Introduction to Stacks and Moduli - Jarod Alper; Playlist
The goal of this course is to establish the following theorem: The moduli space of stable curves of genus g ≥ 2 is a smooth, proper and irreducible Deligne-Mumford stack of dimension 3g-3 which admits a projective coarse moduli space. Along the way we will develop the foundations of algebraic spaces and stacks, and in particular we will precisely define each term in the above theorem.
Check https://sites.math.washington.edu/~jarod/courses/math582C-winter21.html.
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.
Berkovich Spaces, Tropical Geometry and Number Theory - Farbod Shokrieh
The theory of “heights” is essential in studying finiteness questions in diophantine geometry. For example, one of the most spectacular results in number theory is Faltings’s theorem (Mordell’s conjecture): a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. A crucial tool in Faltings’s proof is a notion of height of an abelian variety. Vojta’s proof of Mordell’s conjecture, as well as the more recent uniformity results, use the theory of N´eron–Tate (canonical) heights. In these lectures, we study the interplay between non-archimedean/tropical geometry, arithmetic/Arakelov geometry, and combinatorics/convex geometry arising from the theory of heights on abelian varieties. In the case of Jacobians, these relate to combinatorics, convex geometry, and potential theory (electrical network) of metric graphs. We will make a serious attempt to keep the lectures “self-contained”. We will assume the audience is (mostly) comfortable with combinatorics and tropical geometry, but we will not assume prior experience with Arakelov geometry or with non-archimedean geometry.
Lecture 1/4, Lecture 2/4, Lecture 3/4, Lecture 4/4.
Talk on log curves (incl. introduction to log geometry) - Fumiharu Kato; Link
This is my lecture at the RIMS online seminar "Homotopical Arithmetic Geometry Seminar" on Oct. 7, 2020. The lecture includes a (90 min. long) general introduction to log geometry by Fontaine, Illusie, and Kazuya Kato, as well as the main topic on moduli theory of log stable curves, my doctor thesis work in 1997.
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.
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.
Bass–Serre theory: groups acting on trees - Rylee Lyman; Playlist
Graphs can be cut up into forests along vertices. Surfaces can be cut up into disks along simple closed curves. Haken manifolds can be cut up into 3-balls along 2-sided, embedded incompressible surfaces. Bass–Serre theory provides a common framework for reasoning about cutting up a group along subgroups. As such it is perhaps the main inductive tool in geometric group theory. In this mini-course, we will meet the main features of the theory: graphs of groups, their fundamental groups, their universal covering trees, and their reformulation (due to Scott and Wall) as graphs of spaces. We will also study covering spaces and morphisms of graphs of groups, less well-known but no less useful. At the course we will meet a handful of applications of Bass–Serre theory. Possible examples include JSJ decompositions of hyperbolic groups, the "Nielsen realization problem" for finite groups of outer automorphisms of free groups, and the Bestvina–Feighn combination theorem.
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.
Algebraic geometry I: Varieties - Richard E. Borcherds; Playlist
This playlist is the first part of an online graduate course on algebraic geometry (Berkeley Math 256A Fall 2020). It loosely follows chapter I of Hartshorne's book "Algebraic geometry".
Algebraic geometry II: Schemes - Richard E. Borcherds; Playlist
This playlist is an online graduate algebraic geometry course (the second half of Berkeley Math 256A, 2020 Fall) giving an introduction to schemes, loosely based on chapter II of Hartshorne's book "Algebraic geometry".
Algebraic geometry: extra topics - Richard E. Borcherds; Playlist
This playlist contains several talks on topics in algebraic geometry, including the Riemann-Roch theorem for curves and the Weil conjectures.. They were originally given in discussion sessions of an algebraic geometry course as a supplement to the main lectures.
Commutative algebra - Richard E. Borcherds; Playlist
This playlist is an online course on commutative algebra (Berkeley MATH 250B, Spring 2021), following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud.
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.
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.