Related papers: The model theory of perfectoid fields [after Jahnk…

This survey is a final project of Twopole DRP in fall 20201. In this paper we try to understand a tiny part of the vast theory of perfecoid spaces, called perfectoid fields. We start by giving some motivation and historical background. Then…

In the paper are proved theorems, which amplify the results of my paper "On the difference equation of Poincare type (Part 3)", Max-Plank-Institut fuer Mathematik, Bonn, Preprint Series, 2004, 09, 1-34.

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity…

We relate the structure of the Bloch-Kato groups associated with a de Rham Galois representation over a perfectoid field to the Galois theory of the ring $\mathbf{B}_\mathrm{dR}^+$ of $p$-adic periods. As an application, we answer the…

In this talk results of study in various dimensions of the Boltzmann master field for a subclass of planar diagrams, so called half-planar diagrams, found in the recent work by Accardi, Volovich and one of us (I.A.) are presented.

This version corrects an inportant typographical error in Eq. 17. COMMENTS, FOR THE RECORD: A referees reoprt from Phys. Rev. Lett. read in part ``The first named author has appreciated my exceptionally long report. He has read and well…

It is hard to understand spin-one-half fields without reading Weinberg. This paper is a pedagogical footnote to his formalism with an emphasis on the boost matrix, spinors, and Majorana fields.

We give a proposal for future development of the model theory of valued fields. We also summarize some recent results on p-adic numbers.

These are the notes on two-dimensional conformal field theory, based on a lecture course for graduate math students, given by P.M. in fall 2022 at the University of Notre Dame. These notes are intended to be substantially reworked and…

This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda. The addenda include proofs, in the style of the book,…

This article will appear in the Encyclopedia of Mathematical Physics (Elsevier, 2006).

Review of the two volume set "The Quantum Theory of Fields" by S. Weinberg is presented.

This is the written version of the Bourbaki seminar given in January 2013 and published in 2014 (modulo an additional early reference added subsequently). It describes the first construction of infinite, finitely generated amenable simple…

Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a…

This is the text of my Bourbaki seminar on the proof of the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.

This is the author's PhD thesis, which focuses on the so-called Landau-Ginzburg/conformal field theory correspondence. This correspondence dates from the late 80s and early 90s in the physics literature and in particular, it predicts a…

We study the model theory of deeply ramified fields of positive characteristic. Generalizing the perfect case treated in work by Jahnke and Kartas on the model theory of perfectoid fields, we obtain Ax-Kochen/Ershov principles for certain…

We present a new multisymplectic framework for second-order classical field theories which is based on an extension of the unified Lagrangian-Hamiltonian formalism to these kinds of systems. This model provides a straightforward and simple…

This is an expository note discussing how the Erdos--Ramanujan proof of Bertrand's postulate may be adapted to show the existence of finite fields.

Proposal for contribution to the quantum field theory section in "Encyclopedia of Mathematical Physics".