Winter 2023
The Seminar runs on Tuesdays from 15:00 to 16:00 at MACADAM BLDG MB1.3
17 January, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: No Seminar
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24 January, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Izar Alonso Lorenzo (Oxford)
Title: New examples of $\mathrm{SU}(2)^2$-invariant $\mathrm{G}_2$-instantons
Abstract: $\mathrm{G}_2$-instantons are a special kind of connections on a Riemannian $7$-manifold, analogues of anti-self-dual connections in $4$ dimensions. I will start this talk by describing $\mathrm{G}_2$-instantons and giving an overview of known examples and why are we interested in them. Then, I will explain how we construct $\mathrm{G}_2$-instantons in $\mathrm{SU}(2)^2$-invariant cohomogeneity one manifolds and give new explicit examples of $\mathrm{G}_2$-instantons on $\mathbb{R}^4 \times \mathbb{S}^3$ and $\mathbb{S}^4 \times \mathbb{S}^3$. I will then discuss the bubbling behaviour of sequences of $\mathrm{G}_2$-instantons found.
31 January, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: João Paulo dos Santos (Universidade de Brasília)
Title: Hypersurfaces with constant scalar curvature and Einstein hypersurfaces in product spaces
Abstract: Let $M$ be either the unit $n$-sphere or the $n$-dimensional hyperbolic space, with $n>3$. Using graphs over families of parallel hypersurfaces in $M$, I will present existence results for symmetric hypersurfaces with constant scalar curvature in $M \times \mathbb{R}$. The technique will also be applied to show that an Einstein hypersurface in $M \times \mathbb{R}$ has constant sectional curvature. Finally, employing such graphs, I will provide new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces in $M \times \mathbb{R}$. The talk is based on two joint works: with de Lima and Ramos (regarding the scalar curvature and the sectional curvature) and with Leandro and Pina (on the Einstein hypersurfaces).
7 February, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Jijian Song (Tianjin University)
Title: Irreducible cone spherical metrics and stable vector bundles
Abstract: Cone spherical metrics on compact Riemann surfaces are conformal metrics of constant curvature $+1$ with finitely many conical singularities. They are called $\textit{irreducible}$ if any developing maps of such metrics don’t have monodromy in $\mathrm{U}(1)$. By using projective structures and indigenous bundles on compact Riemann surfaces, we construct a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible cone spherical metrics with cone angles in $2\pi \mathbb{Z}$. We also prove that the map is generically injective in algebro-geometric sense if the Riemann surface has genus $>1$. As an application, we obtain some new existence results about irreducible cone spherical metrics. This is a joint work with Lingguang Li and Bin Xu.
14 February, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: No Seminar
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21 February, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Cristiano Spotti (Aarhus University)
Title: Special locally conformally Kähler metrics
Abstract: In this seminar I will describe a generalization of the Donaldson-Fujiki moment map picture to the locally conformally symplectic case, with the aim of defining a suitable notion of canonical metrics in this setting. The talk is based on joint work with D. Angella, S. Calamai and F. Pediconi.
28 February, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: No Seminar
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7 March, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Dino Festi (University of Milan)
Title: K3 surfaces with two involutions and low Picard number
Abstract: Having an automorphism is a non-trivial property for a complex K3 surface. Indeed, if X is a generic complex K3 surface of degree d >= 4, then the only automorphism of X is the identity. If X is a generic of degree d=2, then X admits only one involution beside the identity map. Hence a natural question arises: given a fixed positive even integer d, how special is it for a K3 surface of degree d to admit an involution? More precisely, what is the minimal Picard number h_d for a K3 surface of degree d in order to admit an involution as automorphism? In this talk we are going to show that h_d=2 for every d\geq 4. This is joint work with Wim Nijgh and Daniel Platt.
14 March, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: John Armstrong (KCL)
Title: Stochastic differentials on manifolds
Abstract: Despite the name, stochastic differential equations (SDEs) are defined using integrals, but the standard notation hints at the existence of a “stochastic differential” in much the same way as the notation for deterministic integrals hints at the existence of differential forms. We will see how a stochastic differential can be rigorously defined and how this can be applied to give an elegant coordinate-free treatment of stochastic differential equations on manifolds
21 March, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Laura Fredrickson (Oregon, US)
Title: The asymptotic geometry of the Hitchin moduli space
Abstract: Hitchin’s equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmüller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkähler metric. An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently. I will discuss some recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.
28 March, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Farhad Babaee (Bristol)
Title: Dynamical tropicalisation
Abstract: In this talk, I will explain some applications of tropical geometry in complex geometry. I will also explain how tropicalisation can be viewed as an equidistribution result.
18 April, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Jordan Hofmann (King’s College London)
Title: Special Spinors and Homogeneous Geometries
Abstract: Spin geometry, and in particular special spinors, is strongly related to many areas of differential geometry, including immersion theory, G-structures, special holonomy, and Einstein metrics. Unfortunately, constructing examples of special spinors is often difficult in practice, and not many examples are known in dimension >8. In this talk I will discuss my PhD research, which approaches this problem in the homogeneous setting using tools from representation theory. In particular, I will discuss our classifications of the invariant spinors carried by the various homogeneous realizations of the sphere and the homogeneous 3-Sasakian spaces, and how these spinors are related to (and in some cases fully determine!) the geometry of the underlying spaces. This talk includes joint work with Ilka Agricola and Marie-Am'elie Lawn.
25 April, 14:00-15:00 (Irregular Time), MACADAM BLDG MB1.3
Speaker: Irene Pasquinelli (Bristol)
Title: Mapping class group orbit closures for non-orientable surfaces
Abstract: The space of measured laminations on a hyperbolic surface is a generalisation of the set of weighted multi curves. The action of the mapping class group on this space is an important tool in the study of the geometry of the surface. For orientable surfaces, orbit closures are now well-understood and were classified by Lindenstrauss and Mirzakhani. In particular, it is one of the pillars of Mirzakhani’s curve counting theorems. For non-orientable surfaces, the behaviour of this action is very different and the classification fails. In this talk I will review some of these differences. I will talk about some of these results and classify mapping class group orbit closures of (projective) measured laminations for non-orientable surfaces. This is joint work with Erlandsson, Gendulphe and Souto.
2 May, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Aleksander Doan (University College London, University of Cambridge)
Title: Symplectic invariants of Calabi-Yau threefolds
Abstract: This talk is about various curve counting invariants of Calabi-Yau threefolds or, more generally, symplectic manifolds of real dimension six. After a brief overview of the subject, I will discuss joint work with E. Ionel and T. Walpuski on the Gopakumar-Vafa finiteness conjecture and an ongoing project with T. Walpuski, whose goal is to combine ideas from gauge theory, holomorphic curve theory, and geometric analysis to define new invariants of symplectic six-manifolds.
9 May, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Alice Kerr (Bristol)
Title: Embedded subgroups in mapping class groups
Abstract: A natural group to study is the group of orientation-preserving diffeomorphisms (or homeomorphisms) of a surface. This is a huge and complicated group, so we often tackle it by either looking at its identity component, or by quotienting it by its identity component. This quotient gives us the mapping class group, which is still a huge group in its own right, but turns out to be far more tractable, especially when using geometric tools. In this talk we will try to understand why the mapping class group has fascinated so many people, give an overview of its properties, and see an example of how the large classes of groups which embed in it benefit from these properties.
16 May, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Ivan Solonenko (King’s College London)
Title: Homogeneous complex hypersurfaces in Hermitian symmetric spaces
Abstract: In his paper from 1968, B. Smyth classified homogeneous complex hypersurfaces (HCHs) in complex space forms: those are C^{n-1} in C^n, CH^{n-1} in CH^n, CP^{n-1} in CP^n, as well as the smooth quadric Q^{n-1} in CP^n (which is the only example that is not totally geodesic). By design, every HCH arises as an orbit of an isometric action of cohomogeneity 1 or 2. Peculiarly, all of the above examples come from C1-actions. This suggests the following question: is it true that any HCH in a Kähler manifold can be realized as a singular orbit of an isometric C1-action?
I am going to talk about Konno’s classification of complex codimension-one embeddings of Kähler C-spaces (a.k.a. (generalized) complex flag manifolds) with b_2 = 1 and explain how it implies the classification of HCHs in irreducible Hermitian symmetric spaces of compact type. By combining that with Kollross’s classification of C1-actions, we will see that the answer to the above questions is ‘yes’ in this case.
Finally, I will show how all non-totally-geodesic HCHs in irreducible compact Hermitian symmetric spaces can be obtained via the complexification construction for projective spaces over normed real division algebras.
23 May, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Ilyas Khan (University of Oxford)
Title: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow (joint with M. Haskins and A. Payne)
Abstract: Riemannian 7-manifolds with holonomy equal to the exceptional Lie group G_2 are intensely studied objects in diverse domains of mathematics and physics. One approach to understanding such manifolds is through natural flows of 3-forms called G_2-structures, the most prominent of which is Bryant’s Laplacian flow. In general, Laplacian flow is expected to encounter finite-time singularities and, as in the case of other flows, self-similar solutions should play a major role in the analysis of these singularities. In this talk, we will discuss recent joint work with M. Haskins and A. Payne in which we prove the uniqueness of asymptotically conical gradient shrinking solitons of the Laplacian flow of closed G_2 structures. We will particularly emphasize the unique difficulties that arise in the setting of Laplacian flow (in contrast to the Ricci flow, where an analogous result due to Kotschwar and Wang is well-known) and how to overcome these difficulties.
30 May, 15:00-16:00, MACADAM BLDG MB1.3
Speaker: Mark Gross (University of Cambridge)
Title: Mirror symmetry and partial compactifications of K3 moduli.
Abstract: I will talk about work with Hacking, Keel and Siebert on using mirror constructions to provide partial compactifications of the moduli of K3 surfaces. Starting with a one-parameter maximally unipotent degeneration of Picard rank 19 K3 surfaces, we construct, using methods of myself and Siebert, a mirror family which is defined in a formal neighbourhood of a union of strata of a toric variety whose fan is defined, to first approximation, as the Mori fan of the original degeneration. This formal family may then be glued in to the moduli space of polarized K3 surfaces to obtain a partial compactification. Perhaps the most significant by-product of this construction is the existence of theta functions in this formal neighbourhood, certain canonical bases for sections of powers of the polarizing line bundle.
6 June, NO SEMINAR
13 June, 15:00-16:00, STRAND BLDG S3.32 (Irregular location)
Speaker: Alexander Esterov (London Institute for Mathematical Sciences [LIMS])
Title: Solvable systems of equations, Galois groups in enumerative geometry, and small lattice polytopes
Abstract: The general polynomial of a degree higher than 4 cannot be solved by radicals. This classical theorem has a multidimensional version: solvable general systems of polynomial equations are in (almost) one-to-one correspondence with lattice polytopes of volume 4, and the latters admit a finite classification. In the narrow sense, I will talk about this xix-century-style result. In a broader sense, we shall look at the Galois groups of problems of enumerative geometry (such as Schubert calculus), and how their study leads to seemingly distant topics such as polyhedral geometry and braid groups.
20 June, 15:00-16:00, STRAND BLDG S2.29 (Irregular location)
Speaker: Gabino Gonzalez Diez (Universidad Autónoma de Madrid)
Title: Dessins d’enfants, filling curves and their associated Riemann surfaces.
Abstract: A filling curve $c$ in a closed oriented surface $X$ determines a complex analytic structure on $X$ in two different ways. One is via Grothendieck’s theory of dessins d’enfants. The other one arises as the hyperbolic structure on $X$ that minimises the length of the curve $c$. We show that these two complex structures agree at least in the case in which the curve $c$ admits a homotopy representative in minimal position such that all self-intersection points have the same self-intersection number and all the faces of the complement $X \setminus c$ have the same degree. This is joint work with E. Girondo and R. Hidalgo.