Texas Tech Quantum Homotopy Seminar

Time and place: Thursday 3:30–5 p.m. Central Time (UTC–05; after November 6: UTC–06), MATH 115.

Online video streaming available, contact the organizer by email for details.

External participants and speakers are welcome, contact the organizer by email for details.

Official mailing list: math.geometry (add ttu edu at the end).

Organizer: Dmitri Pavlov

Departmental seminar website.

Fall 2022

Thursday, 3:30 pm Central Time in MA 115 and online (Zoom credentials provided via email).

Topic: higher geometric (pre)quantization and the geometric cobordism hypothesis. We will explore the notion of a prequantum field theory, which combines traditional action functionals with nontrivial topological data arising from Dirac charge quantization, with the goal of promoting this data to an actual nontopological functorial field theory using the geometric cobordism hypothesis. Necessary prerequisites such as simplicial presheaves, differential cohomology, and higher Lie theory and ∞-Chern–Weil theory will be explored on the way. No knowledge of physics will be assumed.

References:

August 25
Dmitri Pavlov. Introduction to prequantum field theories. Abstract: I will explain the notion of a prequantum field theory using the simplest possible example: the electromagnetic field. The strength tensor of the electromagnetic field together with the topological data responsible for the Aharonov–Bohm effect gives rise to a cocycle in differential cohomology. This cocycle can then be converted to a fully extended functorial field theory using the geometric cobordism hypothesis. The resulting nontopological functorial field theory serves as an input data to the quantization machinery for functorial field theories.
September 1
Jiajun Hoo. Line bundles with connection.
September 8
Emilio Verdooren. Nonabelian Čech cohomology in dimension 0, 1, and 2.
September 15
Gregory Taroyan. Differential cohomology. Handwritten notes.
September 22
Emilio Verdooren. Simplicial sheaves and classifying spaces.
September 29
No seminar.
October 6
Jiajun Hoo. The universal connection on the moduli stack of principal G-bundles with connection.
October 13
Jiajun Hoo. The universal connection on the moduli stack of principal G-bundles with connection. Part II.
October 20
No seminar.
October 27
Dmitri Pavlov. Model structures on simplicial presheaves. Abstract: We will review the projective and injective model structures on simplicial presheaves, as well as their local versions, including practical tools and techniques that allow us to perform computations with simplicial presheaves, in particular, computations for the geometric cobordism hypothesis. Handwritten notes by Gregory Taroyan.
November 3
Dmitri Pavlov. Model structures on simplicial presheaves II. Handwritten notes by Gregory Taroyan.
November 10
Dmitri Pavlov. Geometric structures and the geometric cobordism hypothesis. Abstract: We will review the main site FEmb_d of smooth families of d-manifolds and their fiberwise open embeddings and how it is used in the geometric cobordism hypothesis. Reference: Section 3.3 in arXiv:2111.01095v3. Handwritten notes by Gregory Taroyan.
November 17
Gregory Taroyan. Examples of functorial field theories via the geometric cobordism hypothesis. Handwritten notes.
December 1
Dmitri Pavlov. Bundle (d-1)-gerbes with connection as fully extended nontopological functorial field theories. Handwritten notes by Gregory Taroyan.

Spring 2022

Thursday, 3:30 pm Central Time in MA 115 and online (Zoom credentials provided via email).

January 13
Dmitri Pavlov. Organizational meeting.
January 20
Grigory Taroyan (Faculty of Mathematics, HSE University). 2.1, 2.2, 2.3: Symplectic and oriented derived stacks. Handwritten notes.
January 27
Grigory Taroyan (Faculty of Mathematics, HSE University). 2.4, 2.5: Lagrangian correspondences and oriented cospans. Handwritten notes.
February 3
Grigory Taroyan (Faculty of Mathematics, HSE University). 2.6, 2.7: Higher Lagrangian correspondences and oriented cospans. Handwritten notes.
February 10
Emilio Verdooren. 2.8, 2.9: Higher categories of symplectic and oriented derived stacks.
February 17
Jiajun Hoo. 2.10, 2.11, 2.12: Dualizability and oriented cospans of spaces.
February 24
Emilio Verdooren. 3.1, 3.2: The AKSZ construction on differential forms.
March 3
James Francese. 3.3, 3.4: The AKSZ construction on iterated spans.
March 10
Jiajun Hoo. 4.1, 4.2, 4.3: Higher categories of bordisms.
March 24
Jiajun Hoo. 4.4: Extended functorial field theories. Emilio Verdooren. 5: From Cobordisms to Preoriented Spaces: Idea.
March 31
James Francese. AKSZ-BV Formalism for Sigma Models in the Derived Setting. Abstract: We describe the AKSZ construction as a geometric method for constructing BV action functionals for topological field theories called sigma-models in the setting of derived geometry, including the well-known special case of a symplectic Lie n-algebroid giving rise to higher Chern-Simons theory, and the possibly lesser-known case of a Leibniz algebroid giving rise to supergravity as a low-energy limit of (type II) string theory.
April 7
Daniel Grady. AKSZ and the geometric cobordism hypothesis I. Abstract: We will discuss how to define the AKSZ theory as a fully extended functorial field theory using the geometric cobordism hypothesis.
April 14
Dmitri Pavlov. AKSZ and the geometric cobordism hypothesis II. Abstract: We will discuss how to define the AKSZ theory as a fully extended functorial field theory using the geometric cobordism hypothesis.
April 21
No seminar.
April 28
No seminar.

Fall 2021

Primary reference: Damien Calaque, Rune Haugseng, Claudia Scheimbauer. The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory.

August 26
Dmitri Pavlov. Introduction to functorial field theory. video recording, typeset notes.
September 2
Dmitri Pavlov. The definition of a functorial field theory. video recording, typeset notes. 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: Part of this talk is original work with Daniel Grady.
September 9
Gregory Taroyan (HSE Moscow). Duality between algebra and geometry. handwritten notes.
September 16
Gregory Taroyan (HSE University, Moscow). Sheaves as generalized spaces. handwritten notes.
September 23
No seminar.
September 30
Dmitri Pavlov. Derived duality between geometry and algebra. Abstract: I will introduce derived spaces, motivated by the problem of computing nontransversal intersections. I will then discuss how traditional constructions in differential geometry, such as differential forms and connections on vector bundles, generalize to the setting of derived spaces, such as derived affine schemes or derived C-infinity loci.
October 7
No seminar.
October 14
Emilio Verdooren. Segal spaces.
October 21
No seminar.
October 28
No seminar.
November 4
Emilio Verdooren. Complete Segal spaces.
November 11
Emilio Verdooren. Complete n-fold Segal spaces.
November 18
Emilio Verdooren. Γ-objects and models for smooth symmetric monoidal (∞,n)-categories of bordisms.
November 25
Thanksgiving.

Spring 2021

References and descriptions of talks.

Handwritten notes by Rachel Harris.

January 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.
January 28
Stephen Pena. Simplicial sets and model categories I. Speaker's list of sources:
February 4
Gregory Taroyan (HSE University, Moscow). Simplicial sets and model categories II. Notes.
February 11
James Francese. Simplicial sets and model categories III.
February 18
The Apocalypse.
February 25
Ramiro Ramirez. Homological algebra and model categories of chain complexes.
March 4
James Francese. Differential graded algebras.
March 11
James Francese. Rational homotopy theory. Abstract: Rational homotopy theory is an extremely rich source of algebraic models for geometry and topology. For example, minimal Sullivan models are in 1-1 correspondence with rational homotopy types. In this expository talk we explore this correspondence, along with the rational equivalence between simply connected spaces and connected differential graded Lie algebras.
March 18
Dmitri Pavlov. Differential graded C^∞-rings. Abstract: We will introduce C^∞-rings, which play the same role for smooth manifolds as commutative rings do for schemes. Then we will define differential graded C^∞-rings, introduce a model structure on them, and perform some computations of derived intersections.
March 25
Dmitri Pavlov. Differential graded C^∞-rings: model structures and examples of computations.
April 1
Dmitri Pavlov. Differential graded C^∞-rings: examples of computations.
April 8
Dmitri Pavlov. Derived differentiable stacks.
April 15
Daniel Grady. K-theory of derived differentiable stacks.
April 22
Stephen Peña. The Batalin-Vilkovisky formalism.
April 29
James Francese. BV-BRST Formalism in Derived Differential Geometry. Abstract: We develop deformation theory via the formal neighborhood of a point in a derived stack, which we explain to be a (shifted) L-∞ algebra encoding gauge symmetries of a classical field theory, with working example Chern-Simons theory. We compute the derived critical locus of the classical, free Chern-Simons U(1)-gauge theory via the BV-BRST variational bicomplex, and introduce its BV quantization by means of the quantum master equation. This quantization may admit a further differential twist which is known to obstruct a string structure (refinement of the string Lie 2-group), which we shall observe is automatically described in the derived/L-∞ algebra formalism, unlike in “ordinary” differential geometry, where the special construction of Spin-lifting gerbes is more painstaking.

Fall 2020

August 27
Dmitri Pavlov. Proper orbifold cohomology: an overview.
September 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.
September 10
Daniel Grady. Introduction to equivariant homotopy theory and Elmendorf's theorem. 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.
September 17
Stephen Peña. Introduction to Higher Topos Theory I. 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.
September 24
Stephen Peña. Introduction to Higher Topos Theory II. 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.
October 1
Rachel Harris. Infinity-groups and the internal formulation of groups, actions, and fiber bundles. 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.
October 8
James Francese. Differential Topology via Cohesion in Homotopy- and ∞-Toposes. 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.
October 15
James Francese. Differential Geometry via Elasticity in Homotopy- and ∞-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.
October 22
Nilan Manoj Chathuranga. Formalism for Etale Geometry Internal to Infinity Toposes. 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.
October 29
No seminar.
November 5
Nilan Manoj Chathuranga. Geometry of Singular Cohesive Infinity Toposes. 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.
November 10.
James Francese. A Matinée of Orbispaces and Orbifolds 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.
November 12
James Francese. Structured Orbifold Geometry 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.
November 19
No seminar.

Spring 2020

January 15
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.
January 22
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.
January 29
Dmitri Pavlov. Lie groupoids and simplicial presheaves.
February 5
No seminar (snow storm).
February 12
Daniel Grady. Bundle gerbes with connections.
February 19
James Francese. Précis on Homotopy Type Theory. Abstract: I will review the Curry-Howard-Lambek correspondence, the notion of internal language, the framework of intensional dependent type theory, then Hofmann-Streicher's discovery that identity types have groupoidal structure and Lumsdaine's confirmation that they are actually ∞-groupoids, which via the homotopy hypothesis are identified as topological spaces. I will then describe how HoTT makes this particular idea a theorem, by serving as a synthetic theory of ∞-groupoids which is also apparently “foundational” for mathematics. So mostly a conceptual talk, but I will throw in a range of technical tidbits.
February 26
James Francese. Précis on Homotopy Type Theory II.
March 4
James Francese. Précis on Homotopy Type Theory III.
March 11
James Francese. Précis on Homotopy Type Theory IV.

Fall 2019

August 26
Dmitri Pavlov. What is quantum homotopy?
September 2
No seminar
September 9
Stephen Peña: Introduction to quantum field theory 1. Mechanics on manifolds and classical field theory.
September 16
Stephen Peña: Introduction to quantum field theory 2. Quantum mechanics I.
September 23
Stephen Peña: Introduction to quantum field theory 3. Quantum mechanics II.
September 30
Stephen Peña: Introduction to quantum field theory 4. Quantum mechanics III.
October 7
Stephen Peña: Introduction to quantum field theory 5. Gauge theory I.
October 14
Stephen Peña: Introduction to quantum field theory 6. Gauge theory II.
October 21
Stephen Peña: Introduction to quantum field theory 7. Functorial field theory and algebraic quantum field theory.
October 28
James Francese: Introduction to Lie Theory and Natural Operations.
November 4
James Francese: On the Uniqueness of Lie Theory.
November 11
James Francese: Introduction to Leibniz Algebras.
November 18
James Francese: Generalizations of Lie Theory: Smooth Mal'cev Theories, Formal Group Laws, Fat Points.
November 25
James Francese: Internal Logic of Fat Points.
December 2
James Francese: Internal Logic of Fat Points II: Models for a Leibniz Theory.