Texas Tech Quantum Homotopy Seminar

Time and place: see below. For onsite location in Lubbock, email the organizer.

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

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

Organizer: Dmitri Pavlov

Departmental seminar website.

Fall 2024

Monday at 15:00 UTC (keep track of your time zone and daylight saving time).

Topics: higher Lie theory, higher Lie integration, higher principal bundles with connection, adjusted connections.

References:

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff. Čech cocycles for differential characteristic classes. Advances in Theoretical and Mathematical Physics 16:1 (2012), 149–250. arXiv:1011.4735.
• Christopher L. Rogers, Jesse Wolfson. Lie's third theorem for Lie ∞-algebras. arXiv:2409.08957.

Supplementary resources:

• Andre Henriques. Integrating L-infinity algebras. arXiv:math/0603563.

October 21

Dmitri Pavlov. Introductory meeting. Lie groups, Lie algebras, left-invariant differential geometry of Lie groups. Notes.

October 28

Dmitri Pavlov. Lie's third theorem for ordinary Lie groups and Lie algebras. Notes.

November 4

Jacek Kenig. Simplicial presheaves and higher Lie groups. Notes.

November 11

Emilio Verdooren. Higher Lie integration I. L∞-algebras and the Chevalley-Eilenberg construction: The stuff we integrate. Slides.

November 18

Emilio Verdooren.

November 25

December 2

Spring 2024

Wednesday at 12:00 Central Standard Time (UTC-06; before March 10) or Central Daylight Time (UTC-05; after March 10).

References:

• Grigorios Giotopoulos, Hisham Sati. Field Theory via Higher Geometry I: Smooth Sets of Fields.

Supplementary resources:

• Christian Blohmann. Lagrangian Field Theory. Diffeology, Variational Cohomology, Multisymplectic Geometry.
• Patrick Iglesias-Zemmour. Diffeology.
• Luigi Alfonsi, Charles A. S. Young. Towards non-perturbative BV-theory via derived differential geometry.

2.

Bosonic field spaces as smooth sets: 17 pages

2.1

Intuitive approach to smooth sets

2.2

Field spaces, tangent bundles and diffeomorphisms

2.3

Classifying space of de Rham forms

3

Local Lagrangians: 10 pages

3.1

Infinite jet bundles as locally pro-manifolds

3.2

Local Lagrangians, currents and symmetries

4

Differential geometry on the infinite jet bundle: 11 pages

4.1

Tangent bundle and vector fields

4.2

Horizontal splitting

4.3

Differential forms

5

Euler-Lagrange dynamics via the infinite jet bundle: 19 pages

5.1

The variational bicomplex and EL equations

5.2

On-shell space of fields and conserved currents

5.3

On-shell fields as a smooth critical set

6

Infinitesimal symmetries: 19 pages

6.1

Local infinitesimal symmetries and Noether’s First Theorem

6.2

Infinitesimal gauge symmetries and Noether’s Second Theorem

6.3

Cauchy surfaces and obstructions by gauge symmetries

7

Presymplectic structure of local field theories: 26 pages

7.1

The local bicomplex and its Cartan calculus

7.2

Presymplectic current and induced brackets

7.3

The covariant phase space, off-shell and on-shell Poisson brackets

January 10

No seminar.

January 17

No seminar.

January 24

Organizational meeting.

January 31

JJ Hoo. Chapter 2.

February 7

JJ Hoo. Chapter 2.

February 14

No seminar.

February 21

Tyler. Chapter 3.

February 28

Jacek. Chapter 4.

March 6

Misha. Chapter 5.

March 13

Spring break.

March 20

Misha. Chapter 5.

March 27

Emilio. Chapter 6.

April 3

Alonso. Chapter 7.1.

April 10

No seminar.

April 17

April 24

Tyler Russell. Chapter 7.2.

May 1

(?) Emilio. Chapter 7.3.

Fall 2023

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

Topic: twisted equivariant differential K-theory and condensed matter physics.

References:

• Hisham Sati, Urs Schreiber: Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory.

August 24

Organizational meeting.

August 31

Dmitri Pavlov. K-theory: topological, differential, twisted. Abstract: We will review various models and definitions for K-theory, including differential K-theory and twisted K-theory.

September 7

Dmitri Pavlov. K-theory II.

September 14

Emilio Verdooren. Integer quantum Hall effect and vector bundles with connection.

September 28

Emilio Verdooren. Free topological phases and twisted equivariant topological K-theory.

October 5

Rachel Kinard. Topological Data Analysis (TDA). Abstract: The word “Topology”, best known for its association to the study of invariants of an abstract space, is a branch of Pure Mathematics whose best known applications are found in Physics (Quantum Mechanics, Quantum Field Theory). Very rarely does a Pure Math Field find such as Topology find relevance in a world of Big Data and computer automation. Data Science utilizes these powerful topological invariants to quickly gather information about complex data spaces in a brave new area of study called “Topological Data Analysis” or TDA. Given a set of data points, the nerve construction produces a simplicial complex that can be analyzed to understand important characteristics of the data. I will provide an introduction to TDA and a few examples of surprising Data Science applications.

Spring 2023

Topic: higher geometric (pre)quantization and the geometric cobordism hypothesis.

References:

• Domenico Fiorenza, Christopher L. Rogers, Urs Schreiber. Higher U(1)-gerbe connections in geometric prequantization.
• Domenico Fiorenza, Christopher L. Rogers, Urs Schreiber. L_∞-algebras of local observables from higher prequantum bundles.

January 12

Organizational meeting.

January 19

Dmitri Pavlov. Review of necessary prerequisites from stacks. Handwritten notes by Gregory Taroyan.

January 26

Dmitri Pavlov. Stacks and simplicial presheaves II. Abstract: I will continue to talk about stacks and simplicial presheaves in the context of prequantum field theories.

February 2

Jiajun Hoo: 2.9, 3.1. Handwritten notes by Gregory Taroyan.

February 9

Emilio Verdooren: 3.2.1. Handwritten notes by Gregory Taroyan.

February 16

No seminar.

February 23

Emilio Verdooren: 3.2.1, Part II. Handwritten notes by Gregory Taroyan.

March 2

Gregory Taroyan: 3.2.2, 3.3.1. Handwritten notes by Gregory Taroyan.

March 9

Jiajun Hoo: 3.3.2. Handwritten notes by Gregory Taroyan: a, b.

March 23

James Francese: 4.1, 4.2.

March 30

James Francese: 4.1, 4.2.

April 6

Gregory Taroyan.

April 13

No seminar.

April 20

Emilio Verdooren. L_∞-algebra connections. Typeset notes.

April 27

Emilio Verdooren. L_∞-algebra connections II. Typeset notes.

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:

• Urs Schreiber. Higher prequantum geometry.
• Domenico Fiorenza, Christopher L. Rogers, Urs Schreiber. Higher U(1)-gerbe connections in geometric prequantization.
• Daniel S. Freed. Dirac charge quantization and generalized differential cohomology.
• Araminta Amabel, Arun Debray, Peter Haine. Differential Cohomology. Categories, Characteristic Classes, and Connections.

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:

• 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

Part of this talk is original work with Daniel Grady.

September 9

Gregory Taroyan (HSE Moscow). Duality between algebra and geometry. handwritten notes.

• The Spec functor and its inverse as an equivalence of categories. Smooth manifolds are contravariantly equivalent to a full subcategory of commutative real algebras (Milnor's exercise). Affine schemes are contravariantly equivalent to commutative rings (by definition). Examples: the space R^n, the projective space (as a smooth manifold and as an affine scheme).
• Points of Spec(A) are homomorphisms of algebras A→k. Examples: points of smooth manifolds; points of varieties over an algebraically closed field of characteristic 0. Concrete example: the real line (as a smooth manifold) and its points. Any homomorphism C^∞(R)→R is induced by a point in R.
• Maps Spec(A)→Spec(B) are homomorphisms of algebras B→A. Examples: smooth maps of smooth manifolds; regular maps of affine schemes.
• The Serre-Swan equivalence. On affine schemes or smooth manifolds, dualizable (i.e., finitely generated projective) modules are equivalent to finite-dimensional vector bundles via the functor of global sections. Example: the canonical line bundle on the projective space.
• Pullback of vector bundles in terms of induced modules (extension of scalars). Example: fibers of vector bundles. Example: fibers of the canonical line bundle on the projective space.
• Tangent vectors as derivations A→k. Vector fields as derivations A→A. Example: vector fields on the real line.
• Differential 1-forms as Kähler differentials. Example: vector fields on the real line as C^∞-Kähler differentials.
• Differential forms on Spec(A) as the free commutative differential graded algebra on A. Example: differential forms on R^n. Example: a tautological proof of the de Rham lemma, see the last paragraph of an answer on MathOverflow.
• References:
  • Jet Nestruev: Smooth manifolds and observables

September 16

Gregory Taroyan (HSE University, Moscow). Sheaves as generalized spaces. handwritten notes.

• Examples of sites: the opposite category of commutative rings or algebras. Affine schemes (commutative rings), C^∞-loci (finitely generated germ-determined C^∞-rings), Stein spaces (complex algebras with entire functional calculus). Secondary example: cartesian smooth manifolds (R^n and smooth maps). The Yoneda embedding.
• Sheaves on affine schemes as generalized spaces. Example: internal homs of smooth manifolds. Example: the generalized space of differential n-forms. Example: the Grassmannian via functor of points. Example: traditional schemes in algebraic geometry defined using locally ringed spaces are equivalent to schemes defined using functor of points.
• The tangent space of a sheaf S as the internal hom Hom(Spec(R[x]/x^2), S). Example: the tangent space of the internal hom Hom(M,N) at the point f:M→N is the vector space of sections of f* TN. Example: the Lie algebra of the infinite-dimensional Lie group of diffeomorphisms M→M is precisely the Lie algebra of smooth vector fields on M.
• Differential forms on generalized spaces S as morphisms S → Ω^n. The de Rham complex of S. Example: differential forms on Hom(M,N) are sections on the bundle constructed as follows: at point f∈Hom(M,N) take the exterior algebra of the vector space of sections of f*T*N.
• Sheaves of groupoids (= stacks in groupoids) as generalized spaces. Example: the stack of vector bundles Vect, principal bundles B(G). With connection: Vect_∇, B_∇(G). Vector bundles with connection on a generalized space S are precisely (derived) morphisms S → Vect_∇. Likewise for principal bundles etc. Example: vector and principal bundles with connection on Hom(M,N).
• Example: the Freed--Hopkins theorem says that the de Rham complex of B_∇(G) is precisely the graded abelian group of G-invariant polynomials on the Lie algebra of G, with a vanishing differential. This yields precisely the Chern-Weil homomorphism, including infinite-dimensional manifolds such as Hom(M,N).
• References:
  • Bertrand Toën: A master course on algebraic stacks
  • Urs Schreiber: Geometry of physics
  • Daniel S. Freed, Michael J. Hopkins: Chern–Weil forms and abstract homotopy theory
  • Ieke Moerdijk, Gonzalo E. Reyes: Models for smooth infinitesimal analysis
  • Anders Kock: Synthetic geometry of manifolds

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:

• Friedman has wonderful pictures in his notes, and gives you excellent intuition: An elementary illustrated introduction to simplicial sets.
• Dmitri's notes are more technical, and thus are a reasonable post-Friedman option. There are many detailed examples, which is what makes this reference indispensable once you are ready to dig into the gritty details.
• If you are pretty comfortable with general “abstract nonsense”, then Emily Riehl's notes A leisurely introduction to simplicial sets will be a great reference for you.
• This is nice little reference from OCW, which is nice summary of the basics. Not too detailed, but serves as a nice way to remind yourself of some definitions. An Introduction to Simplicial Sets.
• Now that you are ready to become a titan of homotopy theory you can use Goerss and Jardine exclusively. doi:10.1007/978-3-0346-0189-4, PDF.

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.

• "Higher U(1)-gerbe connections in geometric prequantization," Fiorenza, Rogers, and Schreiber
• "Twisted differential String and Fivebrane structures", Sati, Schreiber, and Stasheff
• "L-∞ algebra connections and applications", Sati, Schreiber, and Stasheff
• "Čech Cocycles for Differential Characteristic Classes: an ∞-Lie theoretic construction", Fiorenza, Schreiber, and Stasheff
• "The Rational Higher Structure of M-theory", Fiorenza, Sati, and Schreiber
• "Spontaneous symmetry breaking: a view from derived geometry", Elliot and Gwilliam
• "Higher Lie Theory," Robert-Nicoud and Vallette
• "Integrating L-∞ algebras", Henriques
• "Chern-Weil Forms and Abstract Homotopy Theory", Freed and Hopkins
• "One-dimensional Chern-Simons Theory and the  genus", Gwilliam and R. Grady
• "On some generalizations of Batalin-Vilkovisky algebras", Akman

• Toward the Mathematics of Quantum Field Theory, Paugam
• Higher Structures in Geometry and Physics, Cattaneo, Giaquinto, and Xu (eds.)
• Algebraic Models in Geometry, Félix, Opera, and Tanré
• Formal Geometry and Bordism Operations (esp. 2.1, 5.6), Peterson
• Loop Spaces, Characteristic Classes, and Geometric Quantization, Brylinski (classical)
• Spin Geometry, Lawson and Michelsohn (classical)

• Quantum Field Theory: Batalin-Vilkovisky Formalism and its Applications, Mnev
• Differential Cohomology in a Cohesive ∞-Topos, Schreiber
• Deformations of Algebras and their Diagrams, Markl
• The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds, Morgan (classical)

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.