Irrational Fans in Physics and Mathematics

To view the abstracts of this program, click here. 

Schedule

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  • Monday October 19th, 2020

    Nonrational Toric Geometry I: Symplectic Toric Quasifolds (Minicourse)

    Dr. Elisa Prato
    Universita Degli Studi Firenze

    8:00am - 9:00am
    Via Zoom
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    Abstract: Toric quasifolds are a class of singular spaces that were introduced first in the symplectic setting (1999) and then, jointly with Battaglia, in the complex setting (2001). They generalize toric varieties to simple convex polytopes that are not rational. 

    We give a brief historical survey of the process that initially led to toric quasifolds in the symplectic setting, with an emphasis on the core notions of quasilattice, quasirationality and quasitorus. We discuss a number of examples: the quasisphere, Penrose tilings and quasicrystals, regular convex polyhedra, and irrational Hirzebruch surfaces. We also touch on some recent results, obtained jointly with Battaglia, Zaffran and Iglesias-Zemmour.

    Minicourse by Dr. Elisa Prato and Dr. Fiametta Battaglia.

    Nonrational Toric Geometry II: Quasifolds, Foliations, Combinatorics and One-parameter Families (Minicourse)

    Dr. Fiametta Battaglia
    Universita Degli Studi Firenze

    9:00am - 10:00am
    Via Zoom
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    Abstract: In this second talk we start by giving an in-depth description of complex toric quasifolds. By analysing their structure we will show that they share key features with rational toric varieties, including the relation between Betti numbers and the polytope combinatorics. Then we continue by extending the contructions introduced in the first talk by Elisa Prato to the case of nonsimple convex polytopes, in the symplectic and complex settings, thus giving a complete generalization of toric varieties to the nonrational case.

    Minicourse by Dr. Elisa Prato and Dr. Fiametta Battaglia.

    Intersection of Quadrics, Moment-Angle Manifolds, Complex and Quaternionic Manifolds and Convex Polytopes

    Dr. Alberto Verjovsky
    National Autonomous University of Mexico

    10:00am - 11:00am
    Via Zoom
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    Abstract: We present an overview of the theory of LVM manifolds and its relation with toric manifolds and present some first steps and comments toward this theory in the quaternionic setting.

    Irrational Toric Varieties and the Secondary Polytope

    Dr. Frank Sottile
    Texas A&M

    11:00am - 12:00pm
    Via Zoom
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    Abstract: Classical toric varieties come in two flavours: Normal toric varieties are given by rational fans in R^n. A (not necessarily normal) affine toric variety is given by finite subset A of Z^n. When A is homogeneous, it is projective. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points A may be arbitrary points in R^n.  For example, in 1963 Birch showed that such an irrational toric variety is homeomorphic to the convex hull of the set A. 

    Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in R^n. These are R^n_>-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set A in R^n is homeomorphic to the secondary polytope of A. 

    This talk will sketch this story of irrational toric varieties. It represents work with Garcia-Puente, Zhu, Postinghel, Villamizar, and Pir.

    To view Dr. Frank Sottile slides, click here. 

  • Tuesday October 20th, 2020

    Enumerative Geometry and the Quantum Torus

    Dr. Michael Polyak
    Israel Institute of Technology

    9:00am - 10:00am
    Via Zoom
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    Abstract: Tropical geometry provides a piece-wise linear approach to algebraic geometry. The role of algebraic curves is played by tropical curves - planar metric graphs with certain requirements of balancing, rationality of slopes and integrality. A number of classical enumerative problems can be easily solved by tropical methods. 

    We consider a generalization of tropical curves, removing requirements of rationality of slopes and integrality and discuss the resulting theory. One of the classical enumerative problems in algebraic geometry is that of counting of complex or real rational curves through a collection of points in a toric variety. We explain this counting procedure as a construction of certain cycles on moduli of rigid pseudotropical curves. Cycles on these moduli turn out to be closely related to Lie algebras. In particular, counting of both complex and real curves is related to the quantum torus Lie algebra. More complicated counting invariants (the so-called Gromov-Witten descendants) are similarly related to the super-Lie structure on the quantum torus.

    Quantum Toric Geometry I (Minicourse)

    Dr. Ernesto Lupercio
    Center for Research and Advanced Studies of the National Polytechnic Institute (Cinvestav-IPN)

    10:00am - 11:00am
    Via Zoom
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    Abstract: In this minicourse I will explain the foundations and motivations behind quantum toric geometry (QTG). QTG is a generalization of classical toric geometry where all tori are replaced by quantum tori thus allowing fans to become irrational. This is joint work with L. Katzarkov, L. Meersseman and A. Verjovsky. Verjovsky’s talk is a prerequisite.

    Non-Simplicial Quantum Toric Varieties

    Mr. Antonie Boivin 
    Universite Angers

    11:00am - 12:00pm
    Via Zoom
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    Abstract: In this talk, I will describe a construction of quantum toric varieties associated to an arbitrary fan on a finitely generated subgroup of $\mathbb{R}^d$.

  • Wednesday October 21st, 2020

    Maximal Torus Actions, Equivariant Principal Bundles and Transverse Equivalence

    Dr. Hiroaki Ishida
    Kagoshima University

    8:00am - 9:00am
    Via Zoom
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    Abstract: In this talk I will explain the one-to-one correspondence between complex manifolds with maximal torus actions (up to an equivalence relation) and irrational fans. I will also explain about the relation between this correspondence and the canonical foliations.

    Basic Cohomology of Moment-Angle Manifolds

    Mr. Roman Krutowski 
    UCLA

    9:00am - 10:00am
    Via Zoom
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    Abstract: For any complete fan $\Sigma$ we may construct a corresponding complex moment-angle $\mathcal{Z}_{\Sigma}$ manifold with canonical holomorphic foliation $\mathcal{F}$ on it. These foliations generalize toric varieties in the case of non-rational fans. From this perspective, basic cohomology rings of these foliations may be regarded as generalized cohomology rings of toric varieties.

    During the talk I am going to present results on calculation of basic de Rham and Dolbeault cohomology of these foliations with the main result being the following Davis-Januszkiewicz type formula

    \[

    H^{*,*}_{\mathcal{F}}(\mathcal{Z}_{\Sigma}) \cong \mathbb{R}[v_1,\ldots,v_m] /(I_{\Sigma}+J),\quad

    v_i \in H^{1,1}_{\mathcal{F}}(\mathcal{Z}_{\Sigma}),

    \] 

    where ideals $I_\Sigma$ and $J$ are reconstructed from the fan data. These results were conjectured by Fiammetta Battaglia and Dan Zaffran and are proved in joint works with Hiroaki Ishida and Taras Panov.

    To view Mr. Roman Krutowski slides, click here. 

    Nonrational Toric Geometry III: Quasifolds, Foliations, Combinatorics and One-parameter Families (Minicourse)

    Dr. Fiametta Battaglia
    Universita Degli Studi Firenze

    10:00am - 11:00am
    Via Zoom
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    Abstract: We illustrate a new viewpoint, introduced jointly with Dan Zaffran: we establish a correspondence between not necessarily rational simplicial fans and certain foliated complex manifolds called LVMB manifolds. Toric quasifolds are then viewed as leaf spaces. In the rational setting, Meersseman and Verjovsky had shown that the leaf space is the classical toric variety. We then give evidence that the rich interplay between toric geometry and combinatorics carries over to our nonrational context, in fact we are able to reformulate Stanley's argument for the proof of the g-Theorem on the combinatorics of simple polytopes.

    We continue by describing, from the viewpoint of foliations, a one-parameter family of toric quasifolds contaning all of the Hirzebruch surfaces. This last part is joint work with Elisa Prato and Dan Zaffran.

    Minicourse by Dr. Elisa Prato and Dr. Fiametta Battaglia.

    Quantum Toric Geometry Minicourse II (Minicourse)

    Dr. Ernesto Lupercio
    Center for Research and Advanced Studies of the National Polytechnic Institute (Cinvestav-IPN)

    11:00am - 12:00pm
    Via Zoom

    Abstract: In this minicourse I will explain the foundations and motivations behind quantum toric geometry (QTG) QTG is a generalization of classical toric geometry where all tori are replaced by quantum tori thus allowing fans to become irrational. This is joint work with L. Katzarkov, L. Meersseman and A. Verjovsky. Verjovsky’s talk is a prerequisite.