LOG3 - MODEL THEORY AND GEOMETRY : RECENT
INTERACTIONS
Organizers
John Alexander Cruz, Max Planck Institut fuer Mathematik, Germany
Timothy Gendron, Instituto de Matemáticas UNAM, México
Andrés Villaveces, Universidad Nacional de Colombia, Colombia
LOG3-1 Leonardo Cano, Universidad Sergio Arboleda, Colombia
Basic aspects of the geometric rigidity of the j function on complex elliptic curves
In the talk we remind that the j-function is the invariant that distinguishes complex structures of surfaces
of genus 1. We point out geometric aspects related with Hurwitz spaces of holomorphic functions between
Riemann surfaces, that could explain the rigidity of uniqueness of j, using the last two words in an intuitive
way.
LOG3-2 John Alexander Cruz, Max Planck Institut fuer Mathematik, Germany
Towards a model theoretic approach to F1 - geometry
The geometric motivation of the object F1 , ”the field with one element” came from the work of Jac Tits in 1956,
where he explained how one can define the Chevalley group of characteristic one to obtain some interesting
geometries such that the symmetric groups happens to be the Weyl group of the corresponding Lie groups. In
the last two decades there have been a lot of development, with motivation coming from Arakelov theory and
some ideas relating the notion to the Riemann zeta function.
We interpret fields of characteristic 1 and algebras over those fields as multiplicative monoids with a shadow
addition. The initial object in the category of F1 -algebras is F1 = {0, 1} the field with one element. We define
the cyclotomic extensions F1n of F1 as the F1 -algebra gievn by the union between {0} and the nth-roots of
unity. We also give a definition for F1alg and formulate a conjecture that such an object is ω − stable. If time
permits I will discuss some relations to the Arakelov geometry modulo n introduced by Kapranov-Smirnov
This is a joint project with Boris Zilber and Lubna Shaheen.
LOG3-3 Timothy Gendron, Instituto de Matemáticas UNAM, México
Ultraschemes and the Universal Modular Invariant
Ultraschemes are geometric objects locally modeled on sheaves of ultrapowers of a structure over a Stone space.
A modular ultrascheme is one whose points parametrize (models of) elliptic curves with additional structure
e.g. a finite subgroup, a foliation, etc. We introduce the universal modular invariant as a function on a modular
ultrascheme and show how the classical and quantum modular invariant (the latter a multi-valued modular
function defined on the moduli space of quantum tori) may be obtained from it as subquotients.
LOG3-4 Jonathan Kirby, University of East Anglia - Norwich, United Kingdom
Exponentially closed fields
The field C of complex numbers is well-known to be algebraically closed; this is the so-called fundamental
theorem of algebra. As a model-theoretic structure, it follows that it is strongly minimal: every subset of C
definable in the ring language is finite or co-finite. If we consider the complex field in the ring language expanded
by the exponential function, much less is known. I will explain the theory of exponentially closed fields,
analogous to algebraically closed fields, the status of the conjecture that C is exponentially closed, and how
that would imply that it is quasiminimal: every subset of C definable in the exponential ring language should be
countable or co-countable. Diophantine geometry appears in two ways. Kummer theory is involved in classifying
and counting the types of exponential field extension which occur, and the Conjecture on Intersections with
Tori and related theorems are involved in several places. Some of this is joint work with Boris Zilber, and some
with Martin Bays.
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LOG3-5 Jorge Plazas, Pontificia Universidad Javeriana, Colombia
Towards a model theoretic framework for Real Multiplication
About 15 years ago Y. Manin proposed addressing the explicit class field theory problem for real quadratic fields
using real multiplication noncommutative tori in a manner analogous to the case of quadratic imaginary fields
where explicit class field theory can be given in terms of elliptic curves with complex multiplication. Despite
various parallels and favorable results an arithmetic theory of noncommutative tori is yet to be developed. In
this talk we review real multiplication of noncmmutative tori in the light of recent results in model theory
paying special attention to results of an arithmetic-geometric nature as those of Harris and Daw-Harris and
their possible connection to results closer to noncommutative geometry as those of Gendron and collaborators.
We close with an outline of a proposed program where the discussed techniques from model theory are used to
approach real multiplication.
LOG3-6 Andrés Villaveces, Universidad Nacional de Colombia, Colombia
Modular invariants, towards real multiplication
Finding a ”non-commutative limit” of the j-invariant (to real numbers, in a way that captures reasonably well
the connection with extensions of number fields) has prompted several approaches (Manin-Marcolli, CastañoGendron, etc.). I will describe some connections between these approaches and model theory. In particular, I
will focus on the role of recent interactions with differentiably closed fields in the model-theoretic analysis of
modular invariants and its potential for the extension to the real limit.
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