Lecture Series

Kiumars Kaveh (University of Pittsburgh)

Title: Introduction to toric varieties, Newton-Okounkov bodies and toric degenerations

Abstract: We give a quick review of toric varieties with emphasis on the Bernstein-Kushnirenko theorem on the number of solutions of a system of Laurent polynomial equations in terms of volumes of the corresponding Newton polytopes. We will then discuss how this can be extended to arbitrary varieties via the construction of Newton-Okoukov bodies. Next, we introduce the notion of a toric degeneration, that is, a flat family over the affine line which is trivial away from the origin and the fiber over the origin is a (possibly non-normal) toric variety. Toric degenerations, in some sense, contain the geometric idea behind the theory of Newton-Okounkov bodies. Finally, we discuss a general result that any embedded projective variety has a flat degeneration to a (possibly non-normal) complexity-one variety. 

Hendrik Süß (University of Jena) 

Title: The geometry of algebraic torus actions

Abstract: An algebraic variety X equipped with an action of a torus T can be described in in terms of combinatorial data living on some quotient variety X/T. Morally this combinatorial data should encode the fibres of the quotient map from X to X/T. After discussing the basics of this approach we will consider the special case of complexity-1 torus actions and the comparison with the symplectic picture in this situation. Finally, we will discuss some applications to canonical Kähler metrics and K-stability.

Susan Tolman (University of Illinois at Urbana-Champaign) 

Title: Highly symmetric symplectic manifolds

Abstract: We will begin this series of lectures with a brief introduction to symplectic manifolds and Hamiltonian group actions. We will then focus on symplectic manifolds with "large" Hamiltonian  torus actions, which are well understood in many cases. In particular Delzant proved that toric manifolds, where the torus is half the dimension of the manifold, are classified by their moment image, which is a rational convex polytope.  This theorem is very powerful because it allows us to answer complicated geometric questions using combinatorial techniques, and vice versa.  Similarly, Karshon completed a combinatorial classification of Hamiltonian circle actions of four-dimensional symplectic manifolds. The next important case, which will be the main focus of my talks, will be the general complexity one case, where the torus is one dimension less than half the dimension of the manifold. Here, the classification is more complicated; while most of the invariants are still combinatorial one, called the ​"painting", is more homotopy theoretic. As a consequence, interesting phenomena arise in this case which do not occur in the previous examples.

Research Talks

Lara Bossinger (UNAM)

Title: Toric degenerations and maps to toric varieties

Abstract: I will report on recent joint work in progress with Takuya Murata. Motivated by the work of Harada—Kaveh and the desire to construct moment map type maps to Newton-Okounkov polytopes, we study maps to toric varieties. As algebro-geometric methods turn out to not be appropriate we use results of Mather involving Whitney stratifications to obtain a collapsing map. Although the result is more general, our main application concerns toric degenerations where we construct a map from the general to the special fibre. Moreover, we generalize a result of Harada-Kaveh constructing an integrable system on a projective variety that admits a toric degeneration induced by the moment map on the toric variety. A first preprint of the work is available on arxiv:2210.13137.

  
Isabelle Charton (University of Haifa)

Title: Monotone Hamiltonian T-Spaces of Complexity One

Abstract: A compact symplectic manifold  (M, ω)​ is called positive monotone if its first Chern class is a positive multiple of  [ω]​  in  H2dR(M)​. A Fano variety is a smooth complex variety that admits a holomorphic embedding into ℂPN  (for some N​). Such a variety can be endowed with a symplectic form, making it a positive monotone symplectic manifold. For this reason, Fano varieties are considered the algebraic counterparts of positive monotone symplectic manifolds. A general outstanding issue in symplectic geometry is the question of whether a positive monotone symplectic manifold is diffeomorphic to a Fano variety. In low dimensions, namely two and four, it has been proven by Gromov, Taubes, McDuff, and Ohta-Ono that any positive monotone symplectic manifold is diffeomorphic to a Fano variety. Analogous results are not known in higher dimensions. In this talk, I will explain what is known about the difference between Fano varieties and positive monotone symplectic manifolds endowed with a Hamiltonian torus action. In particular, I will present new results for the case where the complexity of the action is one. This is joint work with Liat Kessler, Silvia Sabatini, and Daniele Sepe.

Oliver Clarke (University of Edinburgh)

Title: Toric degenerations, Khovanskii bases, and Computation

Abstract: In the combinatorial realm of the Grassmannian, well-known families of toric degenerations such as Gelfand-Tsetlin (GT) and Fang-Fourier-Littleman-Vinberg (FFLV) degenerations are united by matching fields and interrelated by mutations. One of the main tools for constructing toric degenerations are Khovanskii bases, or their classical counterpart SAGBI bases. From the perspective of computation, it turns out that SAGBI bases (for quotient rings) and Khovanskii bases are deeply connected. In this talk, I will report on current understanding and computational methods in these areas.

Peter Crooks (Utah State University)

Title: New incarnations of Gelfand-Cetlin systems in symplectic geometry and representation theory 

Abstract: I will give a brief overview of Guillemin and Sternberg's Gelfand-Cetlin systems at the interface of symplectic geometry and representation theory. This will lead to a discussion of modern developments in the subject, with particular emphasis on Hoffman and Lane's recent generalization of Gelfand-Cetlin systems to arbitrary Lie type. In this context, I will outline new roles for Gelfand-Cetlin systems in the abelianization of symplectic quotients, non-abelian Duistermaat-Heckman measures, and geometric quantization. This represents joint work with Jonathan Weitsman. 

Naoki Fujita (Kumamoto University)

Title: Toric and semi-toric degenerations arising from cluster algebras

Abstract: A Newton-Okounkov body is a convex body that gives a systematic method of constructing toric degenerations of projective varieties. In the case of a compactified cluster variety such as a flag variety, we can construct Newton-Okounkov bodies using the theory of cluster algebras. In this talk, we discuss such Newton-Okounkov bodies in the case of a flag variety, which generalize some previously known Newton-Okounkov bodies such as string polytopes. The associated toric degenerations of a flag variety induce semi-toric degenerations of Schubert and opposite Schubert varieties. We see that these semi-toric degenerations can be regarded as generalizations of Kogan-Miller's semi-toric degeneration that is closely related to Schubert calculus. This talk is partly based on a joint work with Hironori Oya.

 
Yoosik Kim (Pusan National University)

Title. Cluster algebras and monotone Lagrangian tori

Abstract. Motivated by the recent development of the construction of toric degenerations via a cluster algebra by Gross--Hacking--Keel--Kontsevich and Fujita--Oya, we consider Lagrangian torus fibrations constructed from the toric degenerations whose Newton--Okounkov polytopes are related by a composition of combinatorial mutations. When a smooth projective variety X​ is equipped with a monotone Kähler form, we discuss a condition that the Lagrangian torus fibration on X​ carries a monotone Lagrangian torus fiber. Also, we derive a criterion of the existence of infinitely many monotone Lagrangian tori, no two of which are related by any symplectomorphisms. By utilizing the developed criterion and the correspondence between the dual canonical basis and the tropical integer points, we show that most flag manifolds of arbitrary type admit infinitely many monotone Lagrangian tori. This talk is based on a joint work with Yunhyung Cho, Myungho Kim, and Euiyong Park.

Donggun Lee (IBS CCG)

Title: Automorphisms and deformations of Hessenberg varieties 

Abstract: Hessenberg varieties are subvarieties in flag varieties which have interesting symmetric group actions on their cohomology. The positivity of induced representations in the permutation module basis expansion is known to be equivalent to a long-standing conjecture proposed by Stanley-Stembridge in combinatorics.

To enhance our understanding of these representations, one might hope to identify a lift of the action on the cohomology to Hessenberg varieties themselves or to discover useful deformations of them. In this talk, we discuss automorphisms and deformations of Hessenberg varieties when they are hypersurfaces in flag varieties. Especially in type A, we provide a complete classification along with an interpretation in terms of moduli of pointed rational curves. This is a joint work in progress with P. Brosnan, L. Escobar, J. Hong, E. Lee, A. Mellit and E. Sommers.

Joseph Palmer (University of Illinois at Urbana-Champaign)

Title: Lifting Torus Actions to Integrable Systems

Abstract: A complexity-one space on a symplectic 2n-manifold is the Hamiltonian action of a torus of dimension n-1. The momentum map for such an action can be thought of as n-1 real valued functions. On the other hand, an integrable system on such a manifold is the data of n functions. This motivates several natural questions: given a complexity-one space, when can an additional function be found to produce an integrable system? When can the resulting system be chosen to be toric? When can it be chosen to have no degenerate singularities? 

The case of when a circle action on a 4-manifold can be lifted to a toric integrable system was already completely understood by Karshon in 1999, but most other cases have remained open until relatively recently. In this talk, I will discuss answers to various versions of these questions, both in dimension four and higher. Parts of this work are joint with Sonja Hohloch, Susan Tolman, and Jason Liu.

Andrea Petracci (Università di Bologna)

Title: On deformations of toric (Fano) varieties

Abstract: Deforming toric varieties is the opposite of finding toric degenerations of (non-toric) algebraic varieties. In this talk, I will review the theory of Altmann on deformations of toric affine varieties and I will mention a couple of results about deformations (and smoothings) of toric Fano varieties.     

This talk is based on two works: the first one in collaboration with Anne-Sophie Kaloghiros and the second one in collaboration with Alessio Corti and Paul Hacking.

 
Milena Wrobel (University of Oldenburg) 

Title: Smooth Fano Varieties with Torus Action of Complexity Two 

Abstract: We study smooth Fano varieties with torus action by using their description via a specific rational quotient, the so called maximal orbit quotient (MOQ). In the case of torus actions of complexity one, the MOQ turns out to be the projective line having points as its critial configuration. In this talk we focus on torus actions of complexity two, which turns the MOQ into a surface. We shortly discuss the case where the MOQ is a projective plane with a hyperplane arrangement as critical configuration. Then, going one step further, we replace the projective plane with ℙ1 × ℙ1  and more generally a Hirzebruch surface, and give first classification results in this setting. 

5-minute-talks 

Eunjeong Lee (Chungbuk National University) Torus closures in the flag variety

Younghan Yoon (Ajou University) Real toric spaces associated with chordal nestohedra

Xiaobin Li (Southwest Jiaotong University) When torus action with complexity meets Gromov-Witten theory

Shin-young Kim (Ewha Womans University) Diagrams for varieties of minimal tangents of wonderful symmetric varieties

Minseong Kwon (KAIST) Spherical geometry of Hilbert schemes of conics in adjoint varieties