Ailsa Keating


keating -- math.columbia.edu

Office: 606

Department of Mathematics, Columbia University
2990 Broadway
New York, NY 10027

Welcome! I am a Junior Fellow in the Simons Society of Fellows, hosted by the mathematics department at Columbia University. I am also a Title A Fellow (Junior Research Fellow) at Trinity College, Cambridge, currently `on undertaking', which means on leave.

Seminars etc.

Denis Auroux' Eilenberg lectures, Fall 2016, Columbia.

I help co-organize the Symplectic Geometry, Gauge Theory and Categorification Seminar at Columbia.

Research

I study problems in symplectic geometry. I'm particularly interested in combining "modern" invariants, such as Floer cohomology or the Fukaya category, with tools from other areas, such as classical singularity theory or the study of mapping class groups. Here is an introduction to my thesis aimed at an interested lay scientist.

Homological mirror symmetry for hypersurface cusp singularities. We study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. We show that the Milnor fibres of each of these carries a distinguished Lefschetz fibration; its derived directed Fukaya category is equivalent to the derived category of coherent sheaves on a smooth rational surface Y_{p,q,r}. By using localization techniques on both sides, we get an isomorphism between the derived wrapped Fukaya category of the Milnor fibre and the derived category of coherent sheaves on a quasi-projective surface given by deleting an anti-canonical divisr D from Y_{p,q,r}. In the cusp case, the pair (Y_{p,q,r}, D) is naturally associated to the dual cusp singularity, tying into Gross, Hacking and Keel's proof of Looijenga's conjecture.mann surfaces.
Here are some slides from a AMS talk on this.

Lagrangian tori in four-dimensional Milnor fibres Accepted, GAFA. The Milnor fibre of any isolated hypersurface singularity contains 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. I construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities (one level of complexity up from the simple ones).

Dehn twists and free subgroups of symplectic mapping class groups, Journal of Topology (2014) 7 (2): 436-474, doi:10.1112/jtopol/jtt033
Given two Lagrangian spheres in an exact symplectic manifold, I present 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. I also construct families of examples containing such spheres.
Here are some slides from a AMS talk on this.

Past reading groups

Fall 2015: preprint seminar on symplectic topology and related topics.

Spring 2015: preprint seminar on symplectic topology and related topics.

Fall 2014: reading group on Categorical Dynamics and Symplectic Geometry.

Past teaching

At Columbia:

Spring 2016: Introduction to knot theory (Math W4052) -- on CourseWorks

Spring 2015: Math V1202, Section 2 (Calculus IV)

At MIT:

Summer 2013: instructor for 18.089
Spring 2013: recitation instructor for 18.03
Spring 2012: recitation instructor for 18.06 (linear algebra)
     Materials: notes from review session #1, #2, #3, #4; addendum to recitation #9
Summer 2011: mentor for SPUR
Spring 2011: recitation instructor for 18.03 (differential equations)
Fall 2010: TA for 18.905 (algebraic topology)
Summer 2010: mentor for RSI

Warning: while I've tried to proof-read the notes as best I could, there might still be typos! If you find any, it would be great if you could let me know.

 

Past conferences

Topological and Quantitative Aspects of Symplectic Manifolds, a conference in honor of Dusa McDuff's 70th birthday, March 17–20, 2016.

2013 MIT-RTG Mirror Symmetry Workshop, with Mark Gross

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