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)
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5 November, Part I: 15:00-15:40, Part II: 15:45-16:30, 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)
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19 November, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Alastair Craw (Bath)
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26 November, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Ben Sharp (Leeds)
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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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