Aleksandar MilivojevicGraduate studentI am a fourth-year graduate student, interested mostly in algebraic topology. I think about rational homotopy theory and what it can say about complex manifolds. My advisor is Dennis Sullivan. You can find my CV here. ResearchOn the minimal sum of Betti numbers of an almost complex manifold (with Michael Albanese), Differential Geometry and its Applications, 2018. (arxiv: https://arxiv.org/abs/1805.04751)Some short notes (informal)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 sixth k-invariant in the Postnikov tower for BSO(3) 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. And believe it or not, not all homogeneous spaces are formal. 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. Closed Lie groups are rationally products of odd spheres. A symplectic non-Kaehler complex threefold all of whose odd Betti numbers are even, and an almost-complex four manifold 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, 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, titled "Symplectic non-Kaehler manifolds". Notes for a talk I gave at the Stony Brook graduate student seminar 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". |