Autumn 2025
The Seminar runs on Tuesdays at STRAND BLDG S5.20. This term, some talks will run from 15:00 to 16:00, and others will be in two parts – an introductory talk from 15:00 to 15:40 and a research talk from 15:45 to 16:30.
14 October, 15:00-16:00, STRAND BLDG S5.20
Speaker: Luca Tasin (Univeristy of Milan)
Title: Sasaki–Einstein metrics on spheres
Abstract: The differential geometry of spheres has long been a source of central problems in mathematics, and Sasaki–Einstein metrics—odd-dimensional analogues of Kähler–Einstein metrics—offer a particularly rich perspective, with significance both in geometry and in theoretical physics. In joint work with Yuchen Liu and Taro Sano, we construct infinitely many Sasaki–Einstein metrics on odd-dimensional spheres that bound parallelizable manifolds, thereby confirming conjectures of Boyer–Galicki–Kollár and Collins–Székelyhidi. Our approach is based on establishing the K-stability of certain Fano weighted hypersurfaces.
21 October, 15:00-16:00, STRAND BLDG S5.20
Speaker: Hannah Tillmann-Morris (KCL)
Title: Birational mirrors to pairs of Laurent polynomials
Abstract: Under mirror symmetry, deformation classes of Fano varieties are associated to mutation classes of maximally mutable Laurent polynomials (MMLPs). We expect birational relationships between the general members of two deformation classes to be reflected, in the pair of mirror mutation classes, as combinatorial relationships between two MMLPs.
I will present an alternative construction of a Fano variety that is mirror to a given MMLP, which uses mirror theorems of the Gross-Siebert program. The resulting Fano variety is difficult to describe explicitly. However, when two given MMLPs are related by certain combinatorial conditions, the construction can be extended to include the construction of a birational map between the two Fano varieties produced.
Applying all this to rigid MMLPs in two variables recovers all but one of the blow-ups in the chain of smooth del Pezzo surfaces.
28 October, 15:00-16:00, STRAND BLDG S5.20
Speaker: Sasha Veselov (Loughborough)
Title: Markov fractions and the slopes of the exceptional bundles on $\mathbb P^2$
Abstract: We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on $\mathbb P^2$ studied in 1980s by Dr`ezet and Le Potier and by Rudakov. In particular, we provide a simpler proof of Rudakov’s result claiming that the ranks of the exceptional bundles on $\mathbb P^2$ are Markov numbers.
4 November, 15:00-16:00, STRAND BLDG S5.20
Speaker: Roney Santos (King’s College London)
Title: Stability of minimal submanifolds
Abstract: We aim to introduce and discuss some concepts and results on the stability of minimal submanifolds, mainly when the ambient manifold is conformal either to a round sphere or to a convex domain of the Euclidean space. In particular, we will talk about a joint work with Alcides de Carvalho and Federico Trinca, in which we showed that a conformally Euclidean manifold with convex boundary does not admit volume-minimizing free boundary minimal submanifolds.
11 November, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S5.20
Speaker: Cameron Gates Rudd (Oxford)
Title: Almost cycles, almost boundaries, and homological complexity
Abstract: Given a space with some kind of geometry, one can ask how the geometry of the space relates to its homology. This talk will survey some comparisons of geometric notions of complexity with homological notions of complexity. We will then focus on hyperbolic manifolds and discuss how “almost cycles” and “almost boundaries” relate to the “size” of homology.
18 November, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S5.20
Speaker: Dima Panov (KCL)
Title: Polyhedral Kahler metrics, hyperplane arrangements and BMY
Abstract: Polyhedral manifolds are piecewise linear analogues of Riemannian manifolds. They are obtained by taking a collection of Euclidean simplices and identifying their hyperfaces by isometries. For example, the boundary of an n-dimensional Euclidean simplex is a polyhedral n-1-dimensional sphere. In this talk I’ll speak about polyhedral Kahler manifolds, which are even dimensional polyhedral manifolds with unitary holonomy. While any Riemann surface has plenty of polyhedral Kahler metrics, the situation in complex dimension 2 and higher is very different. Such metrics seem to be very rare, and the known ones are related to the most rigid objects, such as complex reflection groups. The talk will be partially based on the recent work with Martin de Borbon, arXiv:2106.13224, arXiv:2411.09573, arXiv:2510.17447.
25 November, 15:00-16:00 STRAND BLDG S5.20
Speaker: Abigail Ward (Cambridge)
Title: A mirror symmetry correspondence in birational geometry
Abstract:
I’ll discuss a construction in homological mirror symmetry and birational geometry which exhibits the group of volume-preserving birational transformations of $\mathbb P^2$ in two ways: as the automorphism group of a certain infinite-type scheme, and as a group of symplectomorphisms of an infinite-type Weinstein manifold. I’ll also give some applications of this work in understanding the symplectic mapping class group of finite-type Weinstein 4-manifolds. This is joint work with Ailsa Keating.
2 December, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S5.20
Speaker: Dan Ciubotaru (Oxford)
Title: Unitary representations and prehomogeneous vector spaces
Abstract: The modern study of unitary representations of reductive groups over local fields has been greatly influenced by Arthur’s conjectures on the parametrisation of the local factors of automorphic forms in the discrete spectrum. For reductive groups over real or p-adic fields, Arthur’s local conjectures admit a precise formulation in the work of Adams, Barbasch, and Vogan, via the microlocal analysis of the geometric parameter space for the admissible dual of the group. In the case of p-adic groups, the geometric parameter space is given by the complex geometric setting of Kazhdan and Lusztig. An important particular case, where everything can be made precise, is the category of unipotent (in the sense of Lusztig) representations of a reductive p-adic group; these are the representations for which the Langlands parameters are unramified, in the sense of being trivial on the inertia subgroup of the Weil group. Once we fix an”infinitesimal character” for the representations, the geometry comes from the action of a complex reductive group on a complex vector space with finitely many orbits; for the Adams-Barbasch-Vogan picture, we are interested in the resulting microlocal packets and the corresponding packets of irreducible representations of the p-adic group. In the talk, I will explain the parametrisations and above constructions, give examples, and concentrate on the integral infinitesimal characters, where some surprisingly strong general conjectures about unitarisability can be formulated.
4 December, 11-12, STRAND BLDG S3.32 (Note the unusual time and place!)
Speaker: Jarek Buczyński (IMPAN)
Title: TBA
Abstract: TBA
9 December, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S5.20
Speaker: Jason Lotay (Oxford)
Title: TBD
Abstract: TBD