Talk Notes

1. Breuil-Kisin Twist

   This talk defines the Breuil-Kisin twist, which is the object which plays the role of a Tate twist in the category of prismatic $F$-crystals. Much of this talk is based on work in this paper by Bhargav Bhatt and Jacob Lurie. The talk was part of a learning seminar on this paper by Bhargav Bhatt and Peter Scholze, and the learning seminar was organized by Alex Youcis.

2. Fourier Transform and Weights

   We complete the proof of the Weil conjectures by using Laumon's Fourier transform for $\ell$-adic sheaves on $\mathbb{A}^1_{\mathbb{F}_q}$. This talk was based on some of the material in this book by Reinhardt Kiehl and Rainer Weissauer. This talk was part of a learning seminar on the Weil conjectures organized by Elden Elmanto and Nick Rozenblyum from January 2025 to April 2025.

3. Unramified Cohomology and the Integral Hodge Conjecture

   We use unramified cohomology to give a counterexample to the integral Hodge conjecture for codimension two cycles. This is based on this paper "Cohomologie non ramifiée et conjecture de Hodge entière " by Jean-Louis Colliot-Thélène and Claire Voisin. This talk was given as part of a learning seminar on unramified cohomology organized by Elden Elmanto during the fall of 2024.

4. Non-Abelian Hodge Theory

   We introduce the ideas of Non-Abelian Hodge theory and we discuss how to study complex variations of Hodge structure on a space by studying its non-abelian cohomology theories. This talk was loosely based on the ideas in Carlos Simpson's amazing paper "Higgs bundles and Local Systems".

5. Six Functor Formalisms

   This talk introduces Grothendieck's six operations. This talk is based on these notes by Martin Gallauer ​and was part of a series of talks for an Étale cohomology learning seminar hosted at the University of Toronto in the summer of 2024.

6. Étale Fundamental Groups

   This talk introduces Étale fundamental groups and computes some basic examples and examples of related objects, such as the prime-to-p part of the Étale fundamental group for curves over fields of characteristic p. This talk is based on this book by James Milne, and was part of a series of talks for an Étale cohomology learning seminar hosted at the University of Toronto in the summer of 2024.

7. Why an Engineer Can Draw Better Than You

   We introduce mechanical linkages and then we draw all (bounded components of) algebraic curves with them. This talk was based on this paper by Michael Kapovich and John J. Millson, and the talk is intended for an audience of graduate students.

8. Counting Curves and Gromov-Witten Invariants

   We discuss how to count curves by Gromov-Witten invariants, and then we discuss the Gromov-Witten theory of the projective plane. This talk was based on these notes by Simon Rose.

9. Character Varieties

   We review the notion of a character variety, and then we compute the tangent space of a smooth point in terms of group cohomology. This talk was based on  this paper by Adam Sikora and was given as a part of a series of talks for a learning seminar on rigid local systems.

10. Symplectic Toric Manifolds

    We review the notion of a symplectic toric manifold, and then we use morse theory to compute the homology of these manifolds. This talk was based on these notes by Ana Cannas da Silva.