Topics in derived algebraic geometry

The picture is taken from here.
This is a seminar of Higher School of Economics offered by prof. Chris Brav, Grigory Kondyrev and myself in the spring semester of 2017.
The seminar is sheduled on Thursday, room 306 at 3 pm.


Derived algebraic geometry is an extension of classical algebraic geometry which provides systematic treatment of "non-transversal situations" in a very satisfactory and workable way. Instead of using various moving lemmas, in DAG one keeps track of some additional information which roughly remembers in what way the intersection was non-transversal. Technically this is achieved by changing basic building blocks of algebraic geometry from ordinary commutative rings to dg-commutative rings or simplicial rings or, more generally, to commutative ring spectra. Consequently, the formalism of DAG turns out to be natural home for deformation theory, intersection theory and various moduli problems. More than that, this theory appears to be useful in various other parts of mathematics such as geometric representation theory, homotopy theory and mathematical physics.

In the first part of our seminar we will give a thorough introduction to the subject aimed at younger students assuming only some background in scheme algebraic geometry, basic homotopy and category theory. In the second part we will proceed to some of the major achievements in the subject such as


  1. Introduction to higher category theory and higher algebra.
  2. Prestacks and stacks, quasi-coherent sheaves.
  3. Derived schemes and Deligne-Mumford stacks, geometric stacks, examples (de Rham stacks, BG, Perf).
  4. Ind-coherent sheaves for schemes and general stacks.
  5. Inf-schemes and cotangent complexes.
  6. Koszul duality. The special case of Lie-coComm duality.
  7. Formal deformation theory in commutative world over arbitrary base.
  8. General formal deformation theory and deformations of associative algebras over a field.
  9. Ringed topoi approach to algebraic geometry, finite presentation and proper morphisms.
  10. Derived Artin's representability theorem. Applications (Weil restriction, Picard functor, ...).
  11. Spectral abelian varieties: duality theory, p-divisible groups, Serre-Tate theorem. Application to topological modular and automorphic forms.
  12. Shifted symplectic structures and quantization.


Talk Speaker Date
Introduction to higher category theory I (definition of higher categories) Dasha Poliakova February, 6
Introduction to higher category theory II (constructions in higher categories) Dasha Poliakova February, 16
Presentable categories
Derived prestacks
Dasha Poliakova
Chirs Brav
February, 23
Stacks, Artin stacksAndrey Konovalov March, 2
More on Artin stacks, quasi-coherent sheaves
Lurie's tensor product
Andrey Konovalov
Grigory Kondyrev
March, 9
Furier-Mukai theory Grigory Kondyrev March, 16
Ind-coherent sheaves I Dasha Poliakova March, 23
Cake Artem Prikhodko March, 30
Beck's monadicity theorem, Ind-coherent sheaves II Artem Prikhodko April, 1
Deformation theory I Chris Brav April, 6
Ind- and Inf-schemes I Artem Prikhodko April, 13
Ind- and Inf-schemes II Artem Prikhodko April, 16
Koszul duality and Lie algebras Grigory Kondyrev April, 27
Blakers-Massey theorem and connective Koszul duality Artme Prikhodko May, 4
Formal deformation theory of (pre)stacks Chris Brav May, 11
Formal deformation theory of (pre)stacks II Artem Prikhodko May, 18
Lie algebroidsIvan Perunov May, 25
Cake IIGrigory Kondyrev June, 15


    Preliminaries on higher category theory and higher algebra

  1. Jacob Lurie, Higher Topos Theory.
  2. Jacob Lurie, Higher Algebra.

    Basics of derived and spectral algebraic geometry

  3. Bertrand Toen, Gabriele Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications.
  4. Dennis Gaisgory, Nick Rozenblyum, A study in derived algebraic geometry, Parts I-III.
  5. Jacob Lurie, Spectral Algebraic Geometry, Parts I and II.

    Koszul duality and formal deformation theory

  6. John Francis, Dennis Gaitsgory, Chiral Koszul duality.
  7. Dennis Gaisgory, Nick Rozenblyum, A study in derived algebraic geometry, Part IV.
  8. Jacob Lurie, Spectral Algebraic Geometry, Part IV.

    DAG and topological modular/automorphic forms

  9. Jacob Lurie, A survey of Elliptic Cohomology.
  10. Jacob Lurie, Elliptic Cohomology I.
  11. Michael Hill, Tyler Lawson, Topological modular forms with level structure.
  12. Mark Behrens, Tyler Lawson, Topological automorphic forms.

    Shifted symplectic structures and quantization

  13. T. Pantev, B. Toen, M. Vaquie, G. Vezzosi Shifted Symplectic Structures.
  14. D. Calaque, T. Pantev, B. Toen, M. Vaquie, G. Vezzosi Shifted Poisson Structures and Deformation Quantization.

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