This is a joint seminar of the Center for Advanced Studies at Skoltech and International laboratory for Mirror Symmetry and Automorphic Forms offered by myself and Vladimir Shaydurov in the 2018-2019 academic year. The lectures are scheduled on

One important problem of the modern algebraic topology is to understand the structure of the ring $\pi_*(\mathbb S)$ of stable homotopy groups of spheres. There are two big classical results in this direction

- Nishida's theorem asserts that every class in $\pi_{>0}(\mathbb S)$ is nilpotent.
- Adams' theorem about the image of $J$-homomorphism provides us with the infinite family of elements in $\pi_*(\mathbb S)$.

It turns out these results of seemingly distinct nature are in fact related to each other and both find natural generalizations in the branch of algebraic topology called * chromatic homotopy theory*. Namely by contrast to the sphere spectrum case, for each odd prime number $p$ Adams constructed a self-map $v_1\colon \Sigma^{2(p-1)}\mathbb S/p \to \mathbb S/p$ which induces multiplication by $\beta^{p-1}$ in $K$-theory and hence can not be nilpotent (the story for $p=2$ works similarly). The $p$-torsion of the image of $J$-homomorphism then can be obtained from iterates of $v_1$ by taking the composition $\mathbb S^{2(p-1)i} \to \mathbb S^{2(p-1)i}/p \stackrel{v_1^i}{\longrightarrow} \mathbb S/p \to \mathbb S^1$. The natural question now is to try to generalize this picture, e.g. can we find a non-nilpotent self-map $v_2\colon \Sigma^r \mathbb S/(p,v_1) \to \mathbb S/(p,v_1)$? To answer such questions it is desirable to have some control over nilpotency. Very satisfactory solution is provided by the following celebrated theorem of Devinatz-Hopkins-Smith:

To explain the power of the nilpotence theorem

, we need first to discuss the geometry of the complex cobordism spectrum $MU$. By the theorems of Miller-Quillen and Landweber-Novikov the Hopf algebroid $(MU_*, MU_*MU)$ corepresents the stack $\mathcal M_{fg}$ of $1$-dimensional commutative formal group laws. Now $p$-locally every formal group has an invariant called * height*: the height of formal multiplicative group is $1$, all formal groups of heights $\le 2$ can be obtained from elliptic curves, with formal groups of heights $2$ corresponding precisely to supersingular elliptic curves, e.t.c. It follows the stack $\mathcal M_{fg,(p)}:= \mathcal M_{fg}\otimes_{\mathbb Z} \mathbb Z_{(p)}$ admits an increasing filtration by open substacks $\mathcal M_{fg,(p)}^{\le h}$ of formal group laws of heights less then $h+1$ with locally closed strata $\mathcal M_{fg,(p)}^h$ being substacks of formal group laws of heights exactly $h$. Given any spectrum $X$ its $MU$-homology $MU_*(X)$ are naturally co-module over $(MU_*, MU_*MU)$, hence descent to give a quasi-coherent sheaf $\mathcal F_X\in \mathrm{QCoh}(M_{fg})$ (in particular the second page of $MU$-based Adams-Novikov spectral sequence for $X$ can be identified with $H^*(\mathcal M_{fg}, \mathcal F_X)$). Given any sheaf $\mathcal F\in \mathrm{QCoh}(\mathcal M_{fg,(p)})$ one can restrict $\mathcal F$ to an open substack $\mathcal M_{fg,(p)}^{\le h}$ or take its completion in $\mathcal M_{fg,(p)}^{h}$. It turns out these operations can be lifted to the category of spectra: for each prime $p$ and natural $n$ there exist spectra $E(n)$ and $K(n)$ (

Since $L_{E(n-1)} \circ L_{E(n)} \simeq L_{E(n-1)}$, for any spectrum $X$ we can then consider its * chromatic tower* $X \to \ldots \to L_{E(n)} X \to L_{E(n-1)} X \to \ldots \to L_{E(0)} X \simeq X\wedge H\mathbb Q_p$. One important consequence of the nilpotence theorem asserts

Back to $v_n$-self maps let us say that $X$ has type $n$ if $L_{E(n-1)} X \simeq 0$ and $L_{E(n)} X \not \simeq 0$. Nilpotence theorem also serves as an important input to the proof of the following result

- Reminder on spectra and Bousfield localizations.
- Complex oriented cohomology theories. Miller-Quillen's theorem. Moduli stack of formal groups. Landweber exact functor theorem. Morava $E$- and $K$-theories.
- $E(n)$- and $K(n)$-local categories. Morava stabilizer group. $v_n$-self maps.
- Nilpotence theorem. Periodicity theorem. Thick subcategory theorem. Smash product theorem. Chromatic convergence theorem. Fracture squares.
- Hopkins-Devinatz theorem. Galois theory of Rognes-Mathew.
- Shimura varieties and automorphic forms.
- Elements of spectral algebraic geometry. Artin's representability theorem. Goerss-Hopkins-Miller-Lurie theorem. Topological automorphic forms.
- Finite resolutions of K(n)-local sphere spectra and congruences in rings of automorphic forms.

Talk | Speaker | Date, place |
---|---|---|

Overview of the course | Artem Prikhodko | September, 18 |

Stable categories and spectra I | Artem Prikhodko | September, 25 |

Introduction to higher categories | Artem Prikhodko | September, 30 |

Presentable categories I | Artem Prikhodko | October, 2 |

Presentable categories II Stable categories and spectra II | Artem Prikhodko | October, 7 |

Stable categories and spectra III | Artem Prikhodko | October, 9 |

Bousfield localisations | Artem Prikhodko | October, 13 |

Complex orientable cohomology theories | Artem Prikhodko | October, 16 |

Steenrod algebra | Vladimir Shaydurov | October, 20 |

Canceled due to Recent developments in Higgs theory conference | October, 23 | |

Exercise session | Artem Prikhodko | October, 27 |

Adams-Novikov spectral sequence I | Artem Prikhodko | October, 30 |

Adams-Novikov spectral sequence II | Artem Prikhodko | November, 3 |

Thom spectra, MU | Artem Prikhodko | November, 6 |

The image of $J$-homomorphism I | Vladimir Shaydurov | November, 10 |

Moduli stacks of formal groups | Kirill Salmagambetov | November, 13 |

The image of $J$-homomorphism II | Vladimir Shaydurov | November, 17 |

Lazard's theorem | Alexander Novikov | November, 20 |

Quillen's theorem | Alexander Novikov | November, 24 |

Landweber's criterion | Dmitrii Zhurbenko | November, 27 |

Lubin-Tate FGL, Morava $E$-theories | Artem Prikhodko | December, 1 |

Morava $K$-theories | Artem Prikhodko | December, 4 |

Lazard's open subgroup theorem | Dmitrii Krekov | December, 8 |

On Borel's equivariant cohomology I | Vasya Rogov | December, 11 |

Morava stabilizer group and its cohomology | Roman Kositsin | December, 15 |

On Borel's equivariant cohomology II | Vasya Rogov | December, 18 |

Nilpotence theorem: implications (thick subcategory theorem, ...) | Pavel Popov | December, 22 |

Nilpotence theorem: proof I | Artem Prikhodko | December, 25 |

Nilpotence theorem: proof II | Artem Prikhodko | December, 29 |

$v_n$-self maps, periodicity theorem I | Egor Shuklin | January, 19 |

Periodicity theorem II | Egor Shuklin | January, 22 |

Periodicity theorem III | Vladimir Shaydurov | January, 26 |

Monochromatic categories | Artem Prikhodko | February, 2 |

Chromatic convergecnce theorem | Artem Prikhodko | February, 9 |

Smash localization theorem | Artem Prikhodko | February, 12 |

Overview of the second part of the course | Artem Prikhodko | February, 16 |

Introduction to operads and monads | Artem Prikhodko | February, 19 |

Examples of operads | Artem Prikhodko | February, 23 |

May recognition | Artem Prikhodko | February, 26 |

Algebraic cobordism: Riemann-Roch theorems, operations and structure | Pavel Sechin | March, 2 |

Aside: (semi-)topological $K$-theory | Andrei Konovalov | March, 9 |

Goerss-Hopkins obstruction theory I | Artem Prikhodko | March, 12 |

Locally symmetric spaces and aproximation theorems | Pavel Popov | March, 19 |

Goerss-Hopkins obstruction theory II | Artem Prikhodko | March, 23 |

Goerss-Hopkins obstruction theory III | Artem Prikhodko | March, 27 |

Introduction to Shimura varietie I | Roman Kositsin | March, 30 |

Introduction to Shimura varietie II | Roman Kositsin | March, 2 |

Introduction to Shimura varietie III | Roman Kositsin | March, 6 |

Cotangent complex I | Egor Shuklin | April, 13 |

Cotangent complex II | Egor Shuklin | April, 20 |

Deformations of commutative algebras I | Artem Prikhodko | April, 23 |

Deformations of commutative algebras II Reminder of DAG | Artem Prikhodko | April, 30 |

Artin's representability theorem | Alexander Novikov | May, 7 |

Serre-Tate theory | Ivan Perunov | May, 14 |

Galois theory and homotopy theory | Catherine Ray | August, 15 |

Some propblems: Problem set 1a, Problem set 1b, Problem set 4.

- Jacob Lurie, Chromatic Homotopy Theory.
- Michael Hopkins. Complex Oriented Cohomology Theories and the Language of Stacks.
- Douglas Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres (green book).
- Douglas Ravenel. Nilpotence and periodicity in stable homotopy theory (orange book).
- Douglas Ravenel. Localization with respect to certain periodic homology theories.
- Ethan Devinatz, Michael Hopkins, Jeffrey Smith. Nilpotence and Stable Homotopy Theory I, II.
- Aldridge Bousfield. The localization of spectra with respect to homology.
- John Rognes, The Adams Spectral Sequence.
### Shimura varieties and automorphic forms

- Pierre Deligne, Travaux de Shimura.
- Alain Genestier, Bao Chau Ngo, Lectures on Shimura varieties.
- Armand Borel, Introduction to the Cohomology of Arithmetic Groups.
- Gunter Harder, Cohomology of Arithmetic Groups.
- Stephen Gelbart, Automorphic Forms on Adele Groups.
### Topological modular and automorphic forms

- Jacob Lurie, A survey of elliptic cohomology.
- Jacob Lurie, Elliptic Cohomology I, II.
- Mark Behrens, A modular description of the K(2)-local sphere at the prime 3.
- Mark Behrens, Tyler Lawson, Topological Automorphic Forms.
- Mark Behrens, Topological modular and automorphic forms.