Time and place: Thursday at 2 p.m., MATH 11. Online streaming via Zoom is available, contact the organizer for credentials.

Organizer: Dmitri Pavlov

Handwritten notes by Rachel Harris are hyperlinked in the titles.

- May 18
- Dmitri Pavlov. Functor of points and derived geometry: new directions and open problems.
- May 27
- Dmitri Pavlov. Derived differentiable stacks. Abstract: We will look at a simplified definition of derived differentiable stacks (using derived cartesian spaces). Just like the last time, several easy-to-write open problems will be communicated.
- June 3
- Dmitri Pavlov. Algebraic structures on derived critical loci. Abstract: We will examine a simplified definition of differential forms in the setting of derived differentiable stacks. We will then relate it to the usual definition by Ben Zvi and Nadler. Finally, we will apply these ideas to define algebraic structures on derived critical loci, such as the shifted symplectic structures of Pantev–Toën–Vaquié–Vezzosi. I will in part follow a recent paper by Vezzosi “Basic structures on derived critical loci”, except that our basic setup is differential geometry, not algebraic geometry.
- June 17
- Rachel Harris. Shifted symplectic structures.
- July 1
- James Francese. Maurer-Cartan stacks for exceptional generalized geometry, Part I. Abstract: In this talk we lay down the groundwork for studying geometric structures on Leibniz algebroids which generalize known integrability conditions of the classical twisted Courant bracket, the setting for Dirac structures. We explain 11-dimensional supergravity as a Leibniz algebroid constructed solely out of natural operations on a generalized tangent bundle, with a bracket twisted by de Rham cohomology classes defined by auxiliary fields satisfying Bianchi-type identities. This is the type of geometric structure we care to study in a derived context by means of Maurer-Cartan stacks controlling higher derived brackets twisted by arbitrary natural operations on a generalized tangent complex.

The first four papers are surveys:

- Toën: Higher and derived stacks: a global overview
- Toën: Derived algebraic geometry
- Pantev, Vezzosi: Symplectic and Poisson derived geometry and deformation quantization
- Toën, Vezzosi: Brave new algebraic geometry and global derived moduli spaces of ring spectra
- Seokbong Seol, Mathieu Stiénon, Ping Xu: Dg manifolds, formal exponential maps and homotopy Lie algebras
- Wai-Kit Yeung: Shifted symplectic and Poisson structures on global quotients
- Marco Benini, Pavel Safronov, Alexander Schenkel: Classical BV formalism for group actions
- Joost Jakob Nuiten: Lie algebroids in derived differential topology
- Vezzosi: Basic structures on derived critical loci
- Ben-Zvi, Nadler: Loop spaces and connections
- Ben-Zvi, Francis, Nadler: Integral transforms and Drinfeld centers in derived algebraic geometry
- Toën, Vezzosi: Chern character, loop spaces and derived algebraic geometry
- Toën, Vezzosi: Caractères de Chern, traces équivariantes et géométrie algébrique dérivée
- Toën, Vezzosi: Algèbres simpliciales S1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs
- Calaque, Grivaux: Formal moduli problems and formal derived stacks
- Pantev, Toën, Vaquié, Vezzosi: Shifted symplectic structures.
- Calaque, Pantev, Toën, Vaquié, Vezzosi: Shifted Poisson structures and deformation quantization
- Calaque: Lagrangian structures on mapping stacks and semi-classical TFTs
- Calaque, Caldararu, Tu: On the Lie algebroid of a derived self-intersection
- Calaque: Derived stacks in symplectic geometry
- Calaque: Shifted cotangent stacks are shifted symplectic
- Raksit: Hochschild homology and the derived de Rham complex revisited.
- Pridham: Shifted Poisson and symplectic structures on derived N-stacks
- Vezzosi: Quadratic forms and Clifford algebras on derived stacks
- Pridham: A differential graded model for derived analytic geometry
- Katzarkov, Pantev, Toën: Algebraic and topological aspects of the schematization functor
- Pridham: Pro-algebraic homotopy types
- Pridham: Presenting higher stacks as simplicial schemes
- Pridham: Representability of derived stacks
- Pridham: Derived moduli of schemes and sheaves
- Pridham: Constructing derived moduli stacks
- Carchedi, Steffens: On the universal property of derived manifolds

A worthy goal would be to understand the work of Pridham and Lurie on the connection between deformation theory and differential graded Lie algebras.

Another worthy goal would be to understand the papers on higher Lie integration of Henriques, Zhu, and others.

- Pridham: Unifying derived deformation theories
- Lurie: Moduli problems for ring spectra
- Henriques: Integrating L_∞-algebras
- Tseng, Zhu: Integrating Lie algebroids via stacks
- Zhu: Lie n-groupoids and stacky Lie groupoids
- Zhu: Lie II theorem for Lie algebroids via higher groupoids
- Zhu: Kan replacements of simplicial manifolds
- Burke: A synthetic version of Lie's second theorem
- Carchedi: Étale stacks as prolongations
- Nuiten: Homotopical algebra for Lie algebroids
- Neuiten: Koszul duality for Lie algebroids
- Grady, Gwilliam: Lie algebroids as L_∞-spaces.
- Gwilliam: Factorization algebras and free field theories, Chapter 7

We studied SDG in the Winter Seminar, we could study it more and establish some connections to Goodwillie calculus.

A worthy goal would be to understand the recent remarkable paper by Bauer, Burke, Ching (see below), which makes rigorous the connection between differential calculus and Goodwillie calculus.

- Kock: Synthetic Geometry of Manifolds, users-math.au.dk/kock/SGM-final.pdf
- Lavendhomme: Basic Concepts of Synthetic Differential Geometry
- Moerdijk, Reyes: Models for Smooth Infinitesimal Analysis
- Bunge, Gago, San Luis: Synthetic Differential Topology
- Burke: A synthetic version of Lie's second theorem
- Leung: Classifying tangent structures using Weil algebras
- Carchedi, Steffens: On the universal property of derived manifolds
- Cockett, Cruttwell: Differential structure, tangent structure, and SDG
- Cockett, Cruttwell: Differential bundles and fibrations for tangent categories
- Cockett, Cruttwell: Connections in tangent categories
- Lucyshyn-Wright: On the geometric notion of connection and its expression in tangent categories
- Blute, Cruttwell, Lucyshyn-Wright: Affine geometric spaces in tangent categories
- Cruttwell, Lucyshyn-Wright: A simplicial foundation for differential and sector forms in tangent categories
- Garner: An embedding theorem for tangent categories
- Lemay: A tangent category alternative to the Faà di Bruno construction
- MacAdam: Vector bundles and differential bundles in the category of smooth manifolds
- Blute, Cockett, Seely: Cartesian differential categories
- Blute, Cockett, Lemay, Seely: Differential categories revisited
- Lemay: Exponential functions in Cartesian differential categories
- Lemay: Convenient antiderivatives for differential linear categories
- Lemay: Differential algebras in codifferential categories
- Cockett, Lemay: Integral categories and calculus categories
- Cockett, Lemay: Cartesian integral categories and contextual integral categories
- Blute, Ehrhard, Tasson: A convenient differential category
- Bauer, Burke, Ching: Tangent ∞-categories and Goodwillie calculus

Time and place: typically Tuesday or Thursday 1–5 p.m., somewhere in Lubbock, Texas.

- May 19
- May 28
- June 2
- June 9
- June 16
- June 23
- July 3
- July 16
- July 24
- July 28
- August 4
- August 13
- August 20