Dehn twists and free subgroups of symplectic mapping class groups

Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres.Comment: 37 pages, 9 figures; v2: corrected proof of Prop. 4.7, and other minor changes following referee report; v3: minor changes only; accepted, Journal of Topolog

Homological mirror symmetry for log Calabi-Yau surfaces

Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M \to \mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b \text{Coh}(Y)$, and the wrapped Fukaya category $\mathcal{W} (M)$ is isomorphic to $D^b \text{Coh}(Y \backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.Comment: Comments welcome

Families of monotone Lagrangians in Brieskorn-Pham hypersurfaces.

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn-Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian S 1 × Σ g in C 3 , distinguished by soft invariants for any genus g ≥ 2 ; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn-Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms.Simons foundation, National Science Foundation, Trinity college, Cambridg

Genetic mechanisms of critical illness in COVID-19.

Host-mediated lung inflammation is present1, and drives mortality2, in the critical illness caused by coronavirus disease 2019 (COVID-19). Host genetic variants associated with critical illness may identify mechanistic targets for therapeutic development3. Here we report the results of the GenOMICC (Genetics Of Mortality In Critical Care) genome-wide association study in 2,244 critically ill patients with COVID-19 from 208 UK intensive care units. We have identified and replicated the following new genome-wide significant associations: on chromosome 12q24.13 (rs10735079, P = 1.65 × 10-8) in a gene cluster that encodes antiviral restriction enzyme activators (OAS1, OAS2 and OAS3); on chromosome 19p13.2 (rs74956615, P = 2.3 × 10-8) near the gene that encodes tyrosine kinase 2 (TYK2); on chromosome 19p13.3 (rs2109069, P = 3.98 ×  10-12) within the gene that encodes dipeptidyl peptidase 9 (DPP9); and on chromosome 21q22.1 (rs2236757, P = 4.99 × 10-8) in the interferon receptor gene IFNAR2. We identified potential targets for repurposing of licensed medications: using Mendelian randomization, we found evidence that low expression of IFNAR2, or high expression of TYK2, are associated with life-threatening disease; and transcriptome-wide association in lung tissue revealed that high expression of the monocyte-macrophage chemotactic receptor CCR2 is associated with severe COVID-19. Our results identify robust genetic signals relating to key host antiviral defence mechanisms and mediators of inflammatory organ damage in COVID-19. Both mechanisms may be amenable to targeted treatment with existing drugs. However, large-scale randomized clinical trials will be essential before any change to clinical practice

Symplectic properties of Milnor fibres

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.69Cataloged from PDF version of thesis.Includes bibliographical references (pages 121-123).We present two results relating to the symplectic geometry of the Milnor fibres of isolated affine hypersurface singularities. First, given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres. In the second half of this thesis, we study exact Lagrangian tori in Milnor fibres. The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones.by Ailsa Macgregor Keating.Ph. D

On the order of Dehn twists

This note records the order of a higher dimensional Dehn twist in a range of topologically significant groups

Symplectomorphisms of exotic discs

We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor--Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration. The Appendix by S. Courte shows that for our symplectic structure the map from compactly supported symplectic mapping classes to compactly supported smooth mapping classes is in fact surjective.R.C. is supported by NSF grant DMS-1608018 and a BBVA Research Fellowship. A.K. was partially supported by NSF grant DMS{1505798, by a Junior Fellow award from the Simons Foundation, and by NSF grant DMS-1128155 whilst at the Institute for Advanced Study. I.S. is partially supported by a Fellowship from the EPSRC

Symplectomorphisms of exotic discs

We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor--Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration. The Appendix by S. Courte shows that for our symplectic structure the map from compactly supported symplectic mapping classes to compactly supported smooth mapping classes is in fact surjective.R.C. is supported by NSF grant DMS-1608018 and a BBVA Research Fellowship. A.K. was partially supported by NSF grant DMS{1505798, by a Junior Fellow award from the Simons Foundation, and by NSF grant DMS-1128155 whilst at the Institute for Advanced Study. I.S. is partially supported by a Fellowship from the EPSRC