Samuel Borza (Vienna). Measure contraction properties for sub-Riemannian structures beyond step 2 Friday 2nd May, 1:30-2:30pm. Huxley 140.

Abstract:  I will introduce Carnot groups, and their quotients, as metric measure spaces. These are examples of Carnot–Carathéodory spaces, or sub-Riemannian manifolds, and include the Heisenberg group, the Engel group, and the Martinet flat structure, to name but a few. Once we have a good grasp of these geometric structures, I will outline some open problems in the field before shifting focus to the study of curvature and the so-called metric contraction properties. These analytic inequalities aim to define, in a synthetic way, a lower bound on the Ricci curvature. The new results I will present show how these properties can be preserved by taking quotients, and how this affects their validity or failure. This is joint work with Luca Rizzi from SISSA.

Masafumi Hattori (Nottingham). K-moduli of quasimaps and quasi-projectivity of K-stable Calabi-Yau fibrations over curves Friday 9th May, 1:30-2:30pm. Huxley 140.

Abstract: K-stability is an important notion in algebraic geometry, which is introduced to detect the existence of constant scalar curvature Kahler metrics, as the Yau-Tian-Donaldson conjecture predicted. On the other hand, this notion is also closely related to GIT and moduli theory. Odaka predicted that we can construct a moduli space (K-moduli) by using K-stability and Xu et. al. constructed K-moduli theory for log Fano pairs with an ample CM line bundle, which is a line bundle canonically defined. However, Odaka’s K-moduli conjecture is still open for general polarized varieties.

In this talk, we introduce uniform adiabatic K-stability for Calabi-Yau fibrations, that is a uniform notion of K-stability when the polarization is very close to the base line bundle, and we construct K-moduli theory of Calabi-Yau fibrations over curves. Moreover, we will construct K-moduli theory for log Fano quasimaps and apply it to the quasi-projectivity for K-moduli of Calabi-Yau fibrations.

Shaked Bader (Oxford). Hyperbolic subgroups of type FP_2(Ring) Friday 16th May, 1:30-2:30pm. Huxley 140.

Abstract: In 1996 Gersten proved that if G is a word hyperbolic group of cohomological dimension 2 and H is a subgroup of type FP_2, then H is hyperbolic as well. In this talk, I will present a project with Robert Kropholler and Vlad Vankov generalising this result to show that the same is true if G is only assumed to have cohomological dimension 2 over some ring R and H is of type FP_2(R).

Matt Booth (Imperial). Reflexive dg categories in algebra and topology Friday 23rd May, 1:30-2:30pm. Huxley 140.

Abstract: Reflexive dg categories, introduced recently by Kuznetsov and Shinder, satisfy strong duality properties which in particular place restrictions on invariants like their Hochschild (co)homology, derived Picard groups, and semiorthogonal decompositions. Examples include various derived categories of proper schemes, as well as finite dimensional algebras. I’ll define what it means for a dg category to be reflexive before giving some examples arising from algebraic geometry, algebraic topology, and symplectic geometry. This talk is based on work in progress joint with Isambard Goodbody and Sebastian Opper.

Yuhan Sun (Imperial).  Rigid fibers in symplectic and contact manifolds Friday 30th May, 1:30-2:30pm. Huxley 140.

Abstract: Given a Hamiltonian integrable system on a closed symplectic manifold, Entov-Polterovich proved it has at least one rigid fiber. We will first survey several ideas of the proof and recent enhancements. Then we will present a rigid fiber theorem for contact manifolds. Based on joint works with Mak, Uljarevic and Varolgunes.

Simon Donaldson (Imperial).  Calabi-Yau threefolds with boundary Friday 6th June, 1:30-2:30pm. Huxley 140.

Abstract: The talk is based on joint work with Fabian Lehmann. By a Calabi-Yau threefold we mean a 3-dimensional complex manifold Z with a nowhere-vanishing holomorphic 3-form Theta. We consider the case of a manifold Z with boundary M, so the real part of Theta  restricts to a closed 3-form on M. Given a closed 3-form on M we consider the problem of finding a (Z,Theta).  Our main result is a solution of the perturbative (i.e. small deformation) version of the problem, extending the well-known local Torelli theorem in the case of closed manifolds. Much of the time will be spent discussing relations of this problem with other fields such as contact, symplectic and affine geometry.

Anya Nordskova (University of Hasselt).  Full exceptional collections on Fano threefolds Friday 13th June, 1:30-2:30pm. Huxley 140.

Abstract: In some cases the derived category D(X) of an algebraic variety X admits a particularly nice description via a so-called full exceptional collection. This essentially means that D(X) can be viewed as being glued from the simplest possible building blocks, each equivalent to the derived category of a point. In this situation there is a braid group acting on the set of all full exceptional collections in D(X). A conjecture by Bondal and Polishchuk (1993) suggested that this braid group action is always transitive for any triangulated category admitting a full exceptional collection. Even though in the original generality the conjecture has been recently disproved, the question is still widely open if one restricts to e.g. derived categories of smooth projective varieties. I will discuss Bondal-Polishchuk’s conjecture for Fano threefolds of Picard rank 1 (in particular, the projective space P^3), which is the first 3-dimensional case where the transitivity has been verified. The talk is based on joint work with Michel Van den Bergh

Daniel Bath (KU Leuven).  A guided tour of (un)Twisted Logarithmic Comparison Theorems Friday 20th June, 1:30-2:30pm. Huxley 340.

Abstract: To compute the cohomology of the complement U of a divisor D on X one computes the cohomology of the de Rham complex of (meromorphic, rational) forms with poles along D. This is a complex whose objects are O_X-modules, whose differentials C_X-linear, but whose the objects have no O_X-finiteness properties. One can ask if whether or not replacing the de Rham complex with a subcomplex of finite O_X-modules computes the same cohomology. There is a natural candidate for this and when this works we say the divisor D satisfies the “Logarithmic Comparison Theorem”. A similar procedure works for arbitrary rank one local systems and then we use the phrase “Twisted Logarithmic Comparison Theorem.” 

I will discuss recent work on this topic, possibly including the following: a proof that hyperplane arrangements satisfy Twisted Logarithmic Comparisons, a conjecture of Terao from the 1970s; a D-module interpretation of Twisted Logarithmic Comparison Theorems; recent work on a 2002 conjecture connecting Logarithmic Comparison Theorems to homogeneity properties of the divisor. The talk is based on, respectively: arXiv:2202.01462 (Bath); arXiv:2203.11716 (Bath, Saito); forthcoming joint work of Bath, Rodriguez, Walther.

Wilhelm Klingenberg (Durhham University)  Proof of the Toponogov Conjecture on Complete Surfaces Friday 27th June, 10:30-11:30am. Huxley 140.

Abstract: In 1995, Victor Andreevich Toponogov [1] authored the following conjecture: “Every smooth non-compact strictly convex and complete surface of genus zero has an umbilic point, possibly at infinity“. In our talk, we will outline the 2024 proof in collaboration with Brendan Guilfoyle [2]. Namely we prove that, should a counter-example to the Conjecture exist, (a) the Fredholm index of an associated  Riemann Hilbert boundary problem for holomorphic discs is negative [3]. Thereby, (b) no such  holomorphic discs survive for a generic perturbation of the boundary condition (these form a Banach manifold under the assumption that the Conjecture is incorrect). Finally, however, (c) the geometrization by a neutral Kaehler metric  [4] of the associated model allows for Mean Curvature Flow [5] with mixed Dirichlet – Neumann boundary conditions to generate a holomorphic disc from an initial spacelike disc. This completes the indirect proof of said conjecture as (b) and (c) are in contradiction.

References :

[1] V.A. Toponogov, (1995) On conditions for existence of umbilical points on a convex surface, Siberian Mathematical Journal, 36  780–784.  

[2] B. Guilfoyle and W. Klingenberg (2024)  Proof of the Toponogov Conjecture on complete surfaces, J. Gokova Geom. Topol. GGT 17 1–50.

[3] Guilfoyle, B., & Klingenberg, W. (2020) Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces. Annales de la Faculté des sciences de Toulouse (En ligne), 29(3), 565-576. 

[4]  B. Guilfoyle and W. Klingenberg (2005) An indefinite Kaehler metric on the space of oriented lines, J. London Math. Soc. 72.2,  497–509.

[5] B. Guilfoyle and W. Klingenberg, Higher codimensional mean curvature flow of compact spacelike submanifolds, Trans. Amer. Math. Soc. 372.9 (2019) 6263–6281.