The lyf so short, the craft so longe to lerne.
I am a fifth-year graduate student, interested mostly in algebraic topology. I think about (rational) homotopy theory and what it can say about (almost) complex manifolds.
My advisor is Dennis Sullivan.
You can find my CV here.
Office: 3-104, Mathematics Department, Stony Brook University
- (with M. Albanese) On the minimal sum of Betti numbers of an almost complex manifold, Differential Geometry and its Applications, 2019. (arxiv: https://arxiv.org/abs/1805.04751)
- (with M. Albanese) Connected sums of almost complex manifolds, products of rational homology spheres, and the twisted spinc Dirac operator, Topology and its Applications, 2019. (arxiv: https://arxiv.org/abs/1905.01760)
- (with S. Cattalani) Verifying the Hilali conjecture up to formal dimension twenty, Journal of Homotopy and Related Structures, 2020. Research done as part of the Directed Reading Program at Stony Brook.
- Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence, a short note in Complex Manifolds, 2020.
Some short notes
The material below is a mix of mostly expository material and some original results. All manifolds are (unless otherwise stated) smooth, closed, and connected. Minimal models are in the sense of rational homotopy theory.
- On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds.
- On the sixth k-invariant in the Postnikov tower for BSO(3)
- Geometric formality is not a rational homotopy invariant
- The rational homotopy type of the classifying space for X-fibrations up to fiber homotopy equivalence, with examples (including one where the fiber space X is non-formal).
- Some calculations with the Frölicher spectral sequence.
- A discussion on almost complex and stably almost complex structures, and
the obstructions to such structures in low dimensions. You can find the
minimal models of some relevant homogeneous spaces SO(2n)/U(n)
- The rational homotopy type of the space of almost complex structures on the six-sphere.
- A note on the difference between the sum of the Hodge numbers and Betti numbers on a non-Kähler complex manifold.
- A symplectic non-Kähler complex threefold all of whose
odd Betti numbers are even, and some almost-complex four manifolds with no
complex structure. Here is an example of a non-integrable almost complex structure connected by a path to an integrable
complex structure on a smooth manifold of even dimension four or greater.
- The minimal models of the complex Grassmannians G(2,4), G(2,5), G(2,6), G(3.6)
Notes for a talk I gave at the City University of New York Graduate Center K-Theory seminar in November 2018, on setting up and calculating the Frölicher spectral sequence.
Notes for a talk I gave at the Stony Brook Symplectic Geometry student seminar in August 2018, titled "Symplectic non-Kähler manifolds".
Notes for a talk I gave at the Stony Brook graduate student seminar in February 2018 as an introduction to rational homotopy theory.
A 1975 paper by Deligne and Sullivan, Complex vector bundles with discrete structure group,
translated from French to English. Here you can find the original.
A brief review of the more topologically-oriented chapters in Freed and
Uhlenbeck's "Instantons and Four Manifolds".
I'm occasionally on MathOverflow.