Conexión de GALoiS:  linking Model Theory and Algebraic Geometry

Meeting weekly 13:00 Bogotá / 12:00 Chicago; 18:00 GMT - Fridays

Google Meet: meet.google.com/oyv-msmv-mos

This semester (Spring 2024) the seminar continues. Jointly organized between UIC (University of Illinois at Chicago) and UNAL (Universidad Nacional de Colombia), our aim is to explore the connections between the two topics in the title. Last semester, we expounded Zilber's general program to capture analytic information  about a canonical structure by an algebraic  $L_{\omega_1,\omega}$-description that is  categorical in power. The seminar will aim at fostering understanding of the viewpoints of model theory and function theory-number theory- algebraic topology- etc.

For further info contact Baldwin (jbaldwin@uic.edu), Cruz (jacruzmo@unal.edu.co) or Villaveces (avillavecesn@unal.edu.co).

Friday 23 February 2024. 12:00 to 13:30 Chicago time

OJO: 13:00 a 14:30 Hora de Bogotá

Meet: http://meet.google.com/oyv-msmv-mos

Joel (Ronnie) Nagloo - UIC:

Model theory and the Schwarzian equations

Abstract: The Schwarzian equations appear as the uniformizing differential equations for Fuchsian covering maps (or uniformizers). In this talk I will survey the work with Blázquez-Sanz, Casale and Freitag, around using model theoretic techniques to study these equations. I will also try and discuss some of the questions asked about them in the previous talks.

Friday 16 February 2024. 12:00 to 13:30 Chicago time

OJO: 13:00 a 14:30 Hora de Bogotá

Meet: http://meet.google.com/oyv-msmv-mos

Joel (Ronnie) Nagloo - UIC:

Model theory and the Schwarzian equations

Abstract: The Schwarzian equations appear as the uniformizing differential equations for Fuchsian covering maps (or uniformizers). In this talk I will survey the work with Blázquez-Sanz, Casale and Freitag, around using model theoretic techniques to study these equations. I will also try and discuss some of the questions asked about them in the previous talks.

Fridays 6, 13, 27 October. John Alexander CRUZ MORALES (UNAL). Mirror maps, differential equations and model theory (I, II, III)

Abstract: 

In this series of talks I will introduce some Schwarzian differential equations that can be obtained from mirror maps for Calabi-Yau manifolds, in particular, hypersurfaces in projective spaces. I will propose a set of expectations about the model theoretic properties of those equations that should extend properties studied in the works of Aslanyan, Freitag, Nagloo, Scanlon and others. The goal is to formulate those expectations as theorems and discuss the strategy of their proof.  At the end, I will mention some possible applications of the model theoretic properties in the study of mirror symmetry. 

Recording: https://drive.google.com/file/d/1gsUfodBZxWhpsIfpkXTpNq-rK15SbLHS/view

Friday 1 September. John Alexander CRUZ MORALES (UNAL). On the modularity of mirror maps: connections with model theory

Abstract: 

Friday 21 April. John BALDWIN (UIC). Towards counting types

Abstract: I will discuss the connection between the domain and variety side-- roughly where Ronnie ended and then describe the types that need to be counted.

Friday 14 April. Joel (Ronnie) NAGLOO (UIC). Towards counting types

Abstract: The connection between Galois representation and counting types will be explored.

Video: https://drive.google.com/file/d/12S5kfxG3RmkWGgbZtl0XtnRDD4lbWVvX/view?usp=sharing

Friday 31 March. Andrés VILLAVECES NIÑO (UNAL). Around the proof of categoricity of modular functions

Abstract: This continues the lecture from a week ago. Within the large-scale framework presented then, a core proof (with variants of different sorts in the 8 situations mentioned) emerged: a connection between categoricity of modular (or Shimura, or...) curves in the setup put forth by Zilber et al, on the one hand, and behavior of the Galois representation (Mumford-Tate Conjecture) on the other hand. I will discuss various aspects of this proof, trying to give a general picture of (part of (one of the directions of)) the proof, and some technical issues that have emerged in our discussions with John Baldwin and Ronnie Nagloo.

Friday 24 March. Andrés VILLAVECES NIÑO (UNAL). Two decades of model theory of universal covers

Abstract: The model theory of universal covers has steadily been surfacing, mainly due to the work of Boris Zilber and other mathematicians who have worked with him (Bays, Daw, Eterović, Harris) and also Hart and Pillay, over the past two decades. Examples notably include model-theoretic approaches to complex exponentiation, covers of multiplicative groups, elliptic curves, Shimura curves and varieties, smooth varieties anchored on o-minimal expansions of the reals. An interesting picture has slowly emerged from these works, and we are at the juncture of being able to posit a unified approach, to signal differences between specific cases, and to point to various possible generalizations. In my two lectures, I will give an overall description of the general picture, and I will emphasize some nodal junctures (and dis-junctures) between several specific cases. In particular, I will describe the new abstract elementary classes arising from some of the constructions. This is joint work with John Baldwin.

Video: https://drive.google.com/file/d/13Rno00ioE6rL1sW_4HPB2MRRRlaoqDbZ/view?usp=sharing

Friday 17 March. John Alexander CRUZ MORALES (UNAL). Does model theory have something to say about mirror symmetry? (2)

Abstract: In this talk we will give a panoramic view on what mirror symmetry is. The purpose of the presentation is to settle down some basic ideas that could help to understand the question in the title which we will tackle directly in the second part of the talk (March 17).

Slides: https://drive.google.com/file/d/1ZC1z9Vjx1UVsMIIOvJTue98zTQCJ3UQT/view?usp=sharing

Video: https://drive.google.com/file/d/1iuTNmAvxLxmhmsnKiZAEzaHIAYEGxDdM/view?usp=sharing

Friday 10 March. John Alexander CRUZ MORALES (UNAL). Does model theory have something to say about mirror symmetry?

Abstract: In this talk we will give a panoramic view on what mirror symmetry is. The purpose of the presentation is to settle down some basic ideas that could help to understand the question in the title which we will tackle directly in the second part of the talk (March 17).

Slides: https://drive.google.com/file/d/1Q6NVycgWA20kIz2IKrD67MoRHPfkY4K3/view?usp=sharing