Mini Courses

Course #1: Introduction to p-adic Analysis

Prof. Trond Digernes
Norwegian University of Science and Technology, Norway
Prof. W. A. Zúñiga-Galindo
The Center for Research and Advanced Studies, Mexico

Duration of the mini-course: 6 hours

ABSTRACT: This mini course aims to provide a fast introduction to p-adic analysis assuming basic knowledge in algebra and analysis. The topics included in this mini course are

  • Construction of the field of p-adic numbers,
  • Topology of the p-adic numbers,
  • Haar measure and integration,
  • Test functions and Fourier transform,
  • Distributions and Fourier transform.
Course #2: Introduction to the local zeta functions

Prof. Edwin León-Cardenal
Mathematics Research Center, CIMAT-Zacatecas, Mexico

Duration of the mini course: 4 hours

ABSTRACT: The mini course aims to introduce the theory of local zeta functions in the p-adic setting. The topics to be presented include

  1. Definition of the Igusa local zeta function
  2. The stationary phase formula
  3. p-Adic manifolds and resolution of singularities.
Course #3: p-Adic models in quantum physics

Prof. Ingmar Saberi
Heidelberg University, Germany

Duration of the mini-course: 5 hours

ABSTRACT: The p-adic numbers have made repeated appearances in mathematical physics over the last half century, being used in various different ways to extend the diverse set of models that make up the theoretical physicist's arsenal. Many (though by no means at all) of these applications hinge on the surprising relevance of ideas from algebraic geometry to physics in two dimensions; others make use of structural commonalities between the p-adics and other local fields. In this minicourse, I'll try to give a selective overview of some of these applications, including (among other things) a bit of p-adic string theory and some connections to the renormalization group. With luck, I hope that a couple of conceptual points will emerge: firstly, on the more formal side, that both the similarities and differences between archimedean and non-archimedean physical models are interesting, and that formulating a model in sufficient generality to include both cases can lead to structural insights; secondly, and perhaps more practically, that the p-adics offer a happy medium between discrete and continuous models, allowing one to recover many of the advantages of lattice discretization while preserving a notion of scale invariance.

Course #4: The p-adic theory of automata

Prof. Vladimir Anashin
Lomonosov Moscow State University
Federal Research Center 'Information and Control' Russian Academy of Sciences

Duration of the mini-course: 8 hours

ABSTRACT: During the mini-course, by the automaton we mean a transducer, i.e., a sequential machine which maps symbols of a finite input alphabet to symbols of a finite output alphabet so that any output symbol depends on corresponding input symbol and of current state of the machine, whereas any input symbol changes current state of the machine. An automaton whose input and output alphabets consist of p symbols can naturally be associated to a mapping from p-adic integers to p-adic integers; the mapping is 1-Lipschitz w.r.t. p-adic metric. Moreover, any 1-Lipschitz mapping from p-adic integers to p-adic integers is a mapping associated with some automaton. Therefore one can study behaviour of automata by studying dynamics of corresponding 1-Lipschitz mappings, the automata functions. The mini-course is devoted to dynamical (especially, ergodic) and other properties of automata functions as functions from p-adic integers to p-adic integers and to various applications of corresponding theory in computer science, cryptography, pseudorandom numbers, physics, etc.

Course #5: p-Adic electrostatics

Prof. Christopher Sinclair
Clark Honors College, University of Oregon

Duration of the mini-course: 6 hours

ABSTRACT: Mathematical statistical physics starts with the axiom that any two states of a physical system with the same energy are equally probable. From this axiom we may derive basic physical quantities (e.g. entropy, free energy, pressure) for physical models. These quantities are encoded in something called the partition function. This is a kind of generating function, and after motivating its definition we will explore how to derive physical information from it.

In classic 1 (and 2)-dimensional electrostatics, the partition function is given by a special kind of integral related to the Selberg integral. For certain temperatures, this integral has a determinantal structure that allows us to discover explicit, nuanced, statistical information about the location of charged particles. In this situation, this partition function also appears as a critical quantity in the study of random matrices.

We will generalize the 1D electrostatic model to the p-adics. In this setting, the partition function is an example of a local zeta function, and gives a p-adic analog of the Selberg integral. We will give a recursive formula for the partition function, and show how we can express statistical quantities of interest using this recursion. By general results about local zeta functions, the partition function is defined in terms of a rational function, and we will determine the zeros and poles of this rational function and see what these tell us about the physical system.

Course #6: Strings Amplitudes, Local Zeta Functions, and log-Coulomb Gases

Prof. Hugo Hugo García-Compeán
Prof. W. A. Zúñiga-Galindo

The Center for Research and Advanced Studies (CINVESTAV), Mexico

Duration of the mini-course: 6 hours

ABSTRACT: The course aims to introduce the connection between local zeta functions, Koba-Nielsen open string amplitudes, and log-Coulomb gases over local fields of characteristic zero. The tentative topics of the mini course include:

  1. Basic ideas on string theory,
  2. Regularization of Koba-Nielsen string amplitudes using local zeta functions,
  3. Computation of p-adic Koba-Nielsen amplitudes for 4 and 5 points,
  4. Local zeta functions and log-Coulomb gases,
  5. Phase transitions at finite temperature.