Abstract: A multitude of interesting objects in number theory may be understood in function of the so-called exponential sums. These sums are intimately related to questions of equidistribution, and the strongest results in this direction come from interpreting them as traces of l-adic sheaves. In this talk, I’ll give an overview of the “classical” works by Deligne and Katz, as well as more recent results based on tannakian methods.
Presentations on the Brazilian Algebraic Geometry Seminar on the 26th August 2022 (video) and on the Séminaire d'Arithmétique of the CMLS on the 13th October 2022.
A presentation about t-structures and perverse sheaves. This is part of a seminar on the Riemann-Hilbert correspondence for D-modules.
A presentation about the use of l-adic cohomology in exponential sums. We see an exponential sum as a sum of traces of Frobenii and then use the main theorem of Weil II, along with a geometrical study of the associated spaces, to bound these sums. This is part of a seminar on Deligne's Weil II.
A presentation about the fundamental machinery behind étale cohomology. We talk about sites and sheaves, about topoi, we study the stalks of the structure sheaf, we see some descent theory and, finally, we calculate some simple cohomology groups. This is part of a seminar on Deligne's Weil II.
A presentation about Brauer groups and Galois cohomology, eventually arriving at the calculation of the étale cohomology groups of smooth projective curves. This is part of a seminar on Deligne's Weil II.
A presentation about Tate's proof of the meromorphic continuation and functional equation of Hecke's L-functions. We focus on the formal analogies with the classical proof of the particular case of the Riemann zeta function.
A presentation about a categorical axiomatization of six-functor formalisms and the plethora of cohomological constructions that come for free.
A presentation about Hilbert's 21st problem, eventually culminating into the Riemann-Hilbert correspondence between D-modules and perverse sheaves. (In portuguese.)
A presentation about six functor formalisms, focusing on Grothendieck duality. We also explain some of the homotopic ideas used by the modern proofs. (In portuguese.)
A summary of the main ideas in T. Szamuely's book with the same title. We also explore more recent subjects such as Bhatt's and Scholze's work on the pro-étale topology. (In portuguese.)
A presentation to the EDMH about my academic history and my thesis project. (In french.)
A presentation about my M2 internship with Javier Fresán. (In french.)
A presentation about my M1 internship with Antoine Ducros. (In french.)
A presentation about my EA project with Benjamin Schraen (which continues another EA project with Gaëtan Chenevier). (In french.)
A presentation about my PSC with Carlos Matheus Silva Santos. (In french.)
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