Related papers: Notes on motivic infinite loop space theory
We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.
We introduce the notion of a relative limit simplicial partial motivic site and a corresponding notion of motivic measurability which specializes to the notion of finite schemic motivic measures of H. Schoutens and the notion of infinite…
In these lectures we review the motivation, principles of and (circumstantial) evidence for the program of unification of the fundamental forces. In an appendix, we review the group theory pertinent to the program.
These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to…
Survey article on loop groups and their representations, following a course of three lectures held at the summer school "algebraic groups" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their…
These are lectures presented at the Les Houches Summer School ``Topology and Geometry in Physics'', July 1998. They provide a simple introduction to non perturbative methods of field theory in 1+1 dimensions, and their application to the…
Brief lecture notes for a course about random matrices given at the University of Cambridge.
We give an interpretation of J-spaces in terms of symmetric spectra in symmetric sequences. As application we show how one can define graded endomorphism objects in a general situation. As example we discuss the motivic bigraded…
This short note provides an overview of some theorems and conjectures obtained by the author and his collaborators. It is an extended abstract for the Oberwolfach workshop "New Trends in Teichm\"uller Theory and Mapping Class Groups", 2…
For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers, and finite fields. We use this to extend Morel's identification of the endomorphism ring…
The aim of the INFINITY workshop is to provide a forum for researchers interested in the development of formal methods and algorithmic techniques for the analysis of systems with infinitely many states, and their application in automated…
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…
These are notes from a three-lecture mini-course on free probability given at MSRI in the Fall of 2010 and repeated a year later at Harvard. The lectures were aimed at mathematicians and mathematical physicists working in combinatorics,…
We provide a gentle introduction to arc spaces, motivic integration and stringy invariants. We explain the basic concepts and first results, including the p-adic number theoretic pre-history, and we provide concrete examples. The text is a…
We survey certain accessible aspects of Grothendieck's theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have recently found applications in quantum field theory. An appendix (by Matilde…
Perturbative superstring theory is revisited, with the goal of giving a simpler and more direct demonstration that multi-loop amplitudes are gauge-invariant (apart from known anomalies), satisfy space-time supersymmetry when expected, and…
These notes are based on lectures at the PSSCMP/PiTP summer school that was held at Princeton University and the Institute for Advanced Study in July, 2015. They are devoted largely to topological phases of matter that can be understood in…
These are notes from a lecture course on symmetric spaces by the second author given at the University of Pittsburgh in the fall of 2010.
The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results…
We survey the recent developments in the area that grew out of attempts to extend an analog of Hilbert's Tenth Problem to the field of rational numbers and rings of integers of number fields. The paper is based on a plenary talk the author…