Related papers: Lecture on Langlands Functoriality Conjecture
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one…
Plenary talk at the conference, LATTICE 99, held in Pisa, Itay from June 29 to July 3, 1999.
These notes provide a short, focused introduction to modelling stochastic gene expression, including a derivation of the master equation, the recovery of deterministic dynamics, birth-and-death processes, and Langevin theory. The notes were…
The present text provides the lecture notes for the course "noise sensitivity and percolation" given at the 2010 Clay Summer School in Buzios, Brazil.
This is an expository article based on my lectures on eigenfunctions of the Laplacian for the 2013 IAS/Park City Mathematics Institute (PCMI) summer school in geometric analysis. Many of the results are based on joint work with H.…
We give an introduction to the structure theory of extended affine Lie algebras, which provide a common framework for finite-dimensional semisimple, affine and toroidal Lie algebras. The notes are based on a lecture series given during the…
Lecture notes for a minicourse to given in the XVII Brazilian School of Geometry, UFAM (Amazonas), Brazil, July 2012.
This is a write-up of the lectures given by the author during the Master Class "Categorification" at {\AA}rhus University, Denmark in October 2010.
Notes of the lectures delivered in Les Houches during the Summer School on Complex Systems (July 2006).
This chapter is based on lectures on Randomized Numerical Linear Algebra from the 2016 Park City Mathematics Institute summer school on The Mathematics of Data.
Sliding friction is usually introduced in high school, but rarely through activities in laboratory. A qualitative introduction to friction is presented by proposing exploration of different kind of materials in order to suggest which…
This is a series of lecture notes explaining topos theory and its application in physics.
The aim of this short lecture series is to expose the students to the beautiful theory of lattices by, on one hand, demonstrating various basic ideas that appear in this theory and, on the other hand, formulating some of the celebrated…
This is an English translation of the expository article written by the author in Japanese for publication in {\em Sugaku}. The author will explain Milnor invariants from the viewpoint of his research.
A brief review of the motivation and the present status of Fractional Superstring is presented. Talk at ``Strings 93'', Berkeley, May 1993.
We study torsion in the homology of arithmetic groups and give evidence that it plays a role in the Langlands program. We prove, among other results, a numerical form of a Jacquet--Langlands correspondence in the torsion setting.
Lecture notes of lectures delivered at the Hodge Theory Summer School in ICTP Trieste, June 2010.
These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor…
These lecture notes are based on lectures given in 2019 Saint-Flour Probability School.
The present notes contain the material of the lectures given by the author at the summer school on ``Modular Forms and their Applications'' at the Sophus Lie Conference Center in the summer of 2004.