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, Math Department, Stony Brook University
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. Spaces have the homotopy type of a finite or countable cell complex. Minimal models are in the
sense of rational homotopy theory.
- The Serre symmetry of Hodge numbers persists through all pages of the Froelicher spectral sequence of a compact complex manifold.
- 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 Froelicher 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-Kaehler complex manifold.
- A nilmanifold is a torus iff all of its triple
Massey products vanish.
- A symplectic non-Kaehler 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),
and those of CP2#CP2 and CP2#-CP2.
Notes for a talk I gave at the CUNY Graduate Center K-Theory seminar in November 2018, on setting up the Froelicher spectral sequence and working with it.
Notes for a talk I gave at the Stony Brook Symplectic Geometry student seminar in August 2018, titled "Symplectic non-Kaehler 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.