Autumn 2023
25 September
No Seminar
3 October, 15:00-16:00, STRAND BLDG S4.29
Speaker: Sebastián Velazquez (King’s College London)
Title: On the deformation theory of L-foliations
Abstract: We will review some general concepts of deformation theory. Then we will apply these ideas in order to explore the geometry of the moduli space $Inv$ of foliations on a given variety $X$ around the points corresponding to foliations induced by Lie group actions. More precisely, let $X$ be a smooth projective variety over the complex numbers and $S(d)$ the scheme parametrizing $d$-dimensional Lie subalgebras of $H^0(X,\mathcal{T} X)$. For every $\mathfrak{g} \in S(d)$ one can consider the corresponding element $\mathcal{F}(\mathfrak{g})\in Inv$, whose generic leaf coincides with an orbit of the action of $\exp(\mathfrak{g})$ on $X$. We will show that under mild hypotheses, after taking a stratification $\coprod_i S(d)_i\to S(d)$ this assignment yields an isomorphism $\coprod_i S(d)_i\to Inv$ locally around $\mathfrak{g}$ and $\mathcal{F}(\mathfrak{g})$.
10 October, 15:00-16:00, STRAND BLDG S4.29
Speaker: Albert Wood (King’s College London)
Title: Cohomogeneity-one Lagrangian Mean Curvature Flow
Abstract: Mean Curvature Flow, the negative gradient flow for the volume functional of submanifolds of Riemannian manifolds, is a well-studied field of modern geometric analysis. Of particular interest are classifications of self-similar solutions (shrinkers, expanders, and translators) and finite-time singularities; projects which when completed will hopefully allow one to apply the flow to prove results in Riemannian geometry and differential topology. Moreover, in a Calabi-Yau manifold the class of Lagrangian submanifolds is preserved by mean curvature flow, a fact which inspired Thomas and Yau to make influential conjectures about existence of special Lagrangians in Calabi-Yau manifolds.
In this talk, we aim to make progress towards an understanding of self-similar solutions and singularities of Lagrangian mean curvature flow, by focusing on Lagrangians in $\mathbb{C}^n$ that are cohomogeneity-one under the action of a compact Lie group. Interestingly, each such Lagrangian lies in a level set $\mu^{-1}(c)$ of the moment map $\mu$, and mean curvature flow preserves this containment. Using this, we classify all shrinking, expanding, and translating solitons, and in the zero level set $\mu^{-1}(0)$, we classify the Type I and Type II blowup models of LMCF singularities. Finally, given any special Lagrangian in $\mu^{-1}(0)$, we’ll show that it arises as a Type II blowup, thereby yielding infinitely many new singularity models of Lagrangian mean curvature flow.
The results presented in this talk are contained in the preprint ‘Cohomogeneity-One Lagrangian Mean Curvature Flow’, which is jointly written with Jesse Madnick, University of Oregon.
17 October, 15:00-16:00, STRAND BLDG S4.29
Speaker: Asma Hassannezhad (University of Bristol)
Title: Steklov eigenvalues of negatively curved manifolds
Abstract: The geometry and topology of negatively curved manifolds are subtly reflected in a geometric bound for the Laplace eigenvalues, a connection that has been explored since the 1980s. Building upon these foundational studies in the case of the Laplacian, we investigate the Steklov eigenvalues of pinched negatively curved manifolds with totally geodesic boundary. These eigenvalues are associated with a first-order elliptic pseudodifferential operator known as the Dirichlet-to-Neumann operator. We discuss how the results for Laplace eigenvalues can be extended to Steklov eigenvalues. In particular, we show a spectral gap for the Steklov eigenvalue problem in negatively curved manifolds with dimensions of at least three. This talk is based on joint work with Ara Basjmaian, Jade Brisson, and Antoine Métras.
24 October, 15:00-16:00, STRAND BLDG S4.29
Speaker: Oscar Randal-Williams (University of Cambridge)
Title: Monodromy and mapping class groups of 3-dimensional hypersurfaces
Abstract: Kreck and Su have recently described, almost completely, the mapping class group of a smooth hypersurface in $\mathbb{C}P^4$. There is a “monodromy” map from the fundamental group of the space of all smooth hypersurfaces in $\mathbb{C}P^4$ to this mapping class group, and I will explain how the image of this map can be described. I will then give some idea of the differential topology methods which go into the proof.
31 October
Reading Week. No Seminar
7 November, 15:00-16:00, STRAND BLDG S4.29
Speaker: Yen-An Chen (National Center for Theoretical Sciences)
Title: MMP for toric foliations
Abstract: In recent years, significant progress has been made in the field of birational geometry for foliations. Notably, the Minimal Model Program (MMP) has been shown to work for foliations on threefolds. In this talk, I will demonstrate that the MMP is applicable to toric foliations as well. Specifically, I will discuss how non-dicritical singularities (and foliated dlt singularities if time permits) are preserved under the MMP. This is a joint work with Chih-Wei Chang.
14 November, 15:00-16:00, STRAND BLDG S4.29
Speaker: Soham Karwa (Imperial College London)
Title: Non-archimedean periods for log Calabi-Yau surfaces
Abstract: Period integrals are a fundamental concept in algebraic geometry and number theory. In this talk, we will study the notion of non-archimedean periods as introduced by Kontsevich and Soibelman. We will give an overview of the non-archimedean SYZ program, which is a close analogue of the classical SYZ conjecture in mirror symmetry. Using the non-archimedean SYZ fibration, we will prove that non-archimedean periods recover the analytic periods for log Calabi-Yau surfaces, verifying a conjecture of Kontsevich and Soibelman. This is joint work with Jonathan Lai.
21 November, 14:00-15:00, STRAND BLDG S-2.25
Note irregular time and location due to department colloquium
Speaker: Steven Sivek (Imperial College London)
Title: Rational homology spheres and $\mathrm{SL}(2,\mathbb{C})$ representations
Abstract: Building on non-vanishing theorems of Kronheimer and Mrowka in instanton Floer homology, Zentner proved that if $Y$ is a homology 3-sphere other than $\mathbb{S}^3$, then its fundamental group admits a homomorphism to $\mathrm{SL}(2,\mathbb{C})$ with non-abelian image. In this talk, I’ll explain how to generalize this to any $Y$ whose first homology is 2-torsion or 3-torsion, other than the connect sum of $n$ copies of $\mathbb{R}\mathrm{P}^3$ for any $n$ or lens spaces of order $3$. This is joint work with Sudipta Ghosh and Raphael Zentner.
28 November, 15:00-16:00, STRAND BLDG S4.29
Speaker: Joel Fine (Université Libre de Bruxelles)
Title: Knots, minimal surfaces and $J$-holomorphic curves
Abstract: Let $K$ be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space $\mathbb{H}^4$. I will describe a programme to count minimal surfaces in $\mathbb{H}^4$ which have $K$ as their asymptotic boundary. This should give an isotopy invariant of the knot. I will explain what has been proved and what remains to be done. Minimal surfaces correspond to $J$-holomorphic curves in the twistor space $Z\to\mathbb{H}^4$, and so this invariant can be seen as a Gromov-Witten type invariant of $Z$. The big difference with the “standard” situation is that the almost complex structure on $Z$ (equivalently, the metric on $\mathbb{H}^4$) blows up at the boundary. This means the $J$-holomorphic equation, or minimal surface equation, becomes degenerate at the boundary of the domain. As a consequence, both the Fredholm and compactness parts of the story need to be reworked by hand. If there is time I will explain how this can be done, relying on results of Mazzeo-Melrose from the 0-calculus, and also some results from the theory of minimal surfaces.
5 December, 14:00-15:00, STRAND BLDG S-2.25
Note irregular time and location due to department colloquium
Speaker: Pavel Tumarkin (Durham University)
Title: Quiver mutations, Coxeter groups and hyperbolic manifolds
Abstract: Mutations of quivers were introduced by Fomin and Zelevinsky in the context of cluster algebras. Since then, mutations appear (sometimes completely unexpectedly) in various domains of mathematics and physics. Using mutations of quivers, Barot and Marsh constructed a series of presentations of finite Coxeter groups as quotients of infinite Coxeter groups. I will discuss a geometric interpretation of this construction: these presentations give rise to a construction of geometric manifolds with large symmetry groups, in particular to some hyperbolic manifolds of relatively small volume with proper actions of Coxeter groups. If time permits, I will discuss a generalization of the construction of Barot and Marsh leading to a new invariant of bordered marked surfaces, and relation to extended affine Weyl groups. The talk is based on joint works with Anna Felikson, John Lawson and Michael Shapiro.