Rational surfaces over nonclosed fields - Brendan Hassett
Clay Mathematics Institute Summer School 2006 on "Arithmetic geometry"
Lecture 1/5, Lecture 2/5, Lecture 3/5, Lecture 4/5, Lecture 5/5.
Toric schemes, semistable degenerations, and tropicalization - Eric Katz
We give an introduction to tropical geometry focusing on the operation of tropicalization which assigns a polyhedral complex to a subvariety of an algebraic torus. Under a smoothness hypothesis, the algebraic variety will have a stratification (in the constant coefficient case) or a semistable degeneration (in the valued field case) modeled on a toric variety or a toric scheme, respectively. Time permitting, we will also discuss how to leverage this description into a computation of cohomology.
Lecture 1/3, Lecture 2/3, Lecture 3/3.
Shimura Varieties - Sophie Morel
Depending on your point of view, Shimura varieties are a special kind of locally symmetric spaces, a generalization of moduli spaces of abelian schemes with extra structures, or the imperfect characteristic 0 version of moduli spaces of shtuka. They play an important role in the Langlands program because they have many symmetries (the Hecke correspondences) allowing us to link their cohomology to the theory of automorphic represnetations, and on the other hand they are explicit enough for this cohomology to be computable. The goal of these lectures is to give an introduction to Shimura varieties, to present some examples, and to explain the conjectures on their cohomology (at least in the simplest case).
Lecture 1/3, Lecture 2/3, Lecture 3/3.
p-adic approaches to rational points on curves - Bjorn Poonen
In these four lectures, I will describe Chabauty's p-adic method for determining the rational points on a curve whose Jacobian has rank less than the genus, hint at Kim's nonabelian generalization, and finally discuss the recent paper of Lawrence and Venkatesh that uses p-adic period maps to give a new proof of Faltings' theorem.
Lecture 1/4, Lecture 2/4, Lecture 3/4, Lecture 4/4.
Course on Mazur's theorem - Andrew Snowden
The purpose of this course is to prove Mazur's theorem on torsion in elliptic curves over the rational numbers. Much of the course is devoted to developing background material on elliptic curves and their moduli.
Check https://websites.umich.edu/~asnowden/teaching/2013/679/.
Lecture 1/23, Lecture 2/23, Lecture 3/23, Lecture 4/23, Lecture 5/23, Lecture 6/23, Lecture 7/23, Lecture 8/23, Lecture 9/23, Lecture 10/23, Lecture 11/23, Lecture 12/23, Lecture 13/23, Lecture 14/23, Lecture 15/23, Lecture 16/23, Lecture 17/23, Lecture 18/23, Lecture 19/23, Lecture 20/23, Lecture 21/23, Lecture 22/23, Lecture 23/23.
Rational Points on Modular Curves - Jan Vonk
In this course, we will study the arithmetic properties of special points on modular curves. The main focus will be on rational points on modular Jacobians arising from the theory of complex multiplication. After a brief discussion of the basics of the theory, we give an overview of some of the most important results of the latter half of the 20th century concerning the question of when such points are non-torsion. Some of the main ideas of two results in particular will be our primary focus: Mazur’s work on torsion on elliptic curves, and the work of Gross-Zagier on heights of Heegner points.
Introduction to Modular Forms - Keith Conrad; Playlist
Topics include Eisenstein series and q-expansions, applications to sums of squares and zeta-values, Hecke operators, eigenforms, and the L-function of a modular form.
Some topics in Diophantine geometry - Elisa Lorenzo García
In this course we present a short introduction to Diophantine Geometry. The main object of study are heights: we study their properties, their constructions and their applications. We start by introducing absolute values and valuations to define heights on projective spaces and later on on varieties, more precisely on abelian varieties via the Weil heights machinery. We revisite Mordell-Weil theorem on the group of rational points on abelian varieties and Falting's theorem on the finitness of rational points on curves of genus greater or equal than 2. We finish the course by discussing some open problems on Diophantine Geometry, as the abc conjecture.
Lecture 1/6, Lecture 2/6, Lecture 3/6, Lecture 4/6, Lecture 5/6, Lecture 6/6.
Toric Varieties - Jürgen Hausen; Playlist
This playlist hosts the video-clips of an introductory course on toric varieties. The course is open to everybody; the prerequisites are basic knowledge in algebraic geometry. The course notes and accompanying exercises are available at https://www.math.uni-tuebingen.de/user/hausen/TV-Video-Course/tv-video-course.pdf.
Étale cohomology and the Weil conjectures - Daniel Litt; Playlist
This course will focus on étale cohomology and the Weil conjectures.
Check https://www.daniellitt.com/tale-cohomology.
Introduction to Elliptic Curves - Álvaro Lozano-Robledo; Playlist
This is an overview of the theory of elliptic curves, discussing the Mordell-Weil theorem, how to compute the torsion subgroup of an elliptic curve, the 2-descent algorithm, and what is currently known about rank and torsion subgroups of elliptic curves.
Introduction to Nonlinear Algebra - Mateusz Michalek & Bernd Sturmfels; Playlist
This course offers an introduction to the concepts and techniques of Nonlinear Algebra. This subject covers tools for Mathematics in the Sciences that go beyond the familiar repertoire of Linear Algebra.
Computational Algebraic Geometry - Emre Sertöz; Playlist
Twelve Lectures on Tropical Geometry - Bernd Sturmfels; Playlist
Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts.