Math Videos
Advanced graduate courses
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.
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.
Graduate courses
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.
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.
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.
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.
Computational Algebraic Geometry - Emre Sertöz; Playlist
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 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.