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Meeting weekly 11:00 to 12:30 Chicago and Bogotá; 16:00 GMT - Fridays
This semester (Fall 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 6 September 2024. 11:00 to 12:30 Chicago and Bogotá time - 16:00 GMT
John T. BALDWIN - University of Illinois at Chicago
Zoom: https://us06web.zoom.us/j/81668158910
Categoricity of universal covers of Fuchsian groups: the non-arithmetic case
Abstract: I sketch an argument for the following theorem.
If a Fuchsian group of type 1 and genus 0 has finite index in its commensurator (i.e., is non-arithmetic), the $L_{\omega_1,\omega}$ of an appropriate formalization of its universal cover is categorical in all infinite cardinalities.
The result here is analogous to the categoricity results for universal covers in by Daw and Harris and by Eterovic, arising from Zilber's program. However, if it holds up, the finite index condition greatly reduces the depth of the geometric results needed and provides a skeleton of the argument, showing where the more powerful results are needed.
Meeting weekly 12:00 to 13:30 Chicago and Bogotá; 17:00 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 26 April 2024. 12:00 to 13:30 Chicago and Bogotá time - 17:00 GMT
Meet: http://meet.google.com/oyv-msmv-mos
Michael HARRIS - Columbia University
p-adic uniformization(s) of Shimura curves
Abstract: I will explain two theories of uniformization of the p-adic points of a Shimura curve. One is based on the theory of moduli of p-divisible groups and goes back to 50-year old work of Cherednik and Drinfeld, and was generalized by Rapoport and Zink to other moduli spaces of abelian varieties. The other is more recent and is based on Scholze's theory of perfectoid spaces. Both of them are analogous, in different ways, to the complex uniformization. I will say nothing new but will rather try to make the constructions clear enough to allow the audience to see whether or not there is room to study these uniformizations from the standpoint of categoricity.
Recording: https://drive.google.com/file/d/12kYFgoZfOTth2WihH3P4zNeIpkNIDvnU/view?usp=sharing
Friday 19 April 2024. 12:00 to 13:30 Chicago and Bogotá time - 17:00 GMT
Meet: http://meet.google.com/oyv-msmv-mos
James FREITAG - University of Illinois at Chicago
Examples of differential equations arising from the action of a discrete group
Abstract: Abstract: Mainly, I'd like to talk about examples of differential equations arising from the action of a discrete group. The main issue I'd like to expose is how o-minimality plays a role. Part of the talk will be about equations on moduli spaces, and part will be about a new look at Manin kernels from this perspective.
Video recording: https://drive.google.com/file/d/10wLshj6YkJmwZqqw-mmJ1fN0zLYn9Z95/view?usp=sharing
Friday 12 April 2024. 12:00 to 13:30 Chicago and Bogotá time - 17:00 GMT
Meet: http://meet.google.com/oyv-msmv-mos
Anatoly LIBGOBER - University of Illinois at Chicago
Problems in the study of infinite degree covers in algebraic geometry
Abstract: I will discuss natural algebro-geometric constructions which lead to infinite degree covers of algebraic varieties including universal covers and period maps as well as some classical conjectures about there structure.
I also discuss how works of Lian-Yau and C.Doran on automorphic and integrality properties of mirror maps fit into this circle of questions.
Video recording: https://drive.google.com/file/d/1BP2O6LO9_TArgbOr1Wl13RzjoFd3srJ3/view?usp=sharing
Friday 5 April 2024. 12:00 to 13:30 Chicago and Bogotá time - 18:00 Oxford - 17:00 GMT
Meet: http://meet.google.com/oyv-msmv-mos
Boris ZILBER - University of Oxford
Definability of topological invariants and some thoughts on mirror symmetry
Abstract: I am going to explain how categorical $L_{\omega_1,\omega}$-theory of cover of a complex variety X can be seen as a complete topological invariant of X.
A further analysis of the axioms reveals two quite different subtheories - one describing the Zariski/holomorphic components of the structure and another - the discrete/metric components.
This suggests an interplay of two distinct geometric categories, similar to the ones conjectured by Kontsevich's HMS.
Video recording: https://drive.google.com/file/d/1WYyoNLcj7hxJSBDSODJ1N2vuUH5TRDYw/view?usp=sharing
Friday 15 March 2024. 12:00 to 13:30 Chicago and Bogotá time
Meet: http://meet.google.com/oyv-msmv-mos
John Alexander CRUZ - UNAL
Can a mirror map be seen as a covering map? (III) + Synthesis Session
Alex Cruz will close his series started this month with further discussion on the questions that have arisen. We will all have an open discussion on possibilities, lines of research, etc. on model theoretic approaches to the understanding of mirror maps.
Friday 8 March 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
John Alexander CRUZ - UNAL
Can a mirror map be seen as a covering map? (II)
Abstract: I will try to argue that a general mirror map can be seen as a covering map from the fact that it satisfies certain differential equation. If my argument works then this should help to use Zilber's program and Freitag-Nagloo program (related to their work with Blázquez-Sanz and Casale) in order to understand mirror maps from a model theoretic point of view.
Recording: https://drive.google.com/file/d/1AMTuj381U6D6q58Zfh9Yz-DNeZC37vNJ/view?usp=sharing
Friday 1st March 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
John Alexander CRUZ - UNAL
Can a mirror map be seen as a covering map?
Abstract: I will try to argue that a general mirror map can be seen as a covering map from the fact that it satisfies certain differential equation. If my argument works then this should help to use Zilber's program and Freitag-Nagloo program (related to their work with Blázquez-Sanz and Casale) in order to understand mirror maps from a model theoretic point of view.
Recording: https://drive.google.com/file/d/13bIwCUaVs2yZ1pUkxRRB2PnpKOJDKq-N/view?usp=sharing
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.
Recording: https://drive.google.com/file/d/18yfoqLBfN0ePl8sTamXRXq4u6bU3bqx0/view?usp=sharing
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
This seminar is run by John Baldwin (UIC), Andrés Villaveces and Alex Cruz (UNAL). Contact: Andrés Villaveces (avillavecesn at unal edu co)
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