SOLNESS: Castles in the air?

HILDA: Yes! Castles in the air — they're so easy to hide away in. And easy to build too.

(Looking contemptuously at him.) Especially for builders who have a dizzy conscience.

— Henrik Ibsen, *The Master Builder*

Department of Mathematics, Columbia University

2990 Broadway, New York, NY 10027, USA

*office:* 627

*email:* aleksander.doan@columbia.edu

I am a Junior Fellow of the Simons Society of Fellows and a Postdoctoral Research Scientist at Columbia University, a position held in conjunction with a Junior Research Fellowship at Trinity College, University of Cambridge. I completed my PhD in 2019 at Stony Brook University, supervised by Simon Donaldson. Here is a picture of me; and another one here.

Spring 2021: Ordinary Differential Equations (MATH 2030)

- differential geometry
- gauge theory
- symplectic geometry
- low-dimensional topology

Research summary: a short, non-technical version and an extended version

Recordings of some of my talks: Institute for Advanced Study, Simons Center, Simons Collaboration on Special Holonomy

In this short interview published by the Simons Foundation I talk about topology and my work in non-technical terms.

*The Gopakumar-Vafa finiteness conjecture* (with E. Ionel and T. Walpuski)

arXiv:2103.08221 (2021)

*Equivariant Brill-Noether theory for elliptic operators and super-rigidity of J-holomorphic maps* (with T. Walpuski)

arXiv:2006.01352 (2020)

*Counting embedded curves in symplectic 6-manifolds* (with T. Walpuski)

arXiv:1910.12338 (2019)

*Castelnuovo's bound and rigidity in almost complex geometry* (with T. Walpuski)

Advances in Mathematics (2020) / arXiv:1809.04731

*Deformation theory of the blown-up Seiberg-Witten equation in dimension three* (with T. Walpuski)

Selecta Mathematica (2020) /
arXiv:1704.02954

*On counting associative submanifolds and Seiberg-Witten monopoles* (with T. Walpuski)

Pure and Applied Mathematics Quarterly (2019) / arXiv:1712.08383

*On the existence of harmonic Z _{2} spinors* (with T. Walpuski)

Journal of Differential Geometry (2019) / arXiv:1710.06781

*Seiberg-Witten monopoles with multiple spinors on a surface times a circle*

Journal of Topology (2019) /
arXiv:1701.07942

*Adiabatic limits and Kazdan-Warner equations*

Calculus of Variations and PDE (2018) / arXiv:1701.07931

My PhD dissertation at Stony Brook University. It incorporates some of the material from my first five papers, three of which were written in collaboration with Thomas Walpuski. The introduction is a short survey of generalized Seiberg-Witten equations, Fueter sections, and their relation to gauge theory in higher dimensions.

Undergraduate projects:

Symplectic cohomology for stable fillings

Part III essay on symplectic fillability

Critical points of one-dimensional Gaussian mixtures

Bachelor's thesis on the Picard-Lefschetz theorem (in Polish)