomology given by ordinary cohesion to one where the constructions of higher Cartan geometry can be carried out. Important examples are given by the categories of jets on Cartesian spaces and ∞-sheaves on jets of Cartesian spaces, which we will show subsumes the classical framework of synthetic differential geometry. 

Abstract. In order to facilitate the notion of local diffeomorphisms in a cohesive infinity topos, one need an additional structure called “elastic subtopos”, where all the cohesion modalities factor thorough this sub-infinity-topos. In this talk, I will discuss how this viewpoint subsumes (some) familiar constructions of classical differential geometry. 

Abstract: Using the singular cohesion one can formulate orbifold geometry, internal to infinity-toposes. In this talk our goal is to define basis notions related to this construction and discuss their properties. We introduce a (2,1)-category that is better suited for globally equivariant homotopy theory, “the global indexing category”, which consists of delooping groupoids of compact Lie groups. Its full subcategory of finite, connected, 1-truncated objects captures singular quotients, and homotopy sheaves on this subcategory valued in a smooth infinity-topos are naturally equipped with a cohesion that reveals various perspectives on singularities. 

Abstract: After establishing clearly a notion of global orbit category (of which there are several variants in the literature), we describe a class of topological stacks locally modeled on action ∞-groupoids with singularities via cohesive shape. In passing to the smooth case to obtain orbifolds as certain differentiable stacks, we describe V-folds as a formulation of étale ∞-groupoids internal to a differentially cohesive ∞-topos, which are also the groundwork for studying e.g. G-structures in this setting.

Abstract: Following through on the promises for Cartan geometry in the first two talks, we formulate Haefliger stacks and G-structures in an elastic ∞-topos, the latter as a special case of the principal ∞-bundle constructions available in any ∞-topos where now the existence of the infinitesimal disk bundle is key. By introducing V-folds with singularities, in the sense of singular (elastic) cohesion, we promote étale ∞-stacks in differential cohesion to higher orbifolds in singular cohesion so as to obtain geometrically structured higher orbifolds, extending the intrinsic étale cohomology of étale ∞-stacks to tangentially twisted proper orbifold cohomology. 

Spring 2020 Schedule

Abstract: This talk will provide an introduction to smooth stacks. The talk will begin with some motivation and continue with several explicit examples of cocycle data which can be obtained via descent. The talk will conclude with an outlook of the general theory. 

Abstract: This talk is a continuation of the first. The talk will begin with a discussion on model structures and Bousfield localization and continue with presentations for the infinity category of smooth stacks. We will use Dugger’s characterization of cofibrant objects to unpackage cocycle data explicitly in several examples.