Time and place: Thursday 3:30–5 p.m., 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

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.

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

- 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.

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 Â 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)

- 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.

- 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.

- 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.