Aleksandar Milivojevic

      Graduate student


      I am a third-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.

      • Email: milivojevic[at]math.stonybrook.edu
      • Office: 3-104, Math Department, Stony Brook University

      Research

      On the minimal sum of Betti numbers of an almost complex manifold (with Michael Albanese), https://arxiv.org/abs/1805.04751 (2018)

      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.

      Some computations 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 word on cohomologically split fibrations.

      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.


      Other

      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".