1:45pm	Registration
	Room: Anatomy Museum

2:10pm	Welcome
	Room: Anatomy Lecture Theatre

2:15pm	Mirror symmetry for Schubert varieties
	Lauren Williams (Harvard University)
	We define a superpotential for a general Schubert variety in the Grassmannian, then use the cluster structure on Schubert varieties to prove that the associated superpotential polytope coincides with a related Newton-Okounkov body. We also show that our superpotential is the superpotential of a Gorenstein Fano toric variety, which provides a partial desingularization of a toric degeneration of the Schubert variety. This is joint work with Konstanze Rietsch.

3:15pm	Stimulating Beverages
	Room: Anatomy Museum

3:45pm	(Generalised) cluster varieties and mirrors of Fanos
	Alessio Corti (Imperial College London)
	I introduce and (generalised) cluster varieties and summarise some key facts about them. If a family of Fano varieties has toric degenerations, then conjecturally it has a mirror that is a cluster variety. I discuss \(\mathbb{P}^3\) and \(\mathrm{Gr}(2,5)\) as examples (some of this joint with Laura Escobar).

4:45pm	Break

5:00pm	Semi-toric degenerations of projective varieties
	Xin Fang (RWTH Aachen)
	In this talk I will introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure allows us to construct (1). a Newton-Okounkov simplicial complex with an extra integral structure; (2). a flat degeneration of the projective variety into a reduced union of toric varieties. For Schubert varieties, Lakshmibai-Seshadri paths got interpretation as successive vanishing orders of certain regular functions within this framework. This talk bases on joint works with Rocco Chirivì and Peter Littelmann.

10:00am	Gorenstein Spherical Fano Varieties
	Johannes Hofscheier (University of Nottingham)
	Gorenstein toric Fano varieties play a crucial role in the famous Batyrev-Borisov mirror construction. In this talk, I will explore new challenges that arise when generalising from toric to spherical varieties — a remarkable class of algebraic varieties equipped with an action of a connected complex reductive group, where a Borel subgroup acts with an open dense orbit. This class includes flag varieties, toric varieties, symmetric spaces, and their equivariant compactifications. I will present joint work in progress with Girtrude Hamm, where we classify all 4-dimensional Gorenstein spherical Fano varieties. Our work extends the renowned Kreuzer-Skarke database and the recent classification by Delcroix-Montagard of 4-dimensional Gorenstein spherical Fano varieties of rank less than or equal to 2. This classification is obtained in terms of polytopes that generalise reflexive polytopes and relies on an algorithm that is inspired by Kasprzyk’s approach to enumerating toric Fano 3-folds with terminal singularities. Similar to the toric case, the geometric properties of the underlying variety, such as the Picard rank and anticanonical degree, can be described combinatorially in terms of these more general polytopes. If time permits, I will conclude with examples illustrating our preliminary observations and ongoing efforts in understanding the mirror symmetry of Gorenstein spherical Fano varieties.

11:00am	Stimulating Beverages
	Room: Anatomy Museum

11:30am	SAGBI bases and mirror constructions for Kronecker moduli spaces
	Elana Kalashnikov (University of Waterloo)
	One way of constructing mirror partners to Fano varieties is via toric degenerations. The case in which this is best understood is the Grassmannian, using the well-known SAGBI basis of the Plucker coordinate ring indexed by semi-standard Young tableaux (SSYT). The mirror construction goes back to work of Eguchi‑Hori‑Xiong, however its geometry and combinatorics still plays an important role in current mirror constructions. In this talk, I will give an overview of this story, then turn to the question of what can be generalized for Kronecker moduli spaces. Like Grassmannians (which they generalize), Kronecker moduli spaces are high Fano index Picard rank 1 smooth Fano varieties. I will introduce linked SSYT pairs, which play the analogous role of SSYT for Grassmannians in understanding the coordinate ring of the Kronecker moduli space. This is joint work with Liana Heuberger.

12:30pm	Lunch

2:30pm	Cluster structures on braid varieties
	Linhui Shen (Michigan State University)
	Let \(G\) be a semisimple group over \(\mathbb{C}\). Let \(\beta\) be a positive braid whose Demazure product is the longest Weyl group element. The braid variety \(M(\beta)\) generalizes many well known varieties, including positroid cells, open Richardson varieties, and double Bruhat cells. We provide a concrete construction of the cluster structures on \(M(\beta)\), using the weaves of Casals and Zaslow and a new combinatorial construction called Lusztig cycles. We show that the coordinate ring of \(M(\beta)\) is a cluster algebra, which confirms a conjecture of Leclerc as special cases. As an application, we show that \(M(\beta)\) admits a natural Poisson structure and can be further quantized. This talk is based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, and Jose Simental.

3:30pm	Break

3:45pm	The twist map on positroid varieties is a quasi-cluster automorphism
	Melissa Sherman-Bennett (Massachusetts Institute of Technology)
	Open positroid varieties are varieties which stratify the Grassmannian. They have a cluster structure with seeds given by Postnikov's plabic graphs. They generalize double Bruhat cells on \(\mathrm{SL}_n\), and are generalized by open Richardson varieties and braid varieties. I will discuss joint work with R. Casals, I. Le and D. Weng in which we show the Muller-Speyer twist map on open positroid varieties interacts nicely with their cluster structures. In particular, the twist map is induced by a green-to-red sequence followed by rescaling by frozen variables. Along the way, we show a close relationship between plabic graphs and Demazure weaves, which were used recently to give cluster structures on braid varieties.

4:45pm	Drinks and Nibbles Reception
	Room: Anatomy Museum

5:30pm	Short Talks

5:30pm	Quantum periods and intrinsic mirror symmetry
	Samuel Johnston (Imperial College London)

5:55pm	Cluster structures on Cox rings and branching algebras
	Luca Francone (Université Claude Bernard Lyon 1)

6:20pm	Categorification: flag combinatorics and quantum minors
	Xiuping Su (University of Bath)

6:45pm	Flags on Fano 3-fold hypersurfaces
	Livia Campo (University of Vienna)
	The existence of Kaehler-Einstein metrics on Fano 3-folds can be determined by studying lower bounds of stability thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. This is deeply intertwined with the geometry of the hypersurfaces: in fact, birational rigidity and superrigidity play a crucial role. The superrigid case had been attacked by Kim-Okada-Won. In this talk I will touch upon the K-stability of strictly rigid Fano hypersurfaces via Abban-Zhuang Theory. This is a joint work with Takuzo Okada.

9:30am	Mutations of polytopes and compactifications of varieties
	Chris Manon (University of Kentucky)
	The theory of Newton-Okounkov bodies allows the combinatorial techniques of toric geometry to be applied to more general projective varieties. Escobar and Harada have defined a notion of wall-crossing for Newton-Okounkov bodies, which involves piecewise-linear mutation maps between different Newton-Okounkov bodies associated to the same variety. Similar phenomena appear in the work of Rietsch and Williams, and Bossinger, Cheung, Magee, and Nájera Chávez on Newton-Okounkov bodies associated to compactifications of cluster varieties. In these settings, the mutations between Newton-Okounkov bodies reflect important aspects of the geometry and combinatorics of the associated variety. In forthcoming work with Laura Escobar and Megumi Harada, we wrap the data of a collection of lattices related by piecewise-linear bijections together into a single semi-algebraic object, equipped with its own notions of convexity and polyhedra. Such a polytope encodes a compactification of an affine variety whose coordinate ring can be equipped with a valuation into one of these objects, and geometry of the compactification can be understood combinatorially. I'll introduce elements of this construction, give examples, and point to some unanswered questions.

10:30am	Stimulating Beverages
	Room: Anatomy Museum

11:00am	Scattering diagrams and Jeffrey-Kirwan residues
	Sara Angela Filippini (Università del Salento)
	Chan, Leung and Ma showed how asymptotic analysis of Maurer-Cartan solutions in an appropriate differential graded Lie algebra gives rise to the consistent scattering diagrams appearing in the construction of mirror families by Kontsevich-Soibelman and Gross-Siebert. This approach was then generalized by Leung, Ma and Young providing a consistent completion of refined scattering diagrams and the associated theta functions. Based on these results we show that the consistent completion of an initial scattering diagram in \(M_\mathbb{R}\) for a finite rank lattice \(M\) can be expressed in terms of the Jeffrey-Kirwan residues of certain explicit meromorphic forms. A similar description holds for the associated theta functions. This is joint work with Jacopo Stoppa.

12:00pm	Break

12:15pm	Symplectic embeddings for cluster surfaces via Newton-Okounkov bodies
	Ben Wormleighton (Washington University in St. Louis)
	Determining when one symplectic manifold embeds in another is a central problem-type in symplectic geometry, leading to mutually beneficial connections with combinatorics, number theory, and algebraic geometry. I will discuss recent and current work with Julian Chaidez and Tim Magee on understanding embeddings into symplectic manifolds arising from two-dimensional cluster varieties. The case of toric 4-manifolds has been extensively studied using the combinatorics of moment polytopes, and we will see that the tropical Newton-Okounkov bodies of Bossinger-Cheung-Magee-Nájera Chávez play an analogous role in the cluster setting.

1:15pm	Free Afternoon

6:30pm	Conference Dinner at the Mulberry Bush

10:00am	Reciprocity for Pairings of Theta Functions
	Greg Muller (University of Oklahoma)
	Given a pair of mirror dual affine log Calabi-Yau varieties \(X\) and \(Y\), the Gross-Siebert program associates a theta function on \(X\) to each boundary valuation on \(Y\). Since mirror duality is a symmetric relation, there are two ways to associate an integer to a pair \((m,n)\) of boundary valuations on \(X\) and \(Y\). 1) Apply the valuation \(m\) to the theta function associated to \(n\). 2) Apply the valuation \(n\) to the theta function associated to \(m\). Resolving a conjecture of Gross-Hacking-Keel-Kontsevich, we show that these two numbers are equal in a generality which covers all cluster algebras (specifically, when the theta functions are given by enumerating broken lines in a scattering diagram generated by finitely-many elementary incoming walls). Time permitting, I will discuss applications to tropicalizations of theta functions, Donaldson-Thomas transformations, and localizations of cluster algebras. This work is joint with Man-Wai Cheung, Tim Magee, and Travis Mandel.

11:00am	Stimulating Beverages
	Room: Anatomy Museum

11:30am	Fundamentals of broken line convex geometry and Batyrev-Borisov duality
	Bosco Frías-Medina (Universidad Michoacana de San Nicolás de Hidalgo)
	Using the Batyrev-Borisov duality for nef-partitions as a motivation, I will discuss a new theory of convex geometry in certain piecewise linear manifolds. The interest of these piecewise linear manifolds (tropical spaces) is that they arise naturally in the study of mirror symmetry for cluster varieties. We expect that this broken line convex geometry will encode the algebraic geometry of compactifications of cluster varieties, just like the algebraic geometry of toric varieties is encoded by usual convex polyhedral geometry. In this talk, the notions of Minkowski sum, polyhedral complexes and duality of polytopes in this setting will be discussed. Finally, I will present the advances of the broken line convex geometry version of the Batyrev-Borisov duality. Based on ongoing joint work with Timothy Magee.

12:30pm	Lunch

3:00pm	Newton-Okounkov bodies of Hilbert schemes
	Ian Cavey (University of Illinois)
	The Hilbert scheme of points on a smooth, projective surface \(X\) is a smooth, projective variety parametrizing finite closed subschemes of \(X\). In this talk, I will describe Newton-Okounkov bodies and valuation semigroups for all ample divisors on the Hilbert schemes of points on \(\mathbb{P}^1\times\mathbb{P}^1\). The valuation semigroups are not finitely-generated, so one does not get toric degenerations of the Hilbert schemes. However, we can give explicit formulas for the corresponding Hilbert series which, thanks to structural results of Ellingsrud, Göttsche, and Lehn, determines the Euler characteristic of any line bundle on the Hilbert scheme of points on any smooth, projective surface.

4:00pm	Snacks
	Room: Anatomy Museum

4:30pm	Towards derived Reid’s recipe for dimer models
	Liana Heuberger (University of Bath)
	Reid's recipe is an equivalent of the McKay correspondence in dimension three. It marks interior line segments and lattice points in the fan of the \(G\)-Hilbert scheme with characters of irreducible representations of \(G\). In joint work with Craw and Tapia Amador, we generalise this by marking the toric fan of a crepant resolution of any affine Gorenstein singularity, in a way that is compatible with both the \(G\)-Hilbert case and its categorical counterpart known as Derived Reid's Recipe. To achieve this, we foray into the combinatorial land of quiver moduli spaces and dimer models. In this talk I will discuss connections between combinatorial and derived Reid's recipe and recent progress concerning low-valency vertices in the quiver. This is joint work with Alastair Craw.

10:00am	Landau-Ginzburg Models for Cominuscule Homogeneous Spaces
	Charles Wang (University of Michigan)
	Rietsch has constructed Landau-Ginzburg (LG) models for all homogeneous spaces \(G/P\) in terms of Lie-theoretic data, and there have been several subsequent works which make use of Plucker coordinate descriptions of Rietsch's LG models in special cases. These models were obtained using heavily type-dependent methods for particular cases such as the Grassmannians \(\mathrm{Gr}(k,n)\) and in this talk, we will present a uniform, type-independent construction of Plucker coordinate LG models agreeing with Rietsch's Lie-theoretic models for all cominuscule homogeneous spaces. One of our main tools is an order-theoretic description of Plucker coordinates using minuscule posets which allows us to avoid many type-specific considerations.

11:00am	Stimulating Beverages
	Room: Anatomy Museum

11:30am	Categorification and Mirror Symmetry for Grassmannians
	Alastair King (University of Bath)
	I will explain a categorical viewpoint on Grassmannian mirror symmetry, in the sense of Rietsch-Williams, and thereby obtain a description of the g-vector cone for the Grassmannian cluster category of Jensen-King-Su. The talk is based on joint work with B.T. Jensen and X.Su [arXiv:2404.14572].