Question for visitor: is there a closed symplectic 2n-manifold such that the i-th Betti number is strictly greater than the (i+2)-nd Betti number, for some i+2 ≤ n?
Aleksandar Milivojević
I am a William T. Tutte postdoctoral fellow at the University of Waterloo. Prior to this I was a postdoc at the Max Planck
Institute for Mathematics in Bonn from 2021 to 2023.
In 2021 I obtained my PhD from Stony Brook University, under the guidance of
Dennis Sullivan.
I mostly think about rational homotopy theory and what it can say about the topology and geometry of manifolds,
with an occasional emphasis on (almost) complex manifolds.
(with Jonas Stelzig) Bigraded notions of formality and
Aeppli-Bott-Chern-Massey products, 2022. arxiv.org/abs/2202.08617. To appear in Communications in Analysis and Geometry.
A list of errata and comments on publications: pdf. (The two of consequence are:
concerning spin^h, where we can only prove
that non-compact orientable 6- and 7-manifolds
are spin^h under an additional assumption, see published corrigendum; and a
corollary of almost complex realization for trivial Euler
characteristic and signature in dimensions 0 mod 4 which had a
missing assumption in the versions prior to the published version.)
Teaching
In the Fall 2024 term I am teaching PMATH965: Topics in Geometry and
Topology - Rational homotopy theory in geometry.
Geometria em Lisboa seminar, IST Lisbon, June 2022 (in person, one hour): link. Here is a related Oberwolfach report.
At the "Higher algebraic structures in algebra, topology and geometry" program at Institut Mittag-Leffler, Formality and non-zero degree maps, February 2022 (in person, half hour): link.
Here is a related Oberwolfach report (same as above).
Stony Brook University capsule talks, thesis overview, May 2021 (online, fifteen minutes plus discussion): link
Some notes
Remark on the Deligne-Griffiths-Morgan-Sullivan formality criterion, 2023. pdf.
(with Scott Wilson) Invariant Dolbeault cohomology for homogeneous almost complex manifolds, 2022. pdf.
(with Bora Ferlengez) A trichotomy of consequences of the existence of holomorphic charts on the six sphere, 2020. pdf; a note related to the paper "On the topology of the space of almost complex structures on the six sphere". Sections 1 and 2 feature alternative arguments for weaker versions of some results in the paper, and Section 3 is disjoint from the paper.
(with Maximilian Keßler and Dmytro Rudenko), On almost complex rational quaternionic and octonionic projective spaces, 2022, pdf. Part of the MPIM Bonn Internship Program (see below under Student mentorship).
On the sixth k-invariant in the Postnikov tower for BSO(3), 2018. pdf
Some calculations of the rational homotopy type of the classifying space for fibrations up to fiber homotopy equivalence, 2018. pdf
A note on the difference between the sum of the Hodge numbers and Betti numbers on a non-Kähler complex manifold, 2018. pdf
Maximilian Keßler and Dmytro Rudenko, MPIM Bonn Internship Program; draft pdf. Hirzebruch proved that no quaternionic projective space HP^n, with its
standard smooth structure, admits an almost complex structure. Keßler and Rudenko gave a partial generalization showing that if HP^n is
rationally homotopy equivalent to a closed almost complex manifold, then n modulo 12 is 0,3,8, or 11. I had shown that for n=3 one can indeed
find such an almost complex manifold; they identified my solution as one in an infinite family of complex cobordism classes, and made some
progress towards the case of n=8.
Masters students:
(unofficial) Assisted Peter Teichner in supervising Anton Ablov's Masters thesis at the University of Bonn, "Formality and coformality in rational homotopy theory"
Notes for talks, older notes,
translations, miscellaneous
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 1956 paper by Haefliger, Sur l'extension du groupe structural d'un espace fibré,
translated from French to English. The original can be found here in the Comptes Rendus archives, pp.558-560. This is the paper where the second Stiefel-Whitney class was first explicitly identified as the (only) obstruction for an oriented bundle to admit a spin structure.