Aleksandar Milivojevic

The lyf so short, the craft so longe to lerne.

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

Dennis Sullivan

here

Email: milivojevic[[at]]mpim-bonn.mpg.[[de]]

Research

Preprints:

(with Jonas Stelzig and Leopold Zoller) Formality is preserved under domination, 2023. arxiv.org/abs/2306.12364.

On the behavior of Massey products under field extension, 2023. arxiv.org/abs/2304.10296.

(with Jonas Stelzig and Leopold Zoller) Poincaré dualization and formal domination, 2022. arxiv.org/abs/2203.15098.

(with Jonas Stelzig) Bigraded notions of formality and Aeppli-Bott-Chern-Massey products, 2022. arxiv.org/abs/2202.08617.

Publications:

(with Gustavo Granja) Topology of almost complex structures on six-manifolds, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) 2022. (arxiv: arxiv.org/abs/2207.12946)

(with Jiahao Hu) Infinite symmetric products of rational algebras and spaces, Comptes Rendus Mathématique. 2022. (arxiv: arxiv.org/abs/2108.09794)

The weak form of Hirzebruch's prize question via rational surgery, Journal of Topology and Analysis, 2022. pdf.

On the characterization of rational homotopy types and Chern classes of closed almost complex manifolds, Complex Manifolds, 2022. pdf. Based on a large part of my thesis; I gave an overview at the 2020 Oberwolfach mini-workshop "Almost complex geometry", with extended abstract here.

(with Bora Ferlengez and Gustavo Granja) On the topology of the space of almost complex structures on the six sphere, New York Journal of Mathematics, 2021. (arxiv: https://arxiv.org/abs/2108.00750)

(with Michael Albanese) Spin^h and further generalisations of spin, Journal of Geometry and Physics, 2021. (arxiv: https://arxiv.org/abs/2008.04934). Corrigendum.

On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds, Proceedings of the American Mathematical Society, 2021. (arxiv: https://arxiv.org/abs/2011.02928)

Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence, a short note in Complex Manifolds, 2020.

(with Spencer Cattalani) Verifying the Hilali conjecture up to formal dimension twenty, Journal of Homotopy and Related Structures, 2020. (arxiv: https://arxiv.org/abs/2210.00607). Research done as part of the undergraduate Directed Reading Program at Stony Brook.

(with Michael Albanese) Connected sums of almost complex manifolds, products of rational homology spheres, and the twisted spin^c Dirac operator, Topology and its Applications, 2019. (arxiv: https://arxiv.org/abs/1905.01760)

(with Michael Albanese) On the minimal sum of Betti numbers of an almost complex manifold, Differential Geometry and its Applications, 2019. (arxiv: https://arxiv.org/abs/1805.04751)

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.)

Recorded talks

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

Teaching
I am currently not teaching. My CV contains a list of courses taught and other teaching activities, including outreach.

Student mentorship
- Bachelors students:
  - Spencer Cattalani, Stony Brook University Directed Reading Program; Verifying the Hilali conjecture up to formal dimension twenty
  - 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"
Older notes, translations, miscellaneous
A symplectic non-Kähler 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.

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.

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 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 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.

I am occasionally on MathOverflow.

Aleksandar Milivojević

Research

Preprints:

Publications:

Recorded talks

Some notes

Teaching

Student mentorship

Older notes, translations, miscellaneous