Online Algebraic Geometry Seminar
This seminar is held online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team. To be added to this team, please email one of Alex Kasprzyk, Livia Campo, or Johannes Hofscheier. For help joining a talk, please follow these instructions. Everybody is welcome.
Follow us on researchseminars.org or subscribe to the online calendar to be kept uptodate about upcoming talks (also available in iCal format). Recordings of talks are available on our YouTube channel.
Note: All times are UK times.
Upcoming Talks

 27 January 2022, 1011am
 Florin Ambro (Simion Stoilow)
 On Seshadri constants
 The Seshadri constant of a polarized variety (X,L) at a point x measures how positive is the polarization L at x. If x is very general, the Seshadri constant does not depend on x, and captures global information on X. Inspired by ideas from the Geometry of Numbers, we introduce in this talk successive Seshadri minima, such that the first one is the Seshadri constant at a point, and the last one is the width of the polarization at the point. Assuming the point is very general, we obtain two results: a) the product of the successive Seshadri minima is proportional to the volume of the polarization; b) if X is toric, the ith successive Seshadri constant is proportional to the ith successive minima of a suitable 0symmetric convex body. Based on joint work with Atsushi Ito.
 Link to Event

 3 February 2022, 1011am
 Wendelin Lutz (Imperial)
 A geometric proof of the classification of Tpolygons
 One formulation of mirror symmetry predicts (omitting a few adjectives) a onetoone correspondence between equivalence classes of lattice polygons and deformation families of del Pezzo surfaces. Lattice polygons that correspond to smooth Del Pezzo surfaces are called Tpolygons and have been classified by KasprzykNillPrince using combinatorial methods, thereby verifying the conjecture in the smooth case. I will give a new geometric proof of their classification result.
 Link to Event

 10 February 2022, 1011am
 Ananyo Dan (Sheffield)
 McKay correspondence for isolated Gorenstein singularities
 The McKay correspondence is a (natural) correspondence between the (nontrivial) irreducible representations of a finite subgroup G of SL(2,C) and the irreducible components of the exceptional divisor of a minimal resolution of the associated quotient singularity C^{2}//G. A geometric construction for this correspondence was given by GonzálezSprinberg and Verdier, who showed that the two sets also correspond bijectively to the set of indecomposable reflexive modules on the quotient singularity. This was generalised to higher dimensional quotient singularities (i.e., quotient of C^{n} by a finite subgroup of SL(n,C)) by ItoReid, where the above sets were substituted by certain smaller subsets. It was further generalised to more general quotient singularities by BridgelandKingReid, IyamaWemyss and others, using the language of derived categories. In this talk, I will survey past results and discuss what happens for the isolated Gorenstein singularities case (not necessarily a quotient singularity). If time permits, I will discuss applications to Matrix factorization. This is joint work in progress with J. F. de Bobadilla and A. RomanoVelazquez.
 Link to Event

 17 February 2022, 1011am
 Qaasim Shafi (Imperial)
 Logarithmic Toric Quasimaps
 Quasimaps provide an alternate curve counting system to GromovWitten theory, related by wallcrossing formulas. Relative (or logarithmic) GromovWitten theory has proved useful for constructions in mirror symmetry, as well as for determining ordinary GromovWitten invariants via the degeneration formula. I’ll discuss how to build a theory of logarithmic quasimaps in the toric case, some restrictions, and why one might want to do so.
 Link to Event
Past Talks

 20 January 2022
 Marvin Hahn (Sorbonne)
 The tropical geometry of monotone Hurwitz numbers
 Hurwitz numbers are important enumerative invariants in algebraic geometry. They count branched maps between Riemann surfaces. Equivalently, they enumerate factorizations in the symmetric group. Hurwitz numbers were introduced in the 1890s by Adolf Hurwitz and became central objects of enumerative algebraic geometry in the 1990s through close connections with the socalled GromovWitten theory. This interplay between Hurwitz and GromovWitten theory is an active field of research and led to, among other things, the celebrated ELSV formula. In the last decade, many variants of Hurwitz numbers have been introduced and studied. In particular, the question of connections between these variants of Hurwitz numbers and GromovWitten theory is of great interest. Socalled monotone Hurwitz numbers , which originate from the theory of random matrices, are among the most studied variants of Hurwitz numbers. This talk is a progress report of our larger program in which we study the connections between monotone Hurwitz numbers and GromovWitten theory by combinatorial methods of tropical geometry, and whose longterm goal is a proof of the still open conjecture of an ELSV  type formula for double monotone Hurwitz numbers. The talk is based in part on joint work with Reinier Kramer and Danilo Lewanski.

 13 January 2022
 Arina Voorhaar (Geneva)
 On the Newton Polytope of the Morse Discriminant
 A famous classical result by Gelfand, Kapranov and Zelevinsky provides a combinatorial description of the vertices of the Newton polytope of the Adiscriminant (the closure of the set of all nonsmooth hypersurfaces defined by polynomials with the given support A). Namely, it gives a surjection from the set of all convex triangulations of the convex hull of the set A with vertices in A (or, equivalently, the set of all possible combinatorial types of smooth tropical hypersurfaces defined by tropical polynomials with support A) onto the set of vertices of this Newton polytope. In my talk, I will discuss a similar problem for the Morse discriminant — the closure of the set of all polynomials with the given support A which are nonMorse if viewed as polynomial maps. Namely, for a 1dimensional support set A, there is a surjection from the set of all possible combinatorial types of socalled Morse tropical polynomials onto the vertices of the Newton polytope of the Morse discriminant.
 Video, Slides

 8 December 2021
 Eleonore Faber (Leeds)
 Matrix factorizations for discriminants of pseudoreflection groups
 In this talk we will give an introduction to the McKay correspondence for complex reflection groups (joint work with Ragnar Buchweitz and Colin Ingalls), and then show how this allows to identify certain matrix factorizations of the discriminants of these reflection groups. We will in particular consider the family of pseudoreflection groups G(r,p,n), for which one can explicitly determine matrix factorizations, using higher Specht polynomials (work in progress with Colin Ingalls, Simon May, and Marco Talarico).
 Video, Slides

 25 November 2021
 Ana Ros Camacho (Cardiff)
 Computational aspects in orbifold equivalence
 LandauGinzburg models are a family of physical theories described by some polynomial (or "potential") characterized by having an isolated singularity at the origin. Often appearing in mirrorsymmetric phenomena, they can be collected in higher categories with nice properties that allow direct computations. In this context, it is possible to introduce an equivalence relation between two different potentials called "orbifold equivalence". We will present some recent examples of this equivalence, and discuss the computational challenges posed by the search of new ones. Joint work with Timo Kluck.
 Video, Slides

 18 November 2021
 Nicholas Anderson (Queen Mary)
 Paving tropical ideals
 Tropical geometry is a powerful tool in algebraic geometry, which offers a multitude of combinatorial approaches to studying algebraic varieties. This talk will focus on the recent development of tropical commutative algebra by Diane Maclagan and Felipe Rincon. The central object of study is the "tropical Ideal", which generalizes the structure of polynomial ideals over fields to be suitable for study in the setting of tropical geometry, that is, in polynomial semirings over semifields. All polynomial ideals over a field can be associated to a "realizable" tropical ideal, and it is a nontrivial fact that "nonrealizable" tropical ideals exist. In this talk, I will demonstrate how the combinatorics of matroid theory allows us to easily generate a subclass of tropical ideals, called paving tropical ideals, which in turn allows us to prove that most zerodimensional tropical ideals are not realizable.
 Video, Slides

 28 October 2021
 Andrea Brini (Sheffield)
 Quantum geometry of logCalabi Yau surfaces
 A logCalabi Yau surface with maximal boundary, or Looijenga pair, is a pair (X,D) with X a smooth complex projective surface and D a singular anticanonical divisor in X. I will introduce a series of physicsmotivated correspondences relating five different classes of enumerative invariants of the pair (X,D):
 the log Gromov–Witten theory of (X,D),
 the Gromov–Witten theory of X twisted by the sum of the dual line bundles to the irreducible components of D,
 the open Gromov–Witten theory of special Lagrangians in a toric Calabi–Yau 3fold determined by (X,D),
 the Donaldson–Thomas theory of a symmetric quiver specified by (X,D), and
 a class of BPS invariants considered in different contexts by Klemm–Pandharipande, Ionel–Parker, and Labastida–Marino–Ooguri–Vafa.
I will also show how the problem of computing all these invariants is closedform solvable. Based on joint works with P. Bousseau, M. van Garrel, and Y. Schueler.
 Video, Slides

 21 October 2021
 AnneSophie Kaloghiros (Brunel)
 The Calabi problem for Fano 3folds
 I will discuss progress on the Calabi problem for Fano 3folds. The 105 deformation families of smooth Fano 3folds, were classified by Iskovskikh, Mori and Mukai. We determine whether or not the general member of each of these 105 families admits a KählerEinstein metric. In some cases, it is known that while the general member of the family admits a KählerEinstein metric, some other member does not. This leads to the problem of determining which members of a deformation family admit a KählerEinstein metric when the general member does. This is accomplished for most of the families, and I will present a conjectural picture for some of the remaining families. This is a joint project with Carolina Araujo, AnaMaria Castravet, Ivan Cheltsov, Kento Fujita, Jesus MartinezGarcia, Constantin Shramov, Hendrik Süss and Nivedita Viswanathan.
 Video, Slides

 14 October 2021
 Alastair Craw (Bath)
 Hyperpolygon spaces: beyond the movable cone
 For n>=4, the hyperpolygon spaces are a collection of Nakajima quiver varieties in dimension 2n6 that have been a useful testing ground for conjectures on conical symplectic varieties. I'll describe joint work in progress with Gwyn Bellamy, Steven Rayan, Travis Schedler and Hartmut Weiss in which we describe completely the birational geometry of these spaces. The case n=5 recovers a wellknown finite quotient singularity in dimension four, and allows us to provide a uniform construction of all 81 projective crepant resolutions studied in previous work of DontenBury–Wiśniewski. I'll also explain the title of the talk by giving a geometric interpretation of the components of the stability parameter even when it doesn't lie in the positive orthant.
 Video, Slides

 7 October 2021
 Tom Coates (Imperial)
 Rigid maximally mutable Laurent polynomials
 I will describe a class of Laurent polynomials which conjecturally corresponds under mirror symmetry to Fano varieties, in any dimension, with mild singularities. This is joint work with Alexander Kasprzyk, Giuseppe Pitton, and Ketil Tveiten.
 Video, Slides

 30 September 2021
 Julius Giesler (Tübingen)
 Kanev and Todorov type surfaces in toric 3folds
 In this talk we show at the example of some surfaces of general type, so called Kanev and Todorov type surfaces, how to construct minimal and canonical models of hypersurfaces in toric varieties. We relate the plurigenera and the Kodaira dimension of the hypersurfaces to a special polytope, known as the Fine interior. Then we study singularities of the canonical models of Kanev/Todorov type surfaces via toric geometry, degenerations of these surfaces and investigate some Hodge theoretic consequences.
 Video, Slides

 23 September 2021
 Nicolas Addington (Oregon)
 Hodge number are not derived invariants in positive characteristic
 Derived categories of coherent sheaves behave a lot like cohomology, so it's natural to ask which cohomological invariants are preserved by derived equivalences. After discussing the motivation and previous results, I'll present a derived equivalence between CalabiYau 3folds in characteristic 3 with different Hodge numbers; this couldn't happen in characteristic 0. The project has a substantial computer algebra component which I'll spend some time on.
 Video, Slides

 16 September 2021
 Nikita Nikolaev (Sheffield)
 Abelianisation of Meromorphic Connections
 There is a natural 11 correspondence between Higgs bundles on a compact complex curve and line bundles on an appropriate branched cover. This abelianisation process goes through the direct image functor and it has been fruitful in addressing a variety of problems relating to bundles on curves. We extend this abelianisation correspondence from Higgs bundles to flat bundles. This generalisation involves choosing a certain graph which translates to cohomology as a natural cocycle that exhibits a local deformation of the direct image functor. Furthermore, our abelianisation correspondence extends to lambdaconnections and recovers the abelianisation of Higgs bundles as lambda goes to 0. Based in part on joint work in progress with Marco Gualtieri.
 Video, Slides

 2 September 2021
 Hunter Spink (Stanford)
 Logconcavity of matroid hvectors and mixed Eulerian numbers
 For any matroid M, we compute the Tutte polynomial using the mixed intersection numbers of certain tautological classes in the combinatorial Chow ring arising from Grassmannians. Using mixed HodgeRiemann relations, we deduce a strengthening of the logconcavity of the hvector of a matroid complex, improving on an old conjecture of Dawson that was resolved contemporaneously by Ardila, Denham, and Huh. Joint with Andrew Berget and Dennis Tseng.
 Video

 26 August 2021
 DongSeon Hwang (Ajou)
 Cascades of singular rational surfaces of Picard number one
 I will introduce the notion of cascades of singular rational surfaces of Picard number one, which consists of a sequence of special birational morphisms, and then discuss some applications in the toric case, Fano case, and (log) general type case. The latter application is closely related to algebraic MontgomeryYang problem, conjectured by Kollár.
 Video, Slides

 12 August 2021
 Ollie Clarke (Bristol and Ghent)
 Combinatorial mutations and block diagonal polytopes
 Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration of the Grassmannian, the polytope of the associated toric variety coincides with the matching field polytope. In this talk I will describe combinatorial mutations of matching field polytopes. We will explore properties of polytopes which are preserved by mutation, and we will see that property of giving rise to a toric degeneration is preserved by mutations. This gives us an easy way to generate new families of toric degenerations of the Grassmannian from old. This talk is based on joint work with Akihiro Higashitani and Fatemeh Mohammadi.
 Video, Slides

 5 August 2021
 Dhruv Ranganathan (Cambridge)
 Toric contact cycles in the moduli space of curves
 The toric contact cycles are loci in the moduli space of curves that parameterize those curves that admit a morphism to a fixed toric variety, with prescribed tangency data with the toric boundary. The cycles are the fundamental building blocks in higher genus logarithmic GromovWitten theory and are higher dimensional analogues of the double ramification cycles, which have been studied intensely in the last decade. In recent work, Sam Molcho (ETH) and I proved that these cycles lie in the tautological part of the Chow ring of the moduli space of curves. A lesson I learned from this project, and earlier work with Navid Nabijou (Cambridge), is that it can be quite profitable to blend Fulton’s analysis of blowups and strict transforms with logarithmic GromovWitten theory and its virtual class. I’ll try to give a sense of the basic geometric phenomena, and point to some other places where they come up.
 Video, Slides

 29 July 2021
 Matthias Nickel (Frankfurt)
 Local positivity and effective Diophantine approximation
 In this talk I will discuss a new approach to prove effective results in Diophantine approximation relying on lower bounds of Seshadri constants. I will then show how to use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation.
 Video, Slides

 22 July 2021
 Kristin DeVleming (UCSD)
 K moduli of quartic K3 surfaces
 We will discuss a family of compactifications of moduli spaces of log Fano pairs coming from Kstability, and discuss an application to moduli of quartic K3 surfaces, with a focus on the locus of hyperelliptic K3s that arise as double covers of P^{1}xP^{1} branched over a (4,4) curve. We will show that Kstability provides a natural way to interpolate between the GIT moduli space and the BailyBorel compactification and will relate this interpolation to VGIT wall crossings. This is joint work with Kenny Ascher and Yuchen Liu.
 Video

 15 July 2021
 Sergey Galkin (PUCRio and HSE)
 Graph potentials and combinatorial nonabelian Torelli
 I will introduce graph potentials and discuss some of their combinatorial aspects, such as small resolution conjecture and combinatorial nonabelian Torelli theorem. The talk is based on the joint works with Pieter Belmans and Swarnava Mukhopadhyay.
 Video, Slides

 8 July 2021
 Chengxi Wang (UCLA)
 Varieties of general type with small volume
 By HaconMcKernan, Takayama, and Tsuji, there is a constant r_{n} such that for every r at least r_{n}, the rcanonical map of every ndimensional variety of general type is birational. In this talk, we show that r_{n} must grow faster than any polynomial in n, by giving examples of general type with small volume in high dimensions. In particular, we construct a klt nfold with ample canonical class whose volume is less than 1/2^{2n}. The klt examples should be close to optimal. This is joint work with Burt Totaro.
 Video, Slides

 1 July 2021
 Pedro Montero (Valparaíso)
 On the liftability of the automorphism group of smooth hypersurfaces of the projective space
 Smooth hypersurfaces are classical objects in algebraic geometry since they are the simplest varieties one can define as they are given by only one equation. As such, they have been intensively studied and their geometry has shaped the development of classic and modern algebraic geometry. In this talk, I will first recall some fundamental results concerning the automorphism group of smooth hypersurfaces of the projective space and then I will present some new results obtained in a joint work with Victor GonzalezAguilera and Alvaro Liendo, which are inspired by the classification groups which faithfully act on smooth cubic and quintic threefolds by Oguiso, Wei and Yu. Finally, I will discuss some perspectives and open problems that arise from this.
 Video, Slides

 24 June 2021
 Yusuke Nakajima (Kyoto)
 Combinatorial mutations and deformations of dimer models
 The combinatorial mutation of a polytope was introduced in the context of the mirror symmetry of Fano manifolds for achieving the classification problem. This operation makes a given polytope another one while keeping some properties. In my talk, I will consider the combinatorial mutation of a polygon associated to a dimer model. A dimer model is a bipartite graph on the real twotorus, and the combinatorics of a dimer model gives rise to a certain lattice polygon. Also, a dimer model enjoys rich information regarding toric geometry associated to that polygon. It is known that for any lattice polygon P there is a dimer model whose associated polygon coincides with P. Thus, there also exists a dimer model giving the lattice polygon obtained as the combinatorial mutation of P. I will observe the relationship between a dimer model giving a lattice polygon P and the one giving the combinatorial mutation of P. In particular, I introduce the operation which I call the deformation of a dimer model, and show that this operation induces the combinatorial mutation of a polygon associated to a dimer model. This talk is based on a joint work with A. Higashitani.
 Video, Slides

 17 June 2021
 Laura Escobar (Washington)
 Wallcrossing phenomenon for NewtonOkounkov bodies
 A NewtonOkounkov body is a convex set associated to a projective variety, equipped with a valuation. These bodies generalize the theory of Newton polytopes and the correspondence between polytopes and projective toric varieties. Work of KavehManon gives an explicit link between tropical geometry and NewtonOkounkov bodies. We use this link to describe a wallcrossing phenomenon for NewtonOkounkov bodies. As an example, we describe wallcrossing formula in the case of the Grassmannian Gr(2,m). This is joint work with Megumi Harada.
 Video, Slides

 10 June 2021
 Anne Lonjou (ParisSaclay)
 Action of Cremona groups on CAT(0) cube complexes
 A key tool to study the plane Cremona group is its action on a hyperbolic space. Sadly, in higher rank such an action is not available. Recently, in geometric group theory, actions on CAT(0) cube complexes turned out to be a powerful tool to study a large class of groups. In this talk, based on a common work with Christian Urech, we will construct such complexes on which Cremona groups of rank n act. Then, we will see which kind of results on these groups we can obtain.
 Video, Slides

 3 June 2021
 Martin Ulirsch (Frankfurt)
 Parabolic Higgs bundles on toric varieties
 In this talk I will explain a version of Simpson’s nonabelian Hodge correspondence on a toric variety X. There is a natural 11 correspondence between stable parabolic Higgs bundles on X and irreducible representations of the fundamental group of the big torus. This correspondence reduces to a correspondence between toric vector bundles and integral unitary representations in a suitable sense. In this story the spherical Tits building will have a surprise appearance. The main result suggests (at least to me) that there is a yettobediscovered logarithmic incarnation of the nonabelian Hodge correspondence.
 Video, Slides

 27 May 2021
 Helge Ruddat (Mainz)
 Polytopes, periods, degenerations
 A lattice polytope describes a projective toric variety and a regular subdivision of the polytope describes a flat degeneration of the toric variety. It is instructive to deform the degenerating family in a way that makes the geometry nontoric and produces a more interesting real torus fibration on the fibres of the degeneration. I am going to explain a simple formula that permits the easy computation of period integrals for the deformed families. This approach to periods doesn't require any differential equations and is flexible enough to give proofs for strong results about GrossSiebert's degenerating families obtained from wall structures. The talk is based on joint work with Bernd Siebert.
 Video, Slides

 20 May 2021
 Hannah Markwig (Tübingen)
 Counting bitangents of plane quartics  tropical, real and arithmetic
 A smooth plane quartic defined over the complex numbers has precisely 28 bitangents. This result goes back to Pluecker. In the tropical world, the situation is different. One can define equivalence classes of tropical bitangents of which there are 7, and each has 4 lifts over the complex numbers. Over the reals, we can have 4, 8, 16 or 28 bitangents. The avoidance locus of a real quartic is the set in the dual plane consisting of all lines which do not meet the quartic. Every connected component of the avoidance locus has precisely 4 bitangents in its closure. For any field k of characteristic not equal to 2 and with a nonArchimedean valuation which allows us to tropicalize, we show that a tropical bitangent class of a quartic either has 0 or 4 lifts over k. This way of grouping into sets of 4 which exists tropically and over the reals is intimately connected: roughly, tropical bitangent classes can be viewed as tropicalizations of closures of connected components of the avoidance locus. Arithmetic counts offer a bridge connecting real and complex counts, and we investigate how tropical geometry can be used to study this bridge. This talk is based on joint work with Maria Angelica Cueto, and on joint work in progress with Sam Payne and Kristin Shaw.
 Video, Slides

 13 May 2021
 Roger Casals (UC Davis)
 Positroid links and braid varieties
 I will discuss a class of affine algebraic varieties associated to positive braids, their relation to open positroid strata in Grassmannians and their cluster structures. First, I will introduce the objects of interest, with the necessary ingredients, and motivate the problem at hand. Then we will discuss in detail how the study of a DGalgebra associated to certain links may allow us to better understand the algebraic (and cluster) geometry of Richardson and positroid varieties. Explicit examples of this interplay between topology and algebraic geometry will be illustrated. At a more conceptual level, the talk brings to bear insight from symplectic topology to better understand positroid varieties. This is joint work with E. Gorsky, M. Gorsky and J. Simental.
 Video, Slides

 5 May 2021
 Travis Mandel (Oklahoma)
 Quantum theta bases for quantum cluster algebras
 One of the central goals in the study of cluster algebras is to better understand various canonical bases and positivity properties of the cluster algebras and their quantizations. GrossHackingKeelKontsevich (GHKK) applied ideas from mirror symmetry to construct socalled "theta bases" for cluster algebras which satisfy all the desired positivity properties, thus proving several conjectures regarding cluster algebras. I will discuss joint work with Ben Davison in which we combine the techniques used by GHKK with ideas from the DT theory of quiver representations to quantize the GHKK construction, thus producing quantum theta bases and proving the desired quantum positivity properties.
 Video, Slides

 29 April 2021
 Michael Wemyss (Glasgow)
 Jacobi algebras on the twoloop quiver and applications
 I will explain recent progress on classifying finite dimensional Jacobi algebras on the two loop quiver. This is a purely algebraic problem, which at first sight is both seemingly hopeless and seemingly detached from any form of reality or wider motivation. There are two surprises: first, the problem is not hopeless, and parts of the answer are in fact very beautiful. Second, this has immediate and surprising consequences to both 3fold flops and 3fold divisortocurve contractions, their curve invariants and their conjectural classification. This is joint work with Gavin Brown.
 Video, Slides

 22 April 2021
 Ben Wormleighton (Washington)
 A tale of two widths: lattice and Gromov
 To a polytope P whose facet normals are rational one can associate two geometric objects: a symplectic toric domain X_{P} and a polarised toric algebraic variety Y_{P}, which can also be viewed as a potentially singular symplectic space. A basic invariant of a symplectic manifold X is its Gromov width: essentially the size of the largest ball that can be 'symplectically' embedded in X. A conjecture of AverkovHofscheierNill proposed a combinatorial bound for the Gromov width of Y_{P}, which I recently verified in dimension two with Julian Chaidez. I’ll discuss the proof, which goes via various symplectic and algebraic invariants with winsome combinatorial interpretations in the toric case. If there’s time, I’ll discuss ongoing work and new challenges for a similar result in higher dimensions.
 Video, Slides

 15 April 2021
 Johannes Nordström (Bath)
 Extratwisted connected sum G_{2}manifolds
 The twisted connected sum construction of Kovalev produces many examples of closed Riemannian 7manifolds with holonomy group G_{2} (a special class of Ricciflat manifolds), starting from complex algebraic geometry data like Fano 3folds. If the pieces admit automorphisms, then adding an extra twist to the construction yields examples with a wider variety of topological features. I will describe the constructions and outline how one can use them to produce example of e.g. closed 7manifolds with disconnected moduli space of holonomy G_{2} metrics, or pairs of G_{2}manifolds that homeomorphic but not diffeomorphic. This is joint work with Diarmuid Crowley and Sebastian Goette.
 Video, Slides

 8 April 2021
 Tim Gräfnitz (Hamburg)
 Tropical correspondence for smooth del Pezzo log CalabiYau pairs
 In this talk I will present the main results of my thesis, a tropical correspondence theorem for log CalabiYau pairs (X,D) consisting of a smooth del Pezzo surface X of degree >=3 and a smooth anticanonical divisor D. The easiest example of such a pair is (P^{2},E), where E is an elliptic curve. I will explain how the genus zero logarithmic GromovWitten invariants of X with maximal tangency to D are related to tropical curves in the dual intersection complex of (X,D) and how they can be read off from the consistent wall structure appearing in the GrossSiebert program. The novelty in this correspondence is that D is smooth but nontoric, leading to log singularities in the toric degeneration that have to be resolved.
 Video, Slides

 1 April 2021
 JiaRui Fei (Shanghai Jiao Tong)
 Tropical Fpolynomials and Cluster Algebras
 The representationtheoretic interpretations of gvectors and Fpolynomials are two fundamental ingredients in the (additive) categorification of cluster algebras. We knew that the gvectors are related to the presentation spaces. We introduce the tropical Fpolynomial f_{M} of a quiver representation M, and explain its interplay with the general presentation for any finitedimensional basic algebra. As a consequence, we give a presentation of the Newton polytope N(M) of M. We propose an algorithm to determine the generic Newton polytopes, and show it works for path algebras. As an application, we give a representationtheoretic interpretation of FockGoncharov's cluster duality pairing. We also study many combinatorial aspects of N(M), such as faces, the dual fan and 1skeleton. We conjecture that the coefficients of a cluster monomial corresponding to vertices are all 1, and the coefficients inside the Newton polytope are saturated. We show the conjecture holds for acyclic cluster algebras. We specialize the above general results to the clusterfinite algebras and the preprojective algebras of Dynkin type.
 Video, Slides

 25 March 2021
 Michał Kapustka (IMPAN and Stavanger)
 Nikulin orbifolds
 The theory of K3 surfaces with symplectic involutions and their quotients is now a well understood classicial subject thanks to foundational works of Nikulin, and van Geemen and Sarti. In this talk we will try to develop an analogous theory in the context of hyperkahler fourfolds of K3^{[2]} type. First, we will present a latttice theoretic classification of such fourfolds which admit a symplectic involution. Then we will investigate the associated quotients that we call Nikulin orbifolds. These are orbifolds which admit a symplectic form on the smooth locus and hence are special cases of so called hyperkahler orbifolds. Finally, we shall discuss families of Nikulin orbifolds and their deformations called hyperkahler orbifolds of Nikulin type. As an application, we will provide a description of the first known example of a complete family of projective hyperkahler orbifolds. This is joint work with A. Garbagnati, C. Camere and G. Kapustka.
 Video, Slides

 18 March 2021
 Taro Sano (Kobe)
 Construction of nonKähler CalabiYau manifolds by log deformations
 CalabiYau manifolds (in the strict sense) form an important class in the classification of algebraic varieties. One can also consider its generalisation by removing the projectivity assumption. It was previously known that there are infinitely many topological types of nonKähler CalabiYau 3folds. In this talk, I will present construction of such examples in higher dimensions by smoothing normal crossing varieties. The key tools of the construction are some isomorphisms between general rational elliptic surfaces which induce isomorphisms between CalabiYau manifolds of Schoen type.
 Video, Slides

 11 March 2021
 Diane Maclagan (Warwick)
 Toric and tropical Bertini theorems in arbitrary characteristic
 The classical Bertini theorem on irreducibility when intersecting by hyperplanes is a standard part of the algebraic geometry toolkit. This was generalised recently, in characteristic zero, by Fuchs, Mantova, and Zannier to a toric Bertini theorem for subvarieties of an algebraic torus, with hyperplanes replaced by subtori. I will discuss joint work with Gandini, Hering, Mohammadi, Rajchgot, Wheeler, and Yu in which we give a different proof of this theorem that removes the characteristic assumption. An application is a tropical Bertini theorem.
 Video, Slides

 4 March 2021
 Fatemeh Rezaee (Loughborough)
 Wallcrossing does not induce MMP
 I will describe a new wallcrossing phenomenon of sheaves on the projective 3space that induces singularities that are not allowed in the sense of the Minimal Model Program (MMP). Therefore, it cannot be detected as an operation in the MMP of the moduli space, unlike the case for many surfaces.
 Slides

 2326 February 2021
 Fano Varieties and Birational Geometry
 An online workshop exploring recent developments in the geometry of Fano varieties.
 Workshop webpage

 18 February 2021
 Francesco Zucconi (Udine)
 Fujita decomposition and Massey product for fibered varieties
 Let f:X > B be a semistable fibration where X is a smooth variety of dimension n >= 2 and B is a smooth curve. We give an interpretation of the second Fujita decomposition of f_{*}ωX/B in terms of local systems of the relative 1forms and of the relative top forms. We show the existence of higher irrational pencils under natural hypothesis on local subsystems.
 Video, Slides

 11 February 2021
 Enrica Mazzon (Bonn)
 Nonarchimedean approach to mirror symmetry and to degenerations of varieties
 Mirror symmetry is a fastmoving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that complex CalabiYau manifolds should come in mirror pairs, in the sense that geometrical information of a CalabiYau manifold can be read through invariants of its mirror. In the first part of the talk, I will introduce some geometrical ideas inspired by mirror symmetry. In particular, I will go through the main steps which relate mirror symmetry to nonarchimedean geometry and the theory of Berkovich spaces. In the second part, I will describe a combinatorial object, the socalled dual complex of a degeneration of varieties. This emerges in many contexts of algebraic geometry, including mirror symmetry where moreover it comes equipped with an integral affine structure. I will show how the techniques of Berkovich geometry give a new insight into the study of dual complexes and their integral affine structure. This is based on a joint work with Morgan Brown and a work in progress with Léonard PilleSchneider.
 Video, Slides

 4 February 2021
 Pieter Belmans (Bonn)
 Hochschild cohomology of partial flag varieties and Fano 3folds
 The HochschildKostantRosenberg decomposition gives a description of the Hochschild cohomology of a smooth projective variety in terms of the sheaf cohomology of exterior powers of the tangent bundle. In all but a few cases it is a nontrivial task to compute this decomposition, and understand the extra algebraic structure which exists on Hochschild cohomology. I will give a general introduction to Hochschild cohomology and this decomposition, and explain what it looks like for partial flag varieties (joint work with Maxim Smirnov) and Fano 3folds (joint work with Enrico Fatighenti and Fabio Tanturri).
 Video, Slides

 28 January 2021
 Matej Filip (Ljubljana)
 The miniversal deformation of an affine toric Gorenstein threefold
 We are going to describe the reduced miniversal deformation of an affine toric Gorenstein threefold. The reduced deformation components correspond to special Laurent polynomials. There is canonical bijective map between the set of the smoothing components and the set of the corresponding Laurent polynomials, which we are going to analyse in more details.
 Video

 21 January 2021
 Michel van Garrel (Birmingham)
 Stable maps to Looijenga pairs
 Start with a rational surface Y admitting a decomposition of its anticanonical divisor into at least two smooth nef components. We associate five curve counting theories to this Looijenga pair: 1) all genus stable log maps with maximal tangency to each boundary component; 2) genus zero stable maps to the local CalabiYau surface obtained by twisting Y by the sum of the line bundles dual to the components of the boundary; 3) the all genus open GromovWitten theory of a toric CalabiYau threefold associated to the Looijenga pair; 4) the DonaldsonThomas theory of a symmetric quiver specified by the Looijenga pair and 5) BPS invariants associated to the various curve counting theories. In this joint work with Pierrick Bousseau and Andrea Brini, we provide closedform solutions to essentially all of the associated invariants and show that the theories are equivalent. I will start by describing the geometric transitions from one geometry to the other, then give an overview of the curve counting theories and their relations. I will end by describing how the scattering diagrams of Gross and Siebert are a natural place to count stable log maps.
 Video, Slides

 14 January 2021
 Lawrence Barrott (Boston College)
 Log geometry and Chow theory
 Log geometry has become a central tool in enumerative geometry over the past years, providing means to study many degenerations situations. Unfortunately much of the theory is complicated by the fact that products of log schemes differ from products of schemes. In this talk I will introduce a gadget which replaces Chow theory for log schemes, reproducing many familiar tools such as virtual pullback in the context of log geometry.
 Video, Slides

 10 December 2020
 Chris Eur (Stanford)
 Tautological bundles of matroids
 Recent advances in matroid theory via tropical geometry broadly fall into two themes: One concerns the Ktheory of Grassmannians, and the other concerns the intersection theory of wonderful compactifications. How do these two themes talk to each other? We introduce the notion of tautological bundles of matroids to unite these two themes. As a result, we give a geometric interpretation of the Tutte polynomial of a matroid that unifies several previous works as its corollaries, deduce new logconcavity statements, and answer few conjectures in the literature. This is an ongoing project with Andrew Berget, Hunter Spink, and Dennis Tseng.
 Video, Slides

 3 December 2020
 Tim Magee (Birmingham)
 Convexity in tropical spaces and compactifications of cluster varieties
 Cluster varieties are a relatively new, broadly interesting class of geometric objects that generalise toric varieties. Convexity is a key notion in toric geometry. For instance, projective toric varieties are defined by convex lattice polytopes. In this talk, I'll explain how convexity generalises to the cluster world, where "polytopes" live in a tropical space rather than a vector space and "convex polytopes" define projective compactifications of cluster varieties. Time permitting, I'll conclude with two exciting applications of this more general notion of convexity: 1) an intrinsic version of NewtonOkounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. Based on joint work with ManWai Cheung and Alfredo Nájera Chávez.
 Video, Slides

 26 November 2020
 Okke van Garderen (Glasgow)
 Refined DonaldsonThomas theory of threefold flops
 DT invariants are virtual counts of semistable objects in the derived category of a CalabiYau variety, which can be calculated at various levels of refinement. An interesting family of CY variety which are of particular interest to the MMP are threefold flopping curves, and in this talk I will explain how to understand their DT theory. The key point is that the stability conditions on the derived categories can be understood via tilting equivalences, which can be seen as the analogue of cluster mutations in this setting. I will show that these equivalences induce wallcrossing formulas, and use this to reduce the DT theory of a flop to a comprehensible set of curvecounting invariants, which can be computed for several examples. These computations produce new evidence for a conjecture of PandharipandeThomas, and show that refined DT invariants are not enough to completely classify flops.
 Video, Slides

 20 November 2020
 Ana PeónNieto (Birmingham/Côte d'Azur)
 Pure codimensionality of wobbly bundles
 Higgs bundles on smooth projective curves were introduced by Hitchin as solutions to gauge equations motivated by physics. They can be seen as points of T*N, where N is the moduli space of vector bundles on the curve. The topology of the moduli space of Higgs bundles is determined by the nilpotent cone, which is a reducible scheme containing the zero section of T*N>N. Inside this section, wobbly bundles are particularly important, as this is the locus where any other component intersects N. In fact, this implies that the geometry of the nilpotent cone can be described in terms of wobbly bundles. In this talk I will explain an inductive method to prove pure codimensionality of the wobbly locus, as announced in a paper by Laumon from the 80's. We expect our method to yield moreover a description of the irreducible components of the nilpotent cone in arbitrary rank.
 Slides

 19 November 2020
 Naoki Fujita (Tokyo)
 NewtonOkounkov bodies arising from cluster structures
 A toric degeneration is a flat degeneration from a projective variety to a toric variety, which can be used to apply the theory of toric varieties to other projective varieties. In this talk, we discuss relations among the following three constructions of toric degenerations: representation theory, NewtonOkounkov bodies, and cluster algebras. More precisely, we construct NewtonOkounkov bodies using cluster structures, and realize representationtheoretic and clustertheoretic toric degenerations from this framework. As an application, we connect two kinds of representationtheoretic polytopes (string polytopes and NakashimaZelevinsky polytopes) by tropicalized cluster mutations. We also discuss relations with combinatorial mutations which was introduced in the context of mirror symmetry for Fano varieties. More precisely, we relate dual polytopes of these representationtheoretic polytopes by combinatorial mutations. This talk is based on joint works with Hironori Oya and Akihiro Higashitani.
 Video, Slides

 12 November 2020
 Arkadij Bojko (Oxford)
 Orientations for DT invariants on quasiprojective CalabiYau 4folds
 DonaldsonThomas type invariants in complex dimension 4 have attracted a lot of attention in the past few years. I will give a brief overview of how one can count coherent sheaves on CalabiYau 4folds. Inherent to the definition of DT4 invariants is the notion of orientations on moduli spaces of sheaves/perfect complexes. For virtual fundamental classes and virtual structure sheaves to be welldefined, one needs to prove orientability. The result of CaoGrossJoyce does this for projective CY 4folds. However, computations are more feasible in the noncompact setting using localization formulae, where the fixed point loci inherit orientations from global ones, and orientations of the virtual normal bundles come into play. I will explain how to use real determinant line bundles of Dirac operators on the double of the original CalabiYau manifold to construct orientations on the moduli stack of compactly supported perfect complexes, moduli schemes of stable pairs and Hilbert schemes. These are controlled by choices of orientations in Ktheory and satisfy compatibility under direct sums. If time allows, I will discuss the connection between the sings obtained from comparing orientations and universal wallcrossing formulae of Joyce using vertex algebras.
 Video, Slides

 11 November 2020
 Enrico Fatighenti (Toulouse)
 Fano varieties from homogeneous vector bundles
 The idea of classifying Fano varieties using homogeneous vector bundles was behind Mukai's classification of prime Fano 3folds. In this talk, we give a survey of some recent progress along the same lines, including a biregular rework of the nonprime MoriMukai 3folds classification and some examples of higherdimensional Fano varieties with special Hodgetheoretical properties.
 Video, Slides

 5 November 2020
 Federico Barbacovi (UCL)
 Understanding the flopflop autoequivalence using spherical functors
 The homological interpretation of the Minimal Model Program conjectures that flips should correspond to embeddings of derived categories, and flops to equivalences. Even if the conjecture doesn't provide us with a preferred functor, there is an obvious choice: the pullpush via the fibre product. When this approach work, we obtain an interesting autoequivalence of either side of the flop, known as the "flopflop autoequivalence". Understanding the structure of this functor (e.g. does it split as the composition of simpler functors?) is an interesting problem, and it has been extensively studied. In this talk I will explain that there is a natural, i.e. arising from the geometry, way to realise the "flopflop autoequivalence" as the inverse of a spherical twist, and that this presentation can help us shed light on the structure of the autoequivalence itself.
 Video, Slides

 29 October 2020
 Catherine Cannizzo (Simons Center)
 Towards global homological mirror symmetry for genus 2 curves
 The first part of the talk will discuss work in arXiv:1908.04227 [math.SG] on constructing a DonaldsonFukayaSeidel type category for the generalized SYZ mirror of a genus 2 curve. We will explain the categorical mirror correspondence on the cohomological level. The key idea uses that a 4torus is SYZ mirror to a 4torus. So if we view the complex genus 2 curve as a hypersurface of a 4torus V, a mirror can be constructed as a symplectic fibration with fiber given by the dual 4torus V^. Hence on categories, line bundles on V are restricted to the genus 2 curve while fiber Lagrangians of V^ are parallel transported over Ushapes in the base of the mirror. Next we describe ongoing work with H. Azam, H. Lee, and CC. M. Liu on extending the result to a global statement, namely allowing the complex and symplectic structures to vary in their real sixdimensional families. The mirror statement for this more general result relies on work of A. Kanazawa and SC. Lau.
 Video, Slides

 22 October 2020
 Tyler Kelly (Birmingham)
 What is an exoflop?
 Aspinwall stated in 2014 that an exoflop "occurs in the gauged linear sigmamodel by varying the Kahler form so that a subspace appears to shrink to a point and then reemerge 'outside' the original manifold." This description may be intangible at first for us to sink our hands into but it turns out to be a great concrete technique that relates to many things we care about as algebraic geometers! We will interpret it in this talk. I will explain in toric geometry concretely what this means for us. Afterwards, I will explain why it’s yet another reason we should listen to our string theoretic friends. Namely, I hope to have enough time to explain how it gives us applications in mirror symmetry and derived categories. Exoflops are a recurring character in my joint work with David Favero (Alberta), Chuck Doran (Alberta), and Dan Kaplan (Birmingham).
 Video

 15 October 2020
 Daniel Kaplan (Birmingham)
 Exceptional collections for invertible polynomials using VGIT
 A sum of n monomials in n variables is said to be invertible if it is quasihomogeneous and quasismooth (i.e. it has a unique singularity at the origin). To an invertible polynomial w, one can associate a maximal symmetry group, and consider the derived category of equivariant matrix factorizations of w. Joint with David Favero and Tyler Kelly, we prove this category has a full exceptional collection, using a variation of GIT result of Ballard—Favero—Katzarkov. Our proof additionally utilizes the KreuzerSkarke classification of invertible polynomials as Thom—Sebastiani sums of Fermat, chain, and loop polynomials. I’ll present a friendly, exampleoriented illustration of our approach, review related literature, and discuss applications to mirror symmetry.
 Video, Slides

 7 October 2020
 Hiroshi Iritani (Kyoto)
 Quantum cohomology of blowups: a conjecture
 In this talk, I discuss a conjecture that a semiorthogonal decomposition of topological Kgroups (or derived categories) due to Orlov should induce a relationship between quantum
cohomology under blowups. The relationship between quantum cohomology can be described in terms of solutions to a RiemannHilbert problem.
 Slides, Handout

 1 October 2020
 Maxim Smirnov (Augsburg)
 Residual categories of Grassmannians
 Exceptional collections in derived categories of coherent sheaves have a long history going back to the pioneering work of A. Beilinson. After recalling the general setup, I will concentrate on some recent developments inspired by the homological mirror symmetry. Namely, I will define residual categories of Lefschetz decompositions and discuss a conjectural relation between the structure of quantum cohomology and residual categories. I will illustrate this relationship in the case of some isotropic Grassmannians. This is a joint work with Alexander Kuznetsov.
 Video, Slides

 24 September 2020
 Navid Nabijou (Cambridge)
 Degenerating tangent curves
 It is wellknown that a smooth plane cubic E supports 9 flex lines. In higher degrees we may ask an analogous question: "How many degree d curves intersect E in a single point?" The problem of calculating such numbers has fascinated enumerative geometers for decades. Despite being an extremely classical and concrete problem, it was not until the advent of GromovWitten invariants in the 1990s that a general method was discovered. The resulting theory is incredibly rich, and the curve counts satisfy a suite of remarkable properties, some proven and some still conjectural. In this talk I will discuss joint work with Lawrence Barrott, in which we study the behaviour of these tangent curves as the cubic E degenerates to a cycle of lines. Using the machinery of logarithmic GromovWitten theory, we obtain detailed information concerning how the tangent curves degenerate along with E. The theorems we obtain are purely classical, with no reference to GromovWitten theory, but they do not appear to admit a classical proof. No prior knowledge of GromovWitten theory will be assumed.
 Video, Slides

 17 September 2020
 Ronan Terpereau (Bourgogne)
 Actions of connected algebraic groups on rational 3dimensional Mori fibrations
 In this talk we will study the connected algebraic groups acting on Mori fibrations X > Y with X a rational threefold and Y a curve or a surface. We will see how these groups can be classified, using the minimal model program (MMP) and the Sarkisov program, and how our results make possible to recover most of the classification of the connected algebraic subgroups of the Cremona group Bir(P^{3}) obtained by Hiroshi Umemura in the 1980's when the base field is the field of complex numbers.
 Video, Slides

 10 September 2020
 Lara Bossinger (Oaxaca)
 Families of Gröbner degenerations, Grassmannians, and universal cluster algebras
 Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over affine mspace that assembles the Gröbner degenerations of V associated with all faces of C. This is a multiparameter generalization of the classical oneparameter Gröbner degeneration associated to a weight. We show that our family can be constructed from KavehManon's recent work on the classification of toric flat families over toric varieties: it is the pullback of a toric family defined by a Rees algebra with base X_{C} (the toric variety associated to C) along the universal torsor A^{m} > X_{C}. If time permits I will explain how to apply this construction to the Grassmannians Gr(2,n) (with Plücker embedding) and Gr(3,6) (with "cluster embedding"). In each case there exists a unique maximal Gröbner cone whose associated initial ideal is the StanleyReisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for Gr(2,n) we show how EscobarHarada's mutation of NewtonOkounkov bodies can be recovered as tropicalized cluster mutation. This is joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez.
 Video, Slides

 4 September 2020
 Andrew Harder (Lehigh)
 Log symplectic pairs and mixed Hodge structures
 A log symplectic pair is a pair (X,Y) consisting of a smooth projective variety X and a divisor Y in X so that there is a nondegenerate log 2form on X with poles along Y. I will discuss the relationship between log symplectic pairs and degenerations of hyperkaehler varieties, and how this naturally leads to a class of log symplectic pairs called log symplectic pairs of "pure weight". I will discuss results which show that the classification of log symplectic pairs of pure weight is analogous to the classification of log CalabiYau surfaces. Time permitting, I'll discuss two classes of log symplectic pairs which are related to real hyperplane arrangements and which admit cluster type structures.
 Video, Slides

 3 September 2020
 Renato Vianna (Rio de Janeiro)
 Sharp ellipsoid embeddings and almosttoric mutations
 We will show how to construct volume filling ellipsoid embeddings in some 4dimensional toric domain using mutation of almost toric compactification of those. In particular we recover the results of
McDuffSchlenk for the ball, FenkelMüller for product of symplectic disks and CristofaroGardiner for E(2,3), giving a more explicit geometric perspective for these results. To be able to represent certain divisors, we develop the idea of symplectic tropical curves in almost toric fibrations, inspired by Mikhalkin's work for tropical curves. This is joint work with Roger Casals.
 Video, Slides

 27 August 2020
 Andrea Petracci (FU Berlin)
 Kmoduli stacks and Kmoduli spaces are singular
 Only recently a separated moduli space for (some) Fano varieties has been constructed by several algebraic geometers: this is the Kmoduli stack which parametrises Ksemistable Fano varieties and has a separated good moduli space. A natural question is: are these stacks and spaces smooth? This question makes sense because deformations of smooth Fano varieties are unobstructed, so moduli stacks of smooth Fano varieties are smooth. In this talk I will explain how to use toric geometry to construct examples of nonsmooth points in the Kmoduli stack and the Kmoduli space of Fano 3folds. This is joint work with AnneSophie Kaloghiros.
 Video, Slides

 20 August 2020
 ManWai "Mandy" Cheung (Harvard)
 Polytopes, wall crossings, and cluster varieties
 Cluster varieties are log CalabiYau varieties which are a union of algebraic tori glued by birational "mutation" maps. Partial compactifications of the varieties, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties. However, it is not clear from the definitions how to characterize the polytopes giving compactifications of cluster varieties. We will show how to describe the compactifications easily by broken line convexity. As an application, we will see the nonintegral vertex in the Newton Okounkov body of Gr(3,6) comes from broken line convexity. Further, we will also see certain positive polytopes will give us hints about the Batyrev mirror in the cluster setting. The mutations of the polytopes will be related to the almost toric fibration from the symplectic point of view. Finally, we can see how to extend the idea of gluing of tori in Floer theory which then ended up with the Family Floer Mirror for the del Pezzo surfaces of degree 5 and 6. The talk will be based on a series of joint works with Bossinger, Lin, Magee, NajeraChavez, and Vienna.
 Video, Slides

 13 August 2020
 Nathan Ilten (Simon Fraser)
 Type D associahedra are unobstructed
 Generalised associahedra associated to any root system were introduced by Fomin and Zelevinsky in their study of cluster algebras. For type A root systems, one recovers the classical associahedron parametrizing triangulations of a regular ngon. For type D root systems, one obtains a polytope parametrizing centrally symmetric triangulations of a 2ngon. In previous work, Jan Christophersen and I showed that the StanleyReisner ring of the simplicial complex dual to the boundary of the classical associahedron is unobstructed, that is, has vanishing second cotangent cohomology. This could be used to find toric degenerations of the Grassmannian G(2,n). In this talk, I will describe workinprogress that generalizes this unobstructedness result to the type D associahedron.
 Video, Slides

 6 August 2020
 YangHui He (City and Oxford)
 Universes as big data: superstrings, CalabiYau manifolds and machinelearning
 We review how historically the problem of string phenomenology lead theoretical physics first to algebraic/diffenretial geometry, and then to computational geometry, and now to data science and AI. With the concrete playground of the CalabiYau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machinelearning can help explore problems of physical and mathematical interest.
 Video, Slides

 30 July 2020
 Benjamin Braun (Kentucky)
 The integer decomposition property and Ehrhart unimodality for weighted projective space simplices
 We consider lattice simplices corresponding to weighted projective spaces where one of the weights is 1. We study the integer decomposition property and Ehrhart unimodality for such simplices by focusing on restrictions regarding the multiplicity of each weight. We introduce a necessary condition for when a simplex satisfies the integer decomposition property, and classify those simplices that satisfy it in the case where there are no more than three distinct weights. We also introduce the notion of reflexive stabilizations of a simpex of this type, and show that higherorder reflexive stabilizations fail to be Ehrhart unimodal and fail to have the integer decomposition property. This is joint work with Robert Davis, Morgan Lane, and Liam Solus.
 Video, Slides

 24 July 2020
 Elana Kalashnikov (Harvard)
 Constructing Laurent polynomial mirrors for quiver flag zero loci
 All smooth Fano varieties of dimension at most three can be constructed as either toric complete intersections (subvarieties of toric varieties) or quiver ﬂag zero loci (subvarieties of quiver ﬂag varieties). Conjecturally, Fano varieties are expected to mirror certain Laurent polynomials. The construction of mirrors of Fano toric complete intersections is wellunderstood. In this talk, I'll discuss evidence for this conjecture by proposing a method of constructing mirrors for Fano quiver flag zero loci. A key step of the construction is via ﬁnding toric degenerations of the ambient quiver ﬂag varieties. These degenerations generalise the GelfandCetlin degeneration of flag varieties, which in the Grassmannian case has an important role in the cluster structure of its coordinate ring.
 Video, Slides

 16 July 2020
 Hendrik Süß (Manchester)
 Normalised volumes of singularities
 The notion of the normalised volume of a singularity has been introduced relatively recently, but plays a crucial role in the context of Einstein metrics and Kstability. After introducing this invariant my plan is to specialise quickly to the case of toric singularities and show that even in this relatively simple setting interesting phenomena occur.
 Video, Slides

 15 July 2020
 Chris Lazda (Warwick)
 A NeronOggShafarevich criterion for K3 surfaces
 The naive analogue of the NéronOggShafarevich criterion fails for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified etale cohomology groups, but which do not admit good reduction over K. Assuming potential semistable reduction, I will show how to correct this by proving that a K3 surface has good reduction if and only if is second cohomology is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain "canonical reduction" of X. This is joint work with B. Chiarellotto and C. Liedtke.
 Video, Slides

 8 July 2020
 Ed Segal (UCL)
 Semiorthogonal decompositions and discriminants
 The derived category of a toric variety can usually be decomposed into smaller pieces, by passing through different birational models and applying the "windows" theory relating VGIT and derived categories. There are many choices involved and the decompositions are not unique. We prove a JordanHolder result, that the multiplicities of the pieces are independent of choices. If the toric variety is CalabiYau then there are no decompositions, instead the theory produces symmetries of the derived category. Physics predicts that these all these symmetries form an action of the fundamental group of the "FI parameter space". I'll explain why our JordanHolder result is necessary for this prediction to work, and state a conjecture (based on earlier work of AspinwallPlesserWang) relating our multiplicities to the geometry of the FI parameter space. This is joint work with Alex Kite.
 Video, Slides

 2 July 2020
 Gregory Smith (Queen’s)
 Geometry of smooth Hilbert schemes
 How can we understand the subvarieties of a fixed projective space? Hilbert schemes provide the geometric answer to this question. After surveying some features of these natural parameter spaces, we will classify the smooth Hilbert schemes. Time permitting, we will also describe the geometry of nonsingular Hilbert schemes by interpreting them as suitable generalisations of partial flag varieties. This talk is based on joint work with Roy Skjelnes (KTH).
 Video, Slides

 25 June 2020
 Klaus Altmann (FU Berlin)
 Displaying the cohomology of toric line bundles
 Line bundles L on projective toric varieties can be understood as formal differences (Δ^{+}Δ^{}) of convex polyhedra in the character lattice. We show how it is possible to use this language for understanding the cohomology of L by studying the settheoretic difference (Δ^{}\Δ^{+}). Moreover, when interpreting these cohomology groups as certain Extgroups, we demonstrate how the approach via (Δ^{}\Δ^{+}) leads to a direct description of the associated extensions. The first part is joint work with Jarek Buczinski, Lars Kastner, David Ploog, and AnnaLena Winz; the second is work in progress with Amelie Flatt.
 Video

 18 June 2020
 Leonid Monin (Bristol)
 Inversion of matrices, a C* action on Grassmannians and the space of complete quadrics
 Let Γ be the closure of the set of pairs (A,A^{1}) of symmetric matrices of size n. In other words, Γ is the graph of the inversion map on the space Sym_{n} of symmetric matrices of size n. What is the cohomology class of Γ in the product of projective spaces? Equivalently, what is the degree of the variety L^{1} obtained as the closure of the set of inverses of matrices from a generic linear subspace L of Sym_{n}? This question is interesting in its own right but it is also motivated by algebraic statistics. In my talk, I will explain how to invert a matrix using a C* action on Grassmannians, relate the above question to classical enumerative problems about quadrics, and give several possible answers. This is joint work in progress with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, Andrzej Weber, and Jaroslaw A. Wisniewski.
 Video

 11 June 2020
 Karin Schaller (FU Berlin)
 Polyhedral Divisors and Orbit Decompositions of Normal Affine Varieties
with Torus Action
 Normal affine varieties of dimension n with an effective action of a kdimensional algebraic torus can be described completely in terms of proper polyhedral divisors living on semiprojective varieties of dimension n−k. We use the language of polyhedral divisors to study the collection of Torbits and Torbit closures of a normal affine Tvariety in terms of its defining ppdivisor. This is based on previous work of Klaus Altmann and Jürgen Hausen complemented by work in progress with Klaus Altmann.
 Video, Slides

 4 June 2020
 Giuliano Gagliardi (Hannover and MPI Bonn)
 The ManinPeyre conjecture for smooth spherical Fano varieties of semisimple rank one
 The ManinPeyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higherdimensional smooth spherical Fano varieties. This is joint work with Valentin Blomer, Jörg Brüdern, and Ulrich Derenthal.
 Video

 28 May 2020
 Tom Sutherland (Lisbon)
 Mirror symmetry for Painlevé surfaces
 This talk will survey aspects of mirror symmetry for ten families of noncompact hyperkähler manifolds on which the dynamics of one of the Painlevé equations is naturally defined. They each have a pair of natural realisations: one as the complement of a singular fibre of a rational elliptic surface and another as the complement of a triangle of lines in a (singular) cubic surface. The two realisations relate closely to a space of stability conditions and a cluster variety of a quiver respectively, providing a perspective on SYZ mirror symmetry for these manifolds.
 Video, Slides

 21 May 2020
 Jesús Martinez Garcia (Essex)
 The moduli continuity method for log Fano pairs
 The moduli continuity method, pioneered by Odaka, Spotti and Sun, allows us to explicitly provide algebraic charts of the GromovHausdorff compactification of (possibly singular) KählerEinstein metrics. Assuming we can provide a homeomorphism to some 'known' algebraic compactification (customarily, a GIT one) the method allows us to determine which Fano varieties (or more generally log Fano pairs) are Kpolystable in a given deformation family. In this talk we provide the first examples of compactification of the moduli of log Fano pairs for the simplest deformation family: that of projective space and a hypersurface, and mention related results for cubic surfaces. This is joint work with Patricio Gallardo and Cristiano Spotti.
 Video, Slides

 19 May 2020
 Timothy Logvinenko (Cardiff)
 Skeintriangulated representations of generalised braids
 Ordinary braid group Br_{n} is a wellknown algebraic structure which encodes configurations of n nontouching strands ("braids") up to continuous transformations ("isotopies"). A classical result of Khovanov and Thomas states that there is a natural categorical action of Br_{n} on the derived category of the cotangent bundle of the variety of complete flags in C^{n}. In this talk, I will introduce a new structure: the category GBr_{n} of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be noninvertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skeintriangulated representation of GBr_{n}. A decadeold conjecture states that there a skeintriangulated action of GBr_{n} on the cotangent bundles of the varieties of full and partial flags in C^{n}. We prove this conjecture for n = 3. We also show that any categorical action of Br_{n} can be lifted to a skeintriangulated action of GBr_{n}, which behaves like a categorical nil Hecke algebra. This is a joint work with Rina Anno and Lorenzo De Biase.
 Video

 14 May 2020
 Alan Thompson (Loughborough)
 Threefolds fibred by sextic double planes
 I will discuss the theory of threefolds fibred by K3 surfaces mirror to the sextic double plane. This theory is unexpectedly rich, in part due to the presence of a polarisationpreserving involution on such K3 surfaces. I will present an explicit birational classification result for such threefolds, along with computations of several of their basic invariants. Along the way we will uncover several (perhaps) surprising links between this theory and Kodaira's theory of elliptic surfaces. This is joint work with Remkes Kooistra.
 Video

 13 May 2020
 Tom Ducat (Imperial)
 A Laurent phenomenon for OGr(5,10) and explicit mirror symmetry for the Fano 3fold V_{12}
 The 5periodic birational map (x, y) > (y, (1+y)/x) can be interpreted as a mutation between five open torus charts in a del Pezzo surface of degree 5, coming from a cluster algebra structure on the Grassmannian Gr(2,5). This can used to construct a rational elliptic fibration which is the LandauGinzburg mirror to dP_{5}. I will briefly recap this, and then explain the following 3dimensional generalisation: the 8periodic birational map (x, y, z) > (y, z, (1+y+z)/x) can be used to exhibit a Laurent phenomenon for the orthogonal Grassmannian OGr(5,10) and construct a completely explicit K3 fibration which is mirror to the Fano 3fold V_{12}, as well as some other Fano 3folds.
 Video, Slides

 6 May 2020
 Livia Campo (Nottingham)
 On a high pliability quintic hypersurface
 In this talk we exhibit an example of a quintic hypersurface with a certain compound singularity that has pliability at least 2. We also show that, while a nontrivial sequence of birational transformations can be constructed between the two elements of the pliability set, the Sarkisov link connecting them is not evident. This is done by studying birational links for codimension 4 index 1 Fano 3folds having Picard rank 2.

 30 April 2020
 Florian Kohl (Aalto)
 Unconditional reflexive polytopes
 A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this talk, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterise unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study a typeB analogue of the Birkhoff polytope. This talk is based on joint work with McCabe Olsen and Raman Sanyal.
 Slides

 16 April 2020
 Alessio Corti (Imperial)
 Volume preserving birational selfmaps of P^{3}
 I describe some results on the structure of the group Bir(P^{3}, D) where D is a quartic surface with mild singularities. Work with Carolina Araujo and Alex Massarenti.

 8 April 2020
 Thomas Prince (Oxford)
 Perturbing torus fibrations on threefold singularities
 Fix a QGorenstein threefold toric singularity X determined by a rational polyhedral cone C in N_{R}, together with a collection of piecewise linear maps (or combinatorial mutations) T_{1},...,T_{k} such that the image of C* under the composition of these linear maps is a half space in R^{3}. We describe how to perturb the torus fibration X to C*, whose fibres are orbits of the (toric) T^{3} action on X, to a torus fibration on a space X' which is a manifold away from a finite collection of singular points. Around each of these singular points X' has the structure of a terminal cyclic quotient singularity. We outline how to globalise this to construct torus fibrations over 3dimensional balls which correspond to (partially) smoothing a threedimensional toric Fano variety to a Fano variety with cyclic quotient terminal singularities. The combinatorial input to this process is closely related to the notion of zeromutable Laurent polynomial which has been recently studied by Corti, Kasprzyk, and Pitton.

 2 April 2020
 Alice Cuzzucoli (Warwick)
 A glimpse at the classification of orbifold del Pezzo surfaces
 In this talk, we will discuss the main ingredients involved in the classification of del Pezzo surfaces with orbifold points, i.e. complex projective varieties of dimension two admitting log terminal singularities.
In the smooth case, we have a wellknown birational classification dating back to the 19th century. In the singular case, we are still missing a classification just as complete. Nevertheless, in the case of cyclic quotient singularities, we have some interesting constructions.
We will introduce the most crucial aspects of such constructions, which are divided into three main steps: firstly, by analysing the graded rings of such surfaces, we can find a bound on the number of singularities and the relative invariants; secondly, with the help of Mori Theory, we can give a first representation of our birational models; then, by having a brief look at the toric case, we will describe how toric degenerations come into play in this classification.
Ultimately, we can recreate analogous constructions to the cascade of blow ups for the smooth case with the representatives of specific deformation classes of our orbifolds.
 Slides
Past Reading Groups

 24 August  30 November 2020, Mondays
 Tropical Combinatorics and Geometry
 A weekly reading group working through M. Joswig's book draft "Essentials of Tropical Combinatorics". In addition, we will cover sections from "Brief Introduction to Tropical Geometry" (E. Brugalle, I. Itenberg, G. Mikhalkin, K. Shaw) and "Tropical Data Science" (R. Yoshida). Material presented by MarieCharlotte Brandenburg, Giulia Codenotti, Maria Dostert, Danai Deligeorgaki, Girtrude Hamm, Aryaman Jal, Katharina Jochemko, Florian Kohl, Fatemeh Mohammadi, Leonid Monin, Petter Restadh, Felix Rydell, and Leonardo Saud.
 Note: This reading group is recognised as an official seminar worth 7.5 credits for students at KTH. For further information, please visit the reading group's KTH webpage.

 1 September  24 November 2020, Tuesdays
 Birational Geometry
 A weekly reading group working through K. Matsuki's book "Introduction to the Mori Program". Material presented by Livia Campo, Lucas Das Dores, Giuliano Gagliardi, Tiago Guerreiro, Thomas Hall, Johannes Hofscheier, Jesus MartinezGarcia, Leonid Monin, and Theodoros Papazachariou.

 31 March  7 July 2020, Tuesdays
 Geometry of Numbers
 A weekly reading group working through Siegel's "Lectures on the Geometry of Numbers". Material presented by Livia Campo, Daniel Cavey, Giulia Codenotti, Oliver Daisey, Thomas Hall, Johannes Hofscheier, Katharina Jochemko, and Leonid Monin.