Autumn 2024
The Seminar runs on Tuesdays at STRAND BLDG S4.29. 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.
1 October, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Stephen Lynch (King’s College London)
Title: Singularities in mean curvature flow
Abstract: Mean curvature flow moves a hypersurface in Euclidean space with velocity equal to its mean curvature vector. This evolution is described by a nonlinear weakly parabolic system. Variationally, it is a formal gradient flow for the volume functional. Solutions to mean curvature flow exhibit a huge variety of different kinds of singularities. For solutions which move monotonically (have nowhere vanishing mean curvature), however, these singularities exhibit enough structure so that they might eventually be completely classified. We will discuss the now essentially complete picture for surfaces in $\mathbb{R}^3$ developed over the last 40 years, and then explore the dramatically more complicated setting of 3-dimensional hypersurfaces in $\mathbb{R}^4$.
3 October, 15:00-16:00, BUSH HOUSE (SE) 1.01 (Note the unusual time and location!)
Speaker: Michael McQuillan (Rome Tor Vergata)
Title: Hyperbolicity of algebraic surfaces.
Abstract: Unlike Riemann surfaces or 3 manifolds, a constant Ricci curvature metric on an algebraic surface is, generically, a rather crude estimate of its geometry. Indeed, from the point of view of an isoperimetric inequality, it is a property of the volume of embedded balls rather than the area of (embedded) discs. The goal of the talk will be to ouline a programme for a complete understanding of an area vs. length inequality for algebraic surfaces. The essential technical tool is the adaption of the techniques introduced by Mori in his study of minimal models of algebraic varieties to the more general context of solutions of differntial equations on an algebraic variety. However, no knowledge of this theory will be supposed, and, indeed, I’ll just aim to explain why this is the natural way to proceed.
8 October, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Bruno Klingler (Humboldt University of Berlin)
Title: Recent progress on Hodge loci
Abstract: Given a quasi projective family S of complex algebraic varieties, its Hodge locus is the locus of points of S where the corresponding fiber admits exceptional Hodge classes (conjecturally: exceptional algebraic cycles). In this talk I will survey the many recent advances in our understanding of such loci, both geometrically and arithmetically, as well as the remaining open questions.
15 October, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Sophie Morier-Genoud (Université Reims Champagne Ardenne)
Title: From deformations of the Farey graph to faithfulness of Burau representations
Abstract: The talk will start from the elementary fact that positive rational numbers can be expanded as finite continued fractions with positive integer coefficients. The positive integer coefficients have combinatorial interpretations in the sense that ‘’they count something’’. I will present combinatorial interpretations based on the Farey graph. Introducing a formal parameter q, I will then make a deformations of the objects and refine the countings. This will bring notions of q-rationals, q-continued fractions, q-SL(2,Z). I will explain the constructions and give the main properties of all these q-analogs. Finally, I will connect this theory to the Burau representation of the braid group B3, and give a partial answer to the question of faithfulness of the complex specialisations of this representation.
22 October, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Laura Wakelin (King’s College London)
Title: Finding characterising slopes for all knots
Abstract: A slope $p/q$ is characterising for a knot $K$ if the oriented homeomorphism type of the 3-manifold obtained by performing Dehn surgery of slope $p/q$ on $K$ uniquely determines the knot $K$. For any knot $K$, there exists a bound $C(K)$ such that any slope $p/q$ with $\vert q\vert \geq C(K)$ is characterising for $K$. This bound has previously been constructed for certain classes of knots, including torus knots, hyperbolic knots and composite knots. In this talk, I will give an overview of joint work with Patricia Sorya in which we complete this realisation problem for all remaining knots.
29 October, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Tommaso Cremaschi (Trinity College Dublin)
Title: Big and small surfaces and Nielsen-Thurston Classification
Abstract: We will give a short overview of the Nielsen-Thurston Classification problem (classifying isotopy classes of homeomorphisms) on finite-type surfaces and then move to infinite-type surface mentioning what is known and pointing out some difficulties. We will then discuss how to approximate, in the compact-open topology, a general self-homeomorphism of an infinite-type surface (joint with Y.Chandran) and potential definitions of pseudo-anosov mapping classes in the infinite-type setting (joint with F.Valdez).
5 November, 15:00-16:00, STRAND BLDG S4.29
Speaker: Francesco Lin (Columbia University)
Title: Divergence-free framings of three-manifolds via eigenspinors
Abstract: Gromov used convex integration to prove that any closed orientable three-manifold equipped with a volume form admits three divergence-free vector fields which are linearly independent at every point. We provide an alternative proof of this using geometric properties of eigenspinors in three dimensions. In fact, our proof shows that for any Riemannian metric, one can find three divergence-free vector fields such that at every point they are orthogonal and have the same non-zero length.
12 November, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Inder Kaur (Glasgow)
Title: A Lefschetz (1, 1) theorem for singular varieties
Abstract: The Lefschetz (1, 1) theorem is a classical result that tells us that for any smooth projective variety, a rational (1, 1) Hodge class comes from a algebraic cycle of codimension 1. In 1994, Barbieri-Viale and Srinivas gave a counter-example to the obvious generalization of this result to singular varieties. Inspired by Totaro, in this talk, I will give a modification of the statement of Lefschetz (1, 1) and show that it is satisfied by several singular varieties such as those with ADE-singularities and rational singularities. In particular, this Lefschetz (1,1) statement is satisfied by the varieties considered by Barbieri-Viale and Srinivas. This is joint work with A. Dan.
19 November, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Alastair Craw (Bath)
Title: Hilbert schemes of points on canonical surfaces, parts I and II
Abstract: Part I will be a gentle introduction to the Hilbert scheme of points in the plane, a moduli space that plays an important role in both algebraic geometry and geometric representation theory. I’ll discuss the notion of “a point” in algebraic geometry, and I’ll introduce a geometric invariant theory quotient construction of the Hilbert scheme of points in the plane. Part I will be a slides talk, and is intended to be accessible.
Part II will introduce recent joint work with Ryo Yamagishi in which we generalise well-known theorems of Fogarty (1968) and Beauville (1983) which show that the Hilbert scheme of points on a non-singular surface (admitting a symplectic form) is a non-singular variety of dimension 2n (admitting a symplectic form). The main result establishes analogous results for surfaces with canonical singularities (respectively, symplectic surface singularities). This part will be a board talk, and as such, is intended to be incomprehensible.
26 November, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Ben Sharp (Leeds)
Title: Index estimates for constant mean curvature surfaces in 3-manifolds
Abstract: In the first part of this talk we will give an introduction to CMC surfaces, define “index”, and discuss by exhibiting examples. Roughly speaking, CMC surfaces are critical with respect to area as long as we restrict to competitor surfaces which ‘enclose the same volume’. The index of a CMC surface is the number of ways we can locally deform it to reduce its area whilst enclosing the same amount of volume. In the second half of the talk we will show that the index of a CMC surface is bounded linearly from above in terms of its genus and a Willmore-type energy. When the mean curvature is of order 1 this Willmore-type energy is the area of the surface.
3 December, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Rachael Boyd (University of Glasgow)
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10 December, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: TBA (TBD)
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