Organiser : Nick Shepherd-Barron
The talks of the Geometry workshop will be held in room K2.31 of the King's Building.
Tea & Coffee will be offered in the Bush House Arcade.
The talks of the Geometry workshop will be held in room K2.31 of the King's Building.
Tea & Coffee will be offered in the Bush House Arcade.
Invited speakers
Anne-Sophie Kaloghiros
Brunel U. UK |
Effie Kalfagianni
Michigan State U. USA |
Tyler Kelly
Birmingham U. UK |
Mark Lackenby
Oxford U. UK |
Johannes Nordström
Bath U. UK |
Peter Topping
Warwick U. UK |
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Tuesday 7th |
Wednesday 8th |
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14.15 - 15.05: Johannes Nordström
15.10 - 16.00: Anne-Sophie Kaloghiros 16.00 - 16.40: Tea & Coffee [BH(N) Arcade] 16.40 - 17.30: Peter Topping |
14.15 - 15.05: Mark Lackenby
15.10 - 16.00: Tyler Kelly 16.00 - 16.40: Tea & Coffee [BH(N) Arcade] 16.40 - 17.30: Effie Kalfagianni |
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Johannes Nordström
Asymptotically conical G2 solitons Abstract. G2 solitons are self-similar solutions to Bryant's Laplacian flow for closed G2-structures on 7-manifolds, a relative of Ricci flow. I will describe examples of G2 solitons that are asymptotically conical (of all three types: expanders, shrinkers and steady solitons) as well as a steady soliton with exponential volume growth. The solitons are defined on the anti-self-dual bundles of CP2 and S4 and have a cohomogeneity one action. This is joint work with Mark Haskins and Rowan Juneman. |
Mark Lackenby
Knot theory and machine learning Abstract. Knot theory is divided into several subfields. One of these is hyperbolic knot theory, which is focused on the hyperbolic structure that exists on many knot complements. Another branch of knot theory is concerned with invariants that have connections to 4-manifolds, for example the knot signature and Heegaard Floer homology. In my talk, I will describe a new relationship between these two fields that was discovered with the aid of machine learning. Specifically, we show that the knot signature can be estimated surprisingly accurately in terms of hyperbolic invariants. We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. Our main result is... that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This theorem has applications to Dehn surgery and to 4-ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that have odd linking number with the knot. My talk will outline the proofs of these results, as well as describing the role that machine learning played in their discovery. This is joint work with Alex Davies, Andras Juhasz, and Nenad Tomasev. |
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Anne-Sophie Kaloghiros
Some examples of K-moduli spaces of Fano 3-folds Abstract. Recent advances in the study of K-stability have shown that there is a projective good moduli space of K-polystable Q-Gorenstein smoothable Q-Fano varieties of dimension n and volume V.
The classification of smooth Fano 3-folds is due to Iskovshikh, Mori and Mukai and dates back to the 80s. While the classification is non-modular, it contains rich information on the geometry of Fano 3-folds. Using the classification, I will show how to explicitly construct and understand some low-dimensional K-moduli spaces of Fano 3-folds.
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Tyler Kelly
Open Mirror Symmetry for Landau-Ginzburg models Abstract. Landau-Ginzburg models are triplets of data (X, G, W) consisting of a quasi-affine variety X, a complex-valued algebraic function W: X → C, and a finite group G acting on X so that W is invariant. Traditionally, Landau-Ginzburg (LG) models have been viewed as mirrors of Fano manifolds (e.g., projective space), however recently they are also viewed as noncommutative deformations of hypersurfaces in projective space where one can use the enumerative geometry of a LG model to learn about the enumerative geometry of the original hypersurface.... Unfortunately, constructing mirrors for LG models is hard and typically done ad hoc using combinatorial means. In joint work with Mark Gross and Ran Tessler, we have started to construct open enumerative theories for LG models and interpret open enumerative invariants as a means to reconstruct their mirrors as generating functions of the invariants. We will explain closed / open enumerative invariants for LG models and then the mirror interpretation of these constructions. |
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Peter Topping
Hamilton's Pinching Conjecture Abstract. I will outline some of my recent work with Man Chun Lee that when combined with work of Lott and Deruelle-Schulze-Simon gives a full resolution of Hamilton’s Pinching Conjecture. As I will explain, this conjecture can be viewed as a scale-invariant version of Bonnet-Myer’s theorem for 3-manifolds. The proof relies on Ricci flow. |
Effie Kalfagianni
Knot crossing numbers and colored jones polynomials. Abstract. It has long been known that the quadratic term in the degree of the colored Jones polynomial invariant of a knot is bounded above in terms of the crossing number of the knot. We will present joint work with Cristine Lee in which we determine the class of knots for which this bound is sharp, and we will discuss how to apply it to determine the crossing numbers of the first infinite families of prime satellite knots. |