Topology and Geometry
Time and place: Tuesday, 3:30PM, Math 011
Spring 2021 Schedule
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
Fall 2020 Schedule
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
Spring 2020 Schedule
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.
Fall 2019 Schedule
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.
Quantum Homotopy Seminar
Time and place: Thursday, 3:30pm, Math 011
Fall 2021 Schedule
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
Spring 2021 Schedule
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.
Fall 2020 Schedule
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.
Spring 2020 Schedule
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.