Schedule for: 22w5022 - Toric Degenerations

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

A basic introduction to symplectic geometry in general and integrable systems in particular through the perspective of the n-body problem. At the very end, I will explain how toric degenerations occupy a liminal space between the Kaehler and symplectic categories. Therefore, they can help solve a number of symplectic mysteries, which I will explain in the remaining talks.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk, we will introduce a class of polytope fibrations, which we call generalized Cayley sums. These fibrations represent a generalization of the Losev-Manin moduli space parametrizing pointed chains of projective lines.

I will discuss how to use the classification of Gorenstein canonical Fano polytopes (and Laurent polynomials supported on them) in the construction of Q-Fano threefolds which admit a toric degeneration. https://arxiv.org/abs/2210.07328

A well-known result of Sturmfels says that there is a Gröbner degeneration of the homogeneous coordinate ring for the Grassmannian G(2,n) to the Stanley-Reisner ring associated to the simplicial complex dual to the n-associahedron. In this talk, I will report on recent work with A. Nájera Chávez and H. Treffinger in which we simultaneously strengthen and generalize this result. We show that for a cluster algebra A of finite cluster type with "enough" frozen coefficients, there is a canonical embedded realization of a certain torus-invariant semi-universal deformation space for the Stanley-Reisner ring of the cluster complex of A. The cluster algebra A (along with its cluster structure) may be recovered as a distinguished fiber of this deformation. In fact, the total space of this deformation may be identified with cluster algebra obtained from A by adding "universal coefficients". Among other consequences, this implies that any cluster algebra of finite cluster type is Gorenstein.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

To start, I will introduce symplectic toric manifolds, classified by Delzant, and other multiplicity free actions. I will then turn to Gelfand-Cetlin systems, as introduced by Guillemin and Sternberg, and explain how they can be understood in terms of toric degenerations.

Let $\mathfrak{g}$ be a simple Lie algebra over $\C$, and $(C, p_1, \ldots, p_n)$ be an $n$-marked, smooth, projective complex curve. Using some representation theory of the affine Kac-Moody algebra associated to $\mathfrak{g}$, the Wess-Zumino-Novikov-Witten model of conformal field theory associates to the data of an $n$-tuple of dominant weights $\lambda_1, \ldots, \lambda_n$ and a non-negative integer $L$ a finite dimensional vector space $V_{C, \vec{p}}(\lambda_1, \ldots, \lambda_n, L)$ called a space of conformal blocks. Computing the dimension of these spaces amounts to finding a method to evaluate the so-called Verlinde formula of the WZNW theory. A striking theorem of Pauly, and Kumar, Narasimhan, and Ramanathan realizes the conformal blocks as the spaces of global sections of line bundles on the moduli $\mathcal{M}_{C, \vec{p}}(G)$ of quasi-parabolic principal $G$ bundles on the marked curve $(C, \vec{p})$; thus the Verlinde formula is linked to the Hilbert functions of line bundles on this moduli problem. The moduli $\mathcal{M}_{C, \vec{p}}(G)$ are themselves quite interesting. For example, if $C$ is the projective line, their geometry is closely related to configurations of $G$-flags, and other spaces which carry a cluster structure. I will give an overview of some known toric degenerations of the moduli $M_{C, \vec{p}}(G)$ when $\mathfrak{g} = sl_2, sl_3, sl_4$. These constructions have the effect of give a diagrammatic way to keep track of a basis of the spaces of conformal blocks. Time permitting, I will also describe a relationship to an integrable system studied by Hurtubise and Jeffries in the case $\mathfrak{g} = sl_2$.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The algebraic matroid of an irreducible variety, embedded into affine space via a specific coordinatization, is the combinatorial structure one gets by keeping track of which coordinate projections are dominant morphisms. Certain problems in statistics and engineering require an understanding of the algebraic matroids of particular families of varieties, such as detrimental varieties. These problems are very hard to crack, partially because there aren't many general techniques. In this talk, I will discuss some successes of tropical geometry as a tool here, highlighting at least one way to think of this in terms of toric degenerations.

Braid varieties are smooth affine varieties associated to any positive braid. Their cohomology is expected to contain information about the Khovanov-Rozansky homology of a related link. Special cases of braid varieties include Richardson varieties, double Bruhat cells, and double Bott-Samelson cells. Cluster algebras are a class of commutative rings with a rich combinatorial structure, introduced by Fomin and Zelevinsky. I'll discuss joint work with P. Galashin, T. Lam and D. Speyer in which we show the coordinate rings of braid varieties are cluster algebras, proving and generalizing a conjecture of Leclerc in the case of Richardson varieties. Seeds for these cluster algebras come from "3D plabic graphs", which are bicolored graphs embedded in a 3-dimensional ball and generalize Postnikov's plabic graphs for positroid varieties.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Symplectic geometry is perhaps best described as a mathematical abstraction of classical mechanics. On the other hand, it is a recurring theme in mathematics that the symmetries of an object should give rise to "quotients" of that object. Marsden-Weinstein reduction is the most fundamental incarnation of this phenomenon in symplectic geometry; it is a systematic approach to taking quotients of symplectic manifolds carrying Hamiltonian symmetries, and has important manifestations in algebraic geometry, mathematical physics, and representation theory. In this context, I will discuss some recent research on "abelianizing" the symplectic quotients taken by any compact connected Lie group. One main ingredient will be the Gelfand-Cetlin integrable systems of Guillemin-Sternberg, as well as recent generalizations to arbitrary Lie type by Hoffman-Lane. This represents joint work with Jonathan Weitsman.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will start by describing symplectomorphisms between Hirzebruch surfaces. Next, I will discuss how results of Harada- Kaveh enable us to see that toric degenerations can be used to explain these symplectomorphisms. Finally, I will explain symplectic cohomological rigidity and my joint work with Pabiniak where we show that toric degenerations can help solve this problem.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

(with Klaus Altmann, Christian Haase, Karin Schaller, and Lena Walter)

Mirror symmetry for Fano varieties is well understood when the variety is a toric manifold. Beyond the toric case, the next best understood example is the type A Grassmannian. In this talk, I’ll survey some mirror symmetry proposals for Grassmannians and flag varieties in type A: mirrors from toric degenerations, mirrors from the Abelian/non-Abelian correspondence, and the Plucker coordinate mirror. I’ll then discuss what’s known and what’s challenging about extending these constructions and results from Grassmannians to flag varieties and beyond.

The open positroid variety in the Grassmannian has both an $\mathcal{A}$ and $\mathcal{X}$ cluster structure. I'll review cluster ensemble maps $p: \mathcal{A} \to \mathcal{X}$ and discuss how such a map induces another cluster ensemble map $p^\vee$ from the mirror of $\mathcal{X}$ to the mirror of $\mathcal{A}$. Using this pair of maps, we will see that we can identify two Landau-Ginzburg mirror constructions for the Grassmannian-- one due to Marsh-Rietsch, and the other based on work of Gross-Hacking-Keel-Kontsevich. When one of these maps identifies the minimal model inputs of the two constructions, the other identifies the LG mirror outputs. As a corollary, we will also relate the polytopes and toric degenerations of Rietsch-Williams with those in the Gross-Hacking-Keel-Kontsevich framework. Based on ongoing joint work with Lara Bossinger, Mandy Cheung, and Alfredo Nájera Chávez.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Together with Jihyeon Jessie Yang, we resurrected an old idea of Raoul Bott for using large torus actions to construct canonical bases for unitary representations of compact Lie groups. Our methods are complex analytic. We apply them to families of Bott-Samelson manifolds parametrized by C^n. This application requires the vanishing of higher cohomology of sheaves of holomorphic sections of certain line bundles over the total spaces of such families; this vanishing is still conjectural. https://arxiv.org/abs/2106.15695

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

The interplay between combinatorics and algebraic geometry has immensely enriched both areas. In this context, the theory of Newton-Okounkov bodies has led to the extension of the geometry-combinatorics dictionary from toric varieties to certain varieties which admit a toric degeneration. In a recent paper with Megumi Harada, we gave a wall-crossing formula for the Newton-Okounkov bodies of a single variety. Our wall-crossing involves a collection of lattices $\{M_i\}_{i\in I}$ connected by piecewise-linear bijections $\{\mu_{ij}\}_{i,j\in I}$. In addition, in previous work Kiumars Kaveh and Christopher Manon analyze valuations into semifields of piecewise linear functions and explore their connections to families of toric degenerations, with particular attention to links to the theory of cluster varieties. Inspired by these ideas in joint work in progress with Megumi Harada and Christopher Manon we propose a generalized notion of polytopes in $\Lambda=(\{M_i\}_{i\in I},\{\mu_{ij}\}_{i,j\in I})$, where the $M_i$ are lattices and the $\mu_{ij}:M_i\to M_j$ are piecewise linear bijections. Roughly, these are $\{P_i\mid P_i\subseteq M_i\otimes \mathbb{R}\}_{I\in I}$ such that $\mu_{ij}(P_i)=P_j$ for all $i,j$. In analogy with toric varieties these generalized polytopes can encode compactifications of affine varieties as well as some of their geometric properties. In this talk, we illustrate these ideas with a concrete class of examples related to classical reflexive polytopes and Fano varieties.