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 up-to-date 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, 10-11am
  • 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 i-th successive Seshadri constant is proportional to the i-th successive minima of a suitable 0-symmetric convex body. Based on joint work with Atsushi Ito.
  • Link to Event
• • 3 February 2022, 10-11am
  • Wendelin Lutz (Imperial)
  • A geometric proof of the classification of T-polygons
  • One formulation of mirror symmetry predicts (omitting a few adjectives) a one-to-one 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 T-polygons and have been classified by Kasprzyk-Nill-Prince 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, 10-11am
  • Ananyo Dan (Sheffield)
  • McKay correspondence for isolated Gorenstein singularities
  • The McKay correspondence is a (natural) correspondence between the (non-trivial) 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 C²//G. A geometric construction for this correspondence was given by González-Sprinberg 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 Cⁿ by a finite subgroup of SL(n,C)) by Ito-Reid, where the above sets were substituted by certain smaller subsets. It was further generalised to more general quotient singularities by Bridgeland-King-Reid, Iyama-Wemyss 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. Romano-Velazquez.
  • Link to Event
• • 17 February 2022, 10-11am
  • Qaasim Shafi (Imperial)
  • Logarithmic Toric Quasimaps
  • Quasimaps provide an alternate curve counting system to Gromov-Witten theory, related by wall-crossing formulas. Relative (or logarithmic) Gromov-Witten theory has proved useful for constructions in mirror symmetry, as well as for determining ordinary Gromov-Witten 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 so-called Gromov-Witten theory. This interplay between Hurwitz and Gromov-Witten 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 Gromov-Witten theory is of great interest. So-called 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 Gromov-Witten theory by combinatorial methods of tropical geometry, and whose long-term 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 A-discriminant (the closure of the set of all non-smooth 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 non-Morse if viewed as polynomial maps. Namely, for a 1-dimensional support set A, there is a surjection from the set of all possible combinatorial types of so-called 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 pseudo-reflection 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 pseudo-reflection 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
  • Landau-Ginzburg 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 mirror-symmetric 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 non-trivial fact that "non-realizable" 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 zero-dimensional tropical ideals are not realizable.
  • Video, Slides
• • 28 October 2021
  • Andrea Brini (Sheffield)
  • Quantum geometry of log-Calabi Yau surfaces
  • A log-Calabi 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 physics-motivated 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 3-fold 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 closed-form solvable. Based on joint works with P. Bousseau, M. van Garrel, and Y. Schueler.
  • Video, Slides
• • 21 October 2021
  • Anne-Sophie Kaloghiros (Brunel)
  • The Calabi problem for Fano 3-folds
  • I will discuss progress on the Calabi problem for Fano 3-folds. The 105 deformation families of smooth Fano 3-folds, were classified by Iskovskikh, Mori and Mukai. We determine whether or not the general member of each of these 105 families admits a Kähler-Einstein metric. In some cases, it is known that while the general member of the family admits a Kähler-Einstein metric, some other member does not. This leads to the problem of determining which members of a deformation family admit a Kähler-Einstein 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, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Jesus Martinez-Garcia, 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 2n-6 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 well-known 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 Donten-Bury–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 3-folds
  • 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 Calabi-Yau 3-folds 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 1-1 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 lambda-connections 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)
  • Log-concavity of matroid h-vectors 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 Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the h-vector 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 Montgomery-Yang 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 Gromov-Witten 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 Gromov-Witten 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 K-stability, 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 P¹xP¹ branched over a (4,4) curve. We will show that K-stability provides a natural way to interpolate between the GIT moduli space and the Baily-Borel 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 (PUC-Rio and HSE)
  • Graph potentials and combinatorial non-abelian Torelli
  • I will introduce graph potentials and discuss some of their combinatorial aspects, such as small resolution conjecture and combinatorial non-abelian 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 Hacon-McKernan, Takayama, and Tsuji, there is a constant r_n such that for every r at least r_n, the r-canonical map of every n-dimensional 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 n-fold with ample canonical class whose volume is less than 1/2²ⁿ. 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 Gonzalez-Aguilera 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 two-torus, 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)
  • Wall-crossing phenomenon for Newton-Okounkov bodies
  • A Newton-Okounkov 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 Kaveh-Manon gives an explicit link between tropical geometry and Newton-Okounkov bodies. We use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies. As an example, we describe wall-crossing formula in the case of the Grassmannian Gr(2,m). This is joint work with Megumi Harada.
  • Video, Slides
• • 10 June 2021
  • Anne Lonjou (Paris-Saclay)
  • 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 non-abelian Hodge correspondence on a toric variety X. There is a natural 1-1 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 yet-to-be-discovered logarithmic incarnation of the non-abelian 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 non-toric 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 Gross-Siebert'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 non-Archimedean 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 DG-algebra 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. Gross-Hacking-Keel-Kontsevich (GHKK) applied ideas from mirror symmetry to construct so-called "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 two-loop 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 3-fold flops and 3-fold divisor-to-curve 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 Averkov-Hofscheier-Nill 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)
  • Extra-twisted connected sum G₂-manifolds
  • The twisted connected sum construction of Kovalev produces many examples of closed Riemannian 7-manifolds with holonomy group G₂ (a special class of Ricci-flat manifolds), starting from complex algebraic geometry data like Fano 3-folds. 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 7-manifolds with disconnected moduli space of holonomy G₂ metrics, or pairs of G₂-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 Calabi-Yau pairs
  • In this talk I will present the main results of my thesis, a tropical correspondence theorem for log Calabi-Yau 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 (P²,E), where E is an elliptic curve. I will explain how the genus zero logarithmic Gromov-Witten 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 Gross-Siebert program. The novelty in this correspondence is that D is smooth but non-toric, leading to log singularities in the toric degeneration that have to be resolved.
  • Video, Slides
• • 1 April 2021
  • JiaRui Fei (Shanghai Jiao Tong)
  • Tropical F-polynomials and Cluster Algebras
  • The representation-theoretic interpretations of g-vectors and F-polynomials are two fundamental ingredients in the (additive) categorification of cluster algebras. We knew that the g-vectors are related to the presentation spaces. We introduce the tropical F-polynomial f_M of a quiver representation M, and explain its interplay with the general presentation for any finite-dimensional 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 representation-theoretic interpretation of Fock-Goncharov's cluster duality pairing. We also study many combinatorial aspects of N(M), such as faces, the dual fan and 1-skeleton. 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 cluster-finite 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 non-Kähler Calabi-Yau manifolds by log deformations
  • Calabi-Yau 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 non-Kähler Calabi-Yau 3-folds. 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 Calabi-Yau 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)
  • Wall-crossing does not induce MMP
  • I will describe a new wall-crossing phenomenon of sheaves on the projective 3-space 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
• • 23-26 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 1-forms 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)
  • Non-archimedean approach to mirror symmetry and to degenerations of varieties
  • Mirror symmetry is a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that complex Calabi-Yau manifolds should come in mirror pairs, in the sense that geometrical information of a Calabi-Yau 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 non-archimedean geometry and the theory of Berkovich spaces. In the second part, I will describe a combinatorial object, the so-called 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 Pille-Schneider.
  • Video, Slides
• • 4 February 2021
  • Pieter Belmans (Bonn)
  • Hochschild cohomology of partial flag varieties and Fano 3-folds
  • The Hochschild-Kostant-Rosenberg 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 non-trivial 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 3-folds (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 Calabi-Yau surface obtained by twisting Y by the sum of the line bundles dual to the components of the boundary; 3) the all genus open Gromov-Witten theory of a toric Calabi-Yau threefold associated to the Looijenga pair; 4) the Donaldson-Thomas 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 closed-form 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 K-theory 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 log-concavity 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 Newton-Okounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. Based on joint work with Man-Wai Cheung and Alfredo Nájera Chávez.
  • Video, Slides
• • 26 November 2020
  • Okke van Garderen (Glasgow)
  • Refined Donaldson-Thomas theory of threefold flops
  • DT invariants are virtual counts of semistable objects in the derived category of a Calabi-Yau 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 wall-crossing formulas, and use this to reduce the DT theory of a flop to a comprehensible set of curve-counting invariants, which can be computed for several examples. These computations produce new evidence for a conjecture of Pandharipande-Thomas, and show that refined DT invariants are not enough to completely classify flops.
  • Video, Slides
• • 20 November 2020
  • Ana Peón-Nieto (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)
  • Newton-Okounkov 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, Newton-Okounkov bodies, and cluster algebras. More precisely, we construct Newton-Okounkov bodies using cluster structures, and realize representation-theoretic and cluster-theoretic toric degenerations from this framework. As an application, we connect two kinds of representation-theoretic polytopes (string polytopes and Nakashima-Zelevinsky 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 representation-theoretic 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 quasi-projective Calabi-Yau 4-folds
  • Donaldson-Thomas 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 Calabi-Yau 4-folds. 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 well-defined, one needs to prove orientability. The result of Cao-Gross-Joyce does this for projective CY 4-folds. However, computations are more feasible in the non-compact 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 Calabi-Yau 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 K-theory and satisfy compatibility under direct sums. If time allows, I will discuss the connection between the sings obtained from comparing orientations and universal wall-crossing 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 3-folds. In this talk, we give a survey of some recent progress along the same lines, including a biregular rework of the non-prime Mori-Mukai 3-folds classification and some examples of higher-dimensional Fano varieties with special Hodge-theoretical properties.
  • Video, Slides
• • 5 November 2020
  • Federico Barbacovi (UCL)
  • Understanding the flop-flop 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 pull-push via the fibre product. When this approach work, we obtain an interesting autoequivalence of either side of the flop, known as the "flop-flop 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 "flop-flop 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 Donaldson-Fukaya-Seidel 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 4-torus is SYZ mirror to a 4-torus. So if we view the complex genus 2 curve as a hypersurface of a 4-torus V, a mirror can be constructed as a symplectic fibration with fiber given by the dual 4-torus V^. Hence on categories, line bundles on V are restricted to the genus 2 curve while fiber Lagrangians of V^ are parallel transported over U-shapes in the base of the mirror. Next we describe ongoing work with H. Azam, H. Lee, and C-C. M. Liu on extending the result to a global statement, namely allowing the complex and symplectic structures to vary in their real six-dimensional families. The mirror statement for this more general result relies on work of A. Kanazawa and S-C. 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 sigma-model 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 quasi-homogeneous and quasi-smooth (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 Kreuzer-Skarke classification of invertible polynomials as Thom—Sebastiani sums of Fermat, chain, and loop polynomials. I’ll present a friendly, example-oriented illustration of our approach, review related literature, and discuss applications to mirror symmetry.
  • Video, Slides
• • 7 October 2020
  • Hiroshi Iritani (Kyoto)
  • Quantum cohomology of blow-ups: a conjecture
  • In this talk, I discuss a conjecture that a semiorthogonal decomposition of topological K-groups (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 Riemann-Hilbert 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 well-known 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 Gromov-Witten 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 Gromov-Witten 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 Gromov-Witten theory, but they do not appear to admit a classical proof. No prior knowledge of Gromov-Witten theory will be assumed.
  • Video, Slides
• • 17 September 2020
  • Ronan Terpereau (Bourgogne)
  • Actions of connected algebraic groups on rational 3-dimensional 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(P³) 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 m-space that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We show that our family can be constructed from Kaveh-Manon'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 Stanley-Reisner 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 Escobar-Harada's mutation of Newton-Okounkov 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 non-degenerate log 2-form 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 Calabi-Yau 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 almost-toric mutations
  • We will show how to construct volume filling ellipsoid embeddings in some 4-dimensional toric domain using mutation of almost toric compactification of those. In particular we recover the results of McDuff-Schlenk for the ball, Fenkel-Müller for product of symplectic disks and Cristofaro-Gardiner 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)
  • K-moduli stacks and K-moduli spaces are singular
  • Only recently a separated moduli space for (some) Fano varieties has been constructed by several algebraic geometers: this is the K-moduli stack which parametrises K-semistable 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 non-smooth points in the K-moduli stack and the K-moduli space of Fano 3-folds. This is joint work with Anne-Sophie Kaloghiros.
  • Video, Slides
• • 20 August 2020
  • Man-Wai "Mandy" Cheung (Harvard)
  • Polytopes, wall crossings, and cluster varieties
  • Cluster varieties are log Calabi-Yau 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 non-integral 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, Najera-Chavez, 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 n-gon. For type D root systems, one obtains a polytope parametrizing centrally symmetric triangulations of a 2n-gon. In previous work, Jan Christophersen and I showed that the Stanley-Reisner 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 work-in-progress that generalizes this unobstructedness result to the type D associahedron.
  • Video, Slides
• • 6 August 2020
  • Yang-Hui He (City and Oxford)
  • Universes as big data: superstrings, Calabi-Yau manifolds and machine-learning
  • 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 Calabi-Yau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning 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 higher-order 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 well-understood. 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 Gelfand-Cetlin 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 K-stability. 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 Neron-Ogg-Shafarevich criterion for K3 surfaces
  • The naive analogue of the Néron-Ogg-Shafarevich 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 semi-stable 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)
  • Semi-orthogonal 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 Jordan-Holder result, that the multiplicities of the pieces are independent of choices. If the toric variety is Calabi-Yau 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 Jordan-Holder result is necessary for this prediction to work, and state a conjecture (based on earlier work of Aspinwall-Plesser-Wang) 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 set-theoretic difference (Δ^-\Δ⁺). Moreover, when interpreting these cohomology groups as certain Ext-groups, 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 Anna-Lena 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 k-dimensional 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 T-orbits and T-orbit closures of a normal affine T-variety in terms of its defining pp-divisor. 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 Manin-Peyre conjecture for smooth spherical Fano varieties of semisimple rank one
  • The Manin-Peyre 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 higher-dimensional 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 non-compact 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 Gromov-Hausdorff compactification of (possibly singular) Kähler-Einstein 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 K-polystable 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)
  • Skein-triangulated representations of generalised braids
  • Ordinary braid group Br_n is a well-known algebraic structure which encodes configurations of n non-touching 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 Cⁿ. 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 non-invertible, 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 skein-triangulated representation of GBr_n. A decade-old conjecture states that there a skein-triangulated action of GBr_n on the cotangent bundles of the varieties of full and partial flags in Cⁿ. We prove this conjecture for n = 3. We also show that any categorical action of Br_n can be lifted to a skein-triangulated 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 polarisation-preserving 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 3-fold V₁₂
  • The 5-periodic 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 Landau-Ginzburg mirror to dP₅. I will briefly recap this, and then explain the following 3-dimensional generalisation: the 8-periodic 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 3-fold V₁₂, as well as some other Fano 3-folds.
  • 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 non-trivial 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 3-folds 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 type-B 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 P³
  • I describe some results on the structure of the group Bir(P³, 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 Q-Gorenstein 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) T₁,...,T_k such that the image of C* under the composition of these linear maps is a half space in R³. We describe how to perturb the torus fibration X to C*, whose fibres are orbits of the (toric) T³ 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 3-dimensional balls which correspond to (partially) smoothing a three-dimensional 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 zero-mutable 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 well-known 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 Marie-Charlotte 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 Martinez-Garcia, ​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.