Guglielmo Nocera




Teaching

Cours d'Homotopie 2, M2 Master de Mathématiques fondamentales 2023-2024 (TD, Jan-Feb 2024) [click for details]

Professor: Geoffroy Horel

Website of the course

The lectures (by Geoffroy Horel) will take place on Mondays 8h30 - 10h30, Université Paris Cité, salle 378 F Halle aux Farines, and Wednesdays 8h30 - 10h30, Université Paris Cité, salle 1009 Sophie Germain.

The TDs (by me) take place on Fridays 9h - 11h, Université Paris Cité, salle 1021 Sophie Germain.

TD sheet A (updated on January 23rd, 2024)
TD sheet B (updated on January 31st, 2024)
TD sheet C (updated on February 6th, 2024)

TD sessions (click to expand):

TD 1, 12 Jan 2024
  • Comments on functorial factorization.
  • Exercise 1 of TD sheet A.
  • Discussion of the proof of Theorem 1.37 in the notes of the course.
TD 2, 19 Jan 2024
  • Further comments on functorial factorization.
  • Exercise 2 of TD sheet A.
TD 3, 26 Jan 2024
  • Exercise 1 of TD sheet B.
  • Exercise 2 of TD sheet B (sketch of points 1,2,3).
TD 4, 2 Feb 2024
  • Comments on Exercise 4 of TD sheet B.
  • Exercise 5 of TD sheet C (sketch of points 1,2,3,4)
TD 5, 9 Feb 2024
  • Fact: the adjunction between Cartesian product and compact-open Hom does not hold for arbitrary topological spaces.
  • Exercises 1,2 of the Exam 2023. Note: the differential for P(B) in Exercise 2 should have $-d\gamma+b'-b$ instead of $d\gamma+b'-b$.
TD 6, 16 Feb 2024
  • Partial solution of exercises 4 and 5 of last years' exam.

A good reference for exercises: the book Modern Classical Homotopy Theory by Jeffrey Strom.
A good reference for theory and exercises: Notes on homotopical algebra by Emanuele Pavia.

Cours d'Algèbre linéaire avancée, M1 Master de Mathématiques fondamentales et appliquées 2023-2024 (TD, Jan-Feb 2024) [click for details]

Professor: Laurent Rigal

Website of the course

Website of last year's course

The TDs (by me) take place on Thursdays 13h45 - 17h, Université Sorbonne Paris Nord.

TD sessions (click to expand):

TD 1, 18 Jan 2024
  • Recall: Endomorphisms of a vector space over a field.
  • Recall: Notion of $k$-algebra. Mentions: euclidean, principal, factorial ring (without definitions).
  • Recall: Prime elements and irreducible elements in a commutative unital ring.
  • Example: Irreducible elements in C[x].
  • Exercise: Irreducible elements in R[x].
TD 2, 25 Jan 2024
  • Recall: $k$-algebra generated by an endomorphism, minimal polynomial of an endomorphism.
  • Exercise 4.3 of Chapter 1 of the polycopie.
TD 3, 31 Jan 2024
  • Recall: lemma of the kernels (lemme des noyaux).
  • Exercises 4.1, 4.2 of Chapter 1 of the polycopie.
  • Exercise 4.4 points 1,2 of Chapter 1 of the polycopie.
TD 4, 8 Feb 2024
  • Cyclic subspaces and cyclic endomorphisms.
  • Cayley-Hamilton Theorem.
  • Statement of the Frobenius decomposition theorem (Theorem 3.5 of the polycopie).
  • Similar endomorphisms.
  • Euclidean rings.
  • Reduction algorithm (page 25 in the polycopie).
TD 5, 15 Feb 2024
  • Exercises from Chapter 3, Section 5.

Research groups

Factorization algebras (2023, online. With Carlo Buccisano, Šarūnas Kaubrys, Alexey Latyntsev, Hugo Pourcelot)

Reading seminars

Factorization homology (Spring 2023, LAGA Université Paris XIII, co-organizer).

Introduction to formal moduli problems (2020-2021, Scuola Normale Superiore and University of Pisa, co-organizer)

This was a reading seminar based on Derived Algebraic Geometry X by Jacob Lurie, and organised by Federico Conti, Julian Demeio, Michele Pernice, Morena Porzio and myself. It ran from December 2020 to July 2021
In this course we introduced the necessary background (starting from basics in deformation theory and ∞-category theory) to understand the link between differentially graded Lie algebras and formal moduli problems, as exposed in Lurie's note, Theorem 2.0.2.
Bibliography: [Calaque-Grivaux] Damien Calaque and Julien Grivaux, Formal moduli problems and formal derived stacks, 2019
[DAG X] Jacob Lurie, Derived Algebraic Geometry X , 2011
[Gepner] David Gepner, An introduction to Higher Categorical Algebra, 2019
[HA] Jacob Lurie, Higher Algebra, 2017
[Hovey] Mark Hovey, Model Categories, AMS, Mathematical Monographs and Surveys 63, 1999
[HTT] Jacob Lurie, Higher Topos Theory, Princeton University Press, 2009
[Porta-Vezzosi] Mauro Porta and Gabriele Vezzosi, Infinitesimal and square-zero extensions of simplicial algebras, 2013
[Pridham] J.P. Pridham, Unifying derived deformation theories, Advances in Mathematics, 2010
[Toën] Bertrand Toën, Problèmes de modules formels, Séminaire Bourbaki, 2016

Fourier-Mukai transforms in Algebraic Geometry (2018-2019, Scuola Normale Superiore and University of Pisa, co-organizer)