Time and place: Tuesday, 3:30PM, Math 011
Joint with Texas Tech University
Wednesday at 16:00 UTC (before March 9); 15:00 UTC (after March 9).
February 12, Luuk Stehouwer. Dalhousie University.
Abstract: 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.
Abstract: 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.
Abstract: 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.
Abstract: 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.
Abstract: 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.
Joint with Texas Tech University
Tuesday at 3:30 pm Central Standard Time (UTC-06; before March 10) or Central Daylight Time (UTC-05; after March 10).
February 13, Grigorios Giotopoulos. NYU Abu Dhabi
Abstract: 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).
Abstract: 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).
Abstract: 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, Adrian Clough (NYU Abu Dhabi).
Abstract: 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, Darrick Lee (University of Oxford).
Abstract: 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.
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).
October 3, Rachel Kinard.
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.
October 5, Rachel Kinard.
Abstract: 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.
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.
November 7, Severin Bunk. University of Oxford.
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.
November 14, Araminta Amabel. Northwestern University.
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.
November 21, Joseph Tooby-Smith.
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.
November 28, Daniel Berwick-Evans. University of Illinois Urbana–Champaign.
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.
Tuesday at 3:30 pm Central Standard Time (UTC-06) or Central Daylight Time (UTC-05).
Joint with Texas Tech University
January 17
January 24
January 31, Dmitri Pavlov.
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.
February 7, Christian Blohmann (Max Planck Institute for Mathematics, Bonn).
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.
February 14
February 21, Aaron Mazel-Gee (Caltech).
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.
February 28
March 7, Domenico Fiorenza (Sapienza University of Rome).
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).
March 21
March 28
April 4, Emilio Minichiello (CUNY GC).
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.
April 11, Robert Fraser (Wichita State).
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.
April 18, Till Heine (Hamburg).
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.
Felix Wierstra (Korteweg-de Vries Institute for Mathematics, University of Amsterdam).
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.
May 2
Feb. 1, Anna Cepek (U. Oregon), The combinatorics of configuration spaces of R^n
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.
Feb. 15, Daniel Bruegmann (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 (Montana State)
March 22, Alexander Schenkel (U. Nottingham)
April 5, John Huerta (U. Lisbon)
April 12, Robin Koytcheff (Louisiana at Lafayette)
Feb. 15 Nilan Chathuranga, Complete distributive inverse semigroups are equivalent to etale localic groupoids
Mar. 2 Stephan Pena, Comparing pAQFT and the Factorization Algebra Formalism I
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.
Mar. 9 Stephan Pena, Comparing pAQFT and the Factorization Algebra Formalism II
Mar. 22 Stephan Pena, Comparing pAQFT and the Factorization Algebra Formalism III
Apr. 6 Stephan Pena, Comparing pAQFT and the Factorization Algebra Formalism IV
Apr. 13 Stephan Pena, Comparing pAQFT and the Factorization Algebra Formalism V
Apr. 20 Stephan Pena, Comparing pAQFT and the Factorization Algebra Formalism VI
Apr. 27 Stephan Pena, Comparing pAQFT and the Factorization Algebra Formalism VII
Aug. 25 James Francese. Restriction categories I
Sep. 1 Nilan Chathuranga, Etale groupoids and semigroups
Sep. 8 James Francese. Restriction categories II
Sep. 15 Daniel Grady, The homotopy type of the cobordism category I Video
Sep. 22 Daniel Grady, The homotopy type of the cobordism category II Video
Jan. 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.
Jan. 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.
Aug. 27, Dmitri Pavlov. What is a geometric cohomology theory?
Sep. 3, Daniel Grady. An introduction to differential cohomology.
Sep. 10, Daniel Grady. The Atiyah–Hirzebruch spectral sequences for differential cohomology theories I.
Sep. 17, Daniel Grady. The Atiyah–Hirzebruch spectral sequences for differential cohomology theories II.
Sep. 24, Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence I.
Oct. 1, Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence II.
Oct. 8, Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence III.
Oct. 15, Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence IV.
Oct. 22, Daniel Grady. Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence V.
Oct. 29 James Francese. Manifolds of Many Holomorphies
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.
Nov. 5, James Francese. Manifolds of Many Holomorphies: Almost-Clifford Moduli Problems with Discussion of Higher Spectral Sequences
Nov. 12 Daniel Grady, Natural operations on differential cohomology.
Nov. 19 James Francese, Manifolds of Many Holomorphies: Almost-Clifford Moduli Problems with Discussion of Higher Spectral Sequences III.
Nov. 26 Rachel Harris, Excision of Skein Categories and Factorization Homology (after Juliet Cooke). Part I.
Dec. 3 Rachel Harris, Excision of Skein Categories and Factorization Homology (after Juliet Cooke). Part II.
Time and place: Thursday, 3:30pm, Math 011
Aug 25, Dmitri Pavlov, Video
Sep. 2., Dmitri Pavlov, The definition of a functorial field theory Video
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
Sep. 9, Grigory Taroyan, Duality between Algebra and Geometry
Sep. 16, Grigory Taroyan, Sheaves as generalized spaces, Video
Jan. 21, Dmitri Pavlov, The four dimensions of modern geometry
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.
Jan 28, Stephan Pena, Introduction to simplicial sets
Feb. 4, Gregory Taroyan, Simplicial sets and Kan complexes
Feb. 11, James Francese, Simplicial sets and model categories III
Feb. 25, Ramiro Ramirez, Homological algebra and model categories of chain complexes
Mar. 11, James Francese, Rational homotopy theory
Mar. 25 Dmitri Pavlov, Model structures on C-infinity rings and examples of derived intersections.
Apr. 8 Dmitri Pavlov, Derived differentiable stacks.
Sep. 3, Dmitri Pavlov, Introduction to simplicial presheaves
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.
Sep. 10, Daniel Grady. Introduction to equivariant homotopy theory. video
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.
Sep. 17, Stephen Pena. Introduction to higher topos theory I video
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.
Sep. 24, Stephen Pena. Introduction to higher topos theory II video
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.
Oct. 1, Rachel Harris. Infinity-groups and the internal formulation of groups, actions, and fiber bundles. video
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.
Oct. 8, James Francese. Differential Topology via Cohesion in Homotopy- and ∞-Toposes. Video
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.
Oct. 15, James Francese. Differential geometry via elasticity in homotopy and infinity toposes.
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.
Oct. 22, Nilan Manoj Chathuranga. Formalism for Etale Geometry Internal to Infinity Toposes. video
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.
Nov. 5, Nilan Manoj Chathuranga. Geometry of Singular Cohesive Infinity Toposes video
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.
Nov. 10, James Francese. A Matinée of Orbispaces and Orbifolds video
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.
Nov 12, James Francese. Structured Orbifold Geometry video
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.
Daniel Grady. Smooth stacks and Čech cocycles 1
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.
Daniel Grady. Smooth stacks and Čech cocycles 2
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.