Geometry section
Invited speakers:
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Programme (all talks are in Abacws/0.01):
Mon 30th March, 2026
| 3:00pm – 4:00pm | Dhruv Ranganathan (Cambridge) |
| “The LMNOP Conjecture” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 5:00pm | Lukas Bertsch (Vienna) |
| “Variation of crepant resolutions of Kleinian singularities” (abstract) | |
| 5:00pm – 6:00pm | Hülya Argüz (Oxford) |
| “b-Complex Manifolds with Generalized Corners and Kato–Nakayama Spaces” (abstract) |
Tue 31st March, 2026
| 3:00pm – 4:00pm | Konstanze Rietsch (KCL) |
| “On mirror symmetry for Schubert varieties” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 5:00pm | Peize Liu (Warwick) |
| “Moduli spaces from the Kuznetsov component of cubic fivefolds” (abstract) | |
| 5:00pm – 6:00pm | Enrico Fatighenti (Bologna) |
| “Modular vector bundles on hyperkähler manifolds” (abstract) |
Wed 1st April, 2026
| 3:00pm – 4:00pm | Robert Hanson (Imperial) |
| “Parabolic induction for Higgs bundles” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 5:00pm | Heath Pearson (Nottingham) |
| “The Mukai conjecture for spherical varieties” (abstract) | |
| 5:00pm – 6:00pm | Piotr Pokora (Krakow) |
| “Curves and surfaces with extreme properties” (abstract) |
Thu 2nd April, 2026
| 1:30pm – 2:30pm | Vladimir Baranovsky (UC Irvine) |
| “Geometry of curved Maurer-Cartan functors” (abstract) | |
| 3:00pm – 4:00pm | Alastair Craw (Bath) |
| “Deformations of Kleinian singularities and their Hilbert schemes” (abstract) | |
| 4:00pm – 4:30pm | Marc Truter (Warwick) |
| “Fano 4-fold hypersurfaces” (abstract) |
Geometry section organisers:
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...with some help of:
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Abstracts:
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Hülya Argüz (Oxford): b-Complex Manifolds with Generalized Corners and Kato–Nakayama Spaces
Manifolds with generalized corners (g-corners), introduced by Joyce, are spaces that locally look like rational polyhedral cones. We describe the notion of b-complex manifolds with g-corners, and establish an analogue of the (formal) Newlander–Nirenberg theorem ensuring the existence of local complex coordinates on such manifolds. We also show that Kato–Nakayama spaces associated to log smooth complex analytic spaces naturally carry the structure of a b-complex manifold with g-corners. Conversely, we provide necessary and sufficient conditions characterizing which b-complex manifolds with g-corners arise as Kato–Nakayama spaces. This establishes a bridge between differential-geometric and algebro-geometric perspectives on spaces with singular boundary behavior. This is joint work in progress with Dominic Joyce.
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Vladimir Baranovsky (UC Irvine): Geometry of curved Maurer-Cartan functors
We will explain some geometric applications of Getzler's theory of Maurer-Cartan simplicial sets of curved complete \(L_{\infty}\)-algebras, in particular to the probem of lifting torsors across extensions of structure groups and construction of sheaves in commutative and noncommutative geometry.
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Lukas Bertsch (Vienna): Variation of crepant resolutions of Kleinian singularities
The McKay correspondence establishes a strong relationship between the classical minimal resolution and the standard orbifold resolution of a Kleinian surface singularity. Based on joint work with Ruth Wye, I will explain how the McKay correspondence extends to a larger class of crepant stacky resolutions of the singularity, and how their Hilbert schemes of points are related through variation of GIT quotients (VGIT). Time permitting, I will also sketch some recent ideas from work in progress with Austin Hubbard on how to relate the resolutions themselves via VGIT by taking into account the variation of monoidal structures on their mutual derived category.
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Alastair Craw (Bath): Deformations of Kleinian singularities and their Hilbert schemes
Every Kleinian surface singularity can be deformed to produce a flat family of symplectic surfaces, each of which is a Nakajima quiver variety, and the simultaneous resolution of these singular surfaces has been well-studied. In this talk, I'll describe work in progress with Ryo Yamagishi where we study the Hilbert schemes of points on these surfaces by investigating the relative Hilbert scheme of points on the flat family of surfaces.
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Enrico Fatighenti (Bologna): Modular vector bundles on hyperkähler manifolds
We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of \(K3^{[2]}\)-type. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperkähler. Interestingly enough, these constructions are almost never infinitesimally rigid, and more precisely we show how to get (infinitely many) 20 and 40 dimensional families. This is a joint work with Claudio Onorati. Time permitting, I will also present a joint work with Alessandro D'Andrea and Claudio Onorati on a connection between discriminants of vector bundles on smooth and projective varieties and representation theory of \(GL(n)\).
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Robert Hanson (Imperial): Parabolic induction for Higgs bundles
Given a reductive group \(G\), a fundamental method of constructing \(G\)-bundles or \(G\)-representations is via parabolic induction, defined by a pull–push construction over the parabolic/Levi subgroups of \(G\). At the geometric, homological, or categorical level, parabolic induction decomposes several objects of interest in algebraic geometry into Levi-structured components, for instance as BPS states in cohomology or as cuspidal–Eisenstein components in the geometric Langlands program. In this talk, we describe instances of this phenomenon on the moduli stack of \(G\)-Higgs bundles over a smooth projective curve, towards potential applications to the Dolbeault geometric Langlands conjecture of Donagi–Pantev. This is based on the article arXiv:2512.24239.
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Peize Liu (Warwick): Moduli spaces from the Kuznetsov component of
cubic fivefolds
The Kuznetsov components of cubic 4-folds and Gushel–Mukai 4-folds are often regarded as non-commutative \(K3\) surfaces, and their Bridgeland moduli spaces provide examples of hyper-Kähler manifolds of \(K3[n]\)-type. Following a recent construction of stability conditions on the Kuznetsov component of cubic 5-folds, we can study the geometry of the moduli spaces and relate them via the restriction functor to the moduli spaces from cubic 4-folds. As a consequence, we obtain a new class of Lagrangian subvarieties, extending constructions of Feyzbakhsh–Guo–Liu–Zhang and Li–Lin–Pertusi–Zhao.
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Heath Pearson (Nottingham): The Mukai conjecture for spherical varieties
The Mukai conjecture introduces an invariant connecting the topology of a Fano variety with the geometry of its rational curves. This invariant is conjecturally positive, and zero for powers of projective spaces. Although the Mukai conjecture remains open, special cases have been proved. In this talk, we will introduce the findings of a proof of this conjecture for the spherical varieties. This is a large class of \(G\)-varieties generalising the toric, flag, and symmetric varieties. We will first give an overview of aspects of the theory, and then explain how in the Mukai conjecture we find a connection to an invariant from birational geometry.
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Piotr Pokora (Krakow): Curves and surfaces with extreme properties
In this talk, I will present constructions of highly singular algebraic curves exhibiting extreme homological behavior. The focus will be on free plane curves and on algebraic surfaces naturally associated with them.
As an application of these constructions, I will describe a family of surfaces in \(\mathbb{P}^3\) with only isolated singularities whose total Tjurina number admits a lower bound given by a cubic polynomial depending solely on the degree of the surface.
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Dhruv Ranganathan (Cambridge): The LMNOP Conjecture
The Gromov-Witten/Donaldson-Thomas correspondence, proposed by Maulik-Nekrasov-Okounkov-Pandharipande (MNOP) in 2003, is a series of conjectures relating two theories of enumerative geometric origin on projective threefolds. The correspondence reflects a mysterious relationship between two ways to study curves: via equations and via parameterizations. I will outline an extension of the conjecture to logarithmic targets, the “LMNOP” conjecture, and then how this helps study the original correspondence. In particular, I will outline a proof of the full descendent GW/PT conjecture for Fano threefolds. Joint work with Davesh Maulik.
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Konstanze Rietsch (KCL): On mirror symmetry for Schubert varieties
I will report on two recent works about the construction of mirror superpotentials for Schubert varieties: one with L. Williams and one with C. Li and M. Yang. The first work concerns Grassmannian Schubert varieties X, where (with L. Williams) we construct a superpotential that is given in terms of Pluecker coordinates and Young diagram combinatorics and that encodes Newton-Okounkov bodies and toric degenerations of \(X\). Our superpotential formula generalises a formula for the full Grassmannian superpotential from earlier work with B. Marsh. In the second work, with Li and Yang, these Grassmannian Schubert superpotentials are then given a Lie-theoretic reinterpretation, and we obtain a conjectural superpotential for a general \(G/P\)-Schubert variety. This work also provides a generalisation of the (Dale) Peterson variety that governs the quantum cohomology rings of the homogeneous spaces \(G/P\), and we state a conjecture about the quantum cohomology rings of smooth Fano Schubert varieties.
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Marc Truter (Warwick): Fano 4-fold hypersurfaces
In 2016, Birkar proved that there are finitely many Fano varieties with terminal singularities in each dimension. This finiteness result motivates the construction of 'periodic tables' of such varieties, analogous to those in dimensions 1 and 2, with great progress made in dimension 3, but where dimension 4 remains largely uncharted territory.
Motivated by this, we study Fano 4-fold hypersurfaces with terminal singularities. A key departure from the 3-dimensional theory is that quasismoothness is no longer a reasonable generality assumption, and the resulting singularities are significantly wilder than in dimension 3. To navigate this nonquasismooth setting, we develop a finite-step criterion for terminality, refining the classical infinite-step criterion of Mori. The power of the toolbox we develop is demonstrated by producing over 100,000 new examples, including previously intractable high degree cases, contributing to the emerging periodic table in dimension 4.
Contact:
All further questions and inquiries can be addressed to Timothy Logvinenko at:
- LogvinenkoT at cardiff dot ac dot uk