Fall Emphasis Semester 2019

Main Workshops

  • Introductory Workshop on HMS, Logic and Sandpiles (09/09/19-09/13/19)

    September 9-14, 2019

    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.

    Anticipated Participants: Xiuxiong Chen (Stony Brook and Shanghai Tech), Juan Carlos Castro Contreras (Cinvestav), Pablo Cesar Cruz Martinez (Cinvestav), Alexander Efimov (Steklov and HSE), Tobias Ekholm (Mittag- Leffler), Yu-Wei Fan (Berkeley), Kenji Fukaya (Simons Center Stony Brook), Xavier Gomez-Mont (Guanajuato), Lino Grama (Unicamp), Sergei Gukov (Cal Tech), Fabian Haiden Oxford), Andrew Harder (Lehigh), Gabriel Kerr (Kansas State), Y.P. Lee (Utah), Bong Lian (Brandeis), Yijia Liu (Kansas State), Ernesto Lupercio (Cinvestav), Jacob Mostovoy (Cinvestav), Yong-Geun Oh (IBS Center Pohang), Gabriela Olmedo (Cinvestav), Ignacio  Otero (Cinvestav), John Pardon (Princeton), Pranav Pandit (ICTS), Aleksander Petkov (Institute of Mathematics Bordeaux), Helge Ruddat (Mainz), Yongbin Ruan (Michigan), Renato Salmeron (Institute of Technology Monterrey), Jose Seade (UNAM Matheamtics Institute), Carlos Simpson (Nice), Yan Soibelman (Kansas State), Hiro Lee Tanaka (Harvard), Jennifer Viafara (Cinvestav), Abigail Ward (Stanford), Ilia Zharkov (Kansas State), Benjamin Gammage (Miami).


    Program

    Monday September 9th
    9:00 AM Carlos Simpson (Nice)
    10:30 AM Sergei Gukov (Cal Tech)
    12:00 PM John Padron (Princeton)
    2:30 PM Yakov Soibelman (Kansas State)
    4:00 PM Yongbi Ruan (Michigan)
    Tuesday September 10th
    9:00 AM Carlos Simpson (Nice)
    10:30 AM Hiro Lee Tanaka (Harvard)
    12:00 PM Xiuxiong Chen (Stony Brook and Shanghai Tech)
    2:30 PM Fabian Haiden (Oxford)
    4:00 PM Abigail Ward (Stanford)
    Wednesday September 11th
    9:00 AM Kenji Fukaya (Simons Center Stony Brook)
    10:30 AM Yong-Geun Oh (IBS Center Pohang)
    12:00 PM Hiro Lee Tanaka (Harvard)
    2:30 PM Ernesto Lupercio (Cinvestav)
    4:00 PM Yuan-Pin Lee (Utah)
    Thursday September 12th
    9:00 AM Tobias Ekholm (Mittag-Leffler)
    10:30 AM Ernesto Lupercio (Civerstav)
    12:00 PM Andrew Harder (Lehigh)
    2:30 PM Benjamin Gammage (Miami)
    4:00 PM Alexander Efimov (Steklov and HSE)
    Friday September 13, 2019
    9:00 AM Gabriela Olmedo (Cinvestav)
    10:30 AM Helge Ruddat (Mainz)
    12:00 PM Gabriel Kerr (Kansas State)
    2:30 PM Lino Grama ( Unicamp))
    4:00 PM Yu-Wei Fan (Berkeley)

  • Quantum Toric Geometry and Chimeras (10/21/2019-10/25/2019)

    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.

     

    Invited Participants: Andrea Sportiello (CNRS), Federico Ardilla (San Francisco State), Aldo Guzman (Watson IBM), Alfredo Najera (UNAM Oaxaca), Lionel Levine (Cornell), Leticia Lopez de Medrano (UNAM Cuernavaca), Fuensanta Aroca (UNAM Cuernavaca), Ignacio Otero (Cinvestav), Enrique Becerra (Cinvestav), Omar Antolin (UNAM CDMX), Alberto Verjovsky (UNAM Cuernavaca), Laurent Meersseman (Angers), Santiago Lopez de Medrano (UNAM CDMX), Jacob Mostovoy (Cinvestav), Dmitry Kaledin (Steklov), Christian Garay (CIMAT), Ernesto Lupercio (Cinvestav)

     


    Program

    Introductory Courses:

     

    1. Quantum Toric Geometry by Laurent Meersseman (University of Angers)

     

    October 21, 22, 24, 25

    1. Chimeric Geometry and Topology by Ernesto Lupercio (Cinvestav)
    October 23, 24, 25

  • Tropical Geometry and Sandpiles (11/17/2019- 11/22/2019)

    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.

     

    Invited Participants: Andrea Sportiello (CNRS), Federico Ardilla (San Francisco State), Aldo Guzman (Watson IBM), Alfredo Najera (UNAM Oaxaca), Lionel Levine (Cornell), Leticia Lopez de Medrano (UNAM Cuernavaca), Fuensanta Aroca (UNAM Cuernavaca), Ignacio Otero (Cinvestav), Enrique Becerra (Cinvestav), Omar Antolin (UNAM CDMX), Alberto Verjovsky (UNAM Cuernavaca), Laurent Meersseman (Angers), Santiago Lopez de Medrano (UNAM CDMX), Jacob Mostovoy (Cinvestav), Dmitry Kaledin (Steklov), Christian Garay (CIMAT), Ernesto Lupercio (Cinvestav)


    Program

    Introductory Courses:

    1. Sandpiles by Nikita Kalinin (via video-conference from St. Peterburg)

    November 17, 18, 19

    1. Introductory Tropical Geometry by Grigory Mikhalkin (University of Geneva)

    November 20, 21, 22

     

    Mon 18 Nov Tue Wed Thu Fri
    9am - 10am Lupercio *10 Kerr Kerr Otero Verjovsky
    10:30am - 11:30am Lang *11:30 Kalinin Lupercio Kalinin Lupercio
    2:30pm - 3:30pm Katzarkov Lopez de Medrano Free Afternoon Uribe Cruz
    4pm - 5pm Gendron Ruiz-Guido Free afternoon Angel Open problems round table