The lyf so short, the craft so longe to lerne.
Aleksandar MilivojevicGraduate student
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.
Some short notesThe 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) here.
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.
OtherNotes 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.