Mathematical Sandpiles and Homological Mirror Symmetry
September 9, 2019 – December 20, 2019
Organizers: Ludmil Katzarkov (Miami) and Ernesto Lupercio (Cinvestav)
In the late 80’s, motivated by the study of char we call now complex systems, physicists stumbled upon a deceptively simple cellular automaton as a toy model for self-organized criticality and stated theorems that from the mathematical perspective are remarkable conjectures. In the last few years a spurt of activity surrounding these conjectures has happened, in particular new connections to the XX! Century field of Tropical Geometry were found. These connections suggest two new avenues of research, in one direction with the advent of quantum toric geometry and quantum tropical geometry, the theory of sandpiles is insensitive to the quantization and provides a guide as to this geometric realm which suggest an amplification to the homological mirror symmetry program, in the opposite direction homological mirror symmetry may provide inspiration to deal with some of the ocnjectures regarding self/organized criticality in sandpiles.
Main Workshops
Introductory Workshop on HMS, Logic and Sandpiles (09/09/19 - 09/13/19)
Schedule and Program
Homological Mirror Symmetry (HMS) has been employed to develop some rather unexpected parallels between classical geometric constructions and category theory. The Emphasis Semester will take these ideas to a natural limit in order to obtain significant applications in physics and important connections to logic. The first portion of program will focus on generalizations of HMS for toric varieties. HMS for toric varieties has been established as a Strominger Yau Zaslov (SYZ) correspondence on families of Lagrangian tori over polytopes. Here a quantum analogue is proposed leading to a quantum version of SYZ, developed as HMS over non-standard fields. In this way, HMS extends over stacks and non-symplectic manifolds connecting with logic. The mathematics behind HMS over stacks segues into tropical geometry, a piece-wise linear or skeletonized version of algebraic geometry which has found a wide range of applications. Kalinin, Luperico and Shkolnikov and collaborators have discovered remarkable relationship between tropical geometry and self-organized criticality in complex systems, in which a self-organized complex system (SOC) is realized inside tropical geometry as the re-scaling limit under the re-normalization group of the canonical sand-pile model in classical complexity theory. They propose new applications to the study of cities, allometry in biology and neural networks in the brain.
Presentations
Dr. Gabriela Olmedo - The Hidden Diversity of Bacteria and Emergent Properties in Synthetic Communities
Dr. Carlos Simpson - Stability Conditions and Spectral Networks
Participants: Jørgen Andersen (Aarhus University), Xiuxiong Chen (Stony Brook and Shanghai Tech), Alexander Efimov (Steklov and HSE), Tobias Ekholm (Mittag- Leffler), Yu-Wei Fan (Berkeley), Kenji Fukaya (Simons Center Stony Brook), Benjamin Gammage (Miami), Lino Grama (Unicamp), Sergei Gukov (Cal Tech), Fabian Haiden Oxford), Andrew Harder (Lehigh), Gabriel Kerr (Kansas State), Y.P. Lee (Utah), Ernesto Lupercio (Cinvestav), Yong-Geun Oh (IBS Center Pohang), Gabriela Olmedo (Cinvestav), John Pardon (Princeton), Yongbin Ruan (Michigan), Helge Ruddat (Mainz), Carlos Simpson (Nice), Yan Soibelman (Kansas State), Hiro Lee Tanaka (Texas State University), Abigail Ward (Stanford), Ilia Zharkov (Kansas State)
Quantum Toric Geometry and Chimeras (10/21/2019 - 10/25/2019)
Program
Quite independently and originating in classical themes of complex non-Kahler geometry, generalizations of Calabi-Eckmann fibrations were introduced and studied as LVMB manifolds, a line of thought that under the influence of mirror symmetry evolved into the field of Quantum Toric Geometry (which is a non-commutative quantization of classical toric geometry).
Quantum toric geometry, while a beautiful self-contained field of non-commutative geometry, is missing some features required for a full-fledged unification with mirror symmetry. It turns out that there is a further generalization of Quantum Toric Geometry discovered in 1998 that uses beautiful ideas from mathematical logic: chimeric algebraic geometry.
Chimeric toric geometry generalizes quantum toric geometry and contains all the necessary cases produced by the sandpile models incorporating scale-invariant self-organized criticality.
The purpose of this workshop and conference is to explore these nascent fields and to investigate their consequences for mirror symmetry.
Mini Courses
Quantum Toric Geometry by Laurent Meersseman (University of Angers)
October 21, 22, 24, 25
Chimeric Geometry and Topology by Ernesto Lupercio (Cinvestav)
October 23, 24, 25
Participants: Andres Angel (U Norte, Barranquilla), Enrique Becerra (Cinvestav), Paul Bressler (U Andes, Bogota), Tim Gedron (UNAM, Cuernavaca), Nikon Kurnosov (UGA, USA), Santiago Lopez de Medrano (UNAM CDMX), Ernesto Lupercio (Cinvestav), Ludmil Katzarkov (U Miami), Laurent Meersseman (Angers), Ignacio Otero (Cinvestav), Carlos Ruiz (CIMAT, Guanajuato)
Tropical Geometry and Sandpiles (11/17/2019 - 11/22/2019)
Program
Classical algebraic geometry is a central field of classical and modern mathematics. In the twenty first century its relation to combinatorial and computational themes has grown rapidly in importance, primarily in its connection to tropical geometry. Tropical geometry (which can also be thought as an outgrowth of toric geometry) has had a very large impact in combinatorics and also through its connection to non-archimidean geometry in the exciting developments in mirror symmetry. A new striking and unexpected bridge between the two fields occurred in recent work connecting the study of complex systems and self-organized criticality with tropical geometry: this connection is realized in the study of abelian sandpile models. Enigmatically some sandpile models seem to suggest a very wide generalization of toric and tropical geometry as they are not included in the classical cases yet seem to be in the same class as them.
Participants: Andres Angel (U Norte, Barranquilla), Pablo, Cruz (Cinvestav, Mexico), Tim Gedron (UNAM, Cuernavaca), Moritz Lang (IST, Vienna), Lucia Lopez de Medrano (UNAM, Curnavaca), Ernesto Lupercio (Cinvestav), Nikita Kalinin (HSE, St. Petersburg), Ludmil Katzarkov (U Miami), Gabriel Kerr (Kansas State), Carlos Ruiz-Guido, Ignacio Otero (Cinvestav), Bernardo Uribe (U Norte, Barranquilla), Alberto Verjovsky (UNAM Cuernavaca)
Mini Courses
Sandpiles by Nikita Kalinin (via video-conference from St. Peterburg)
November 17, 18, 19
