Aleksandar Milivojevic

I am a mathematician, currently working in finance. Previously I was a William T. Tutte postdoctoral fellow at the University of Waterloo, and 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. Before that I was at the University of Zagreb.

My academic work lies in the interactions of rational homotopy theory with the topology and geometry of manifolds, with an occasional emphasis on (almost) complex manifolds.

Research

Submitted:

(with Frederik Benirschke) Punctures and formality for non-simply connected manifolds, 2024, pdf
(with Jonas Stelzig and Leopold Zoller) Formality is preserved under domination, 2023, arxiv

Accepted:

(with Gustavo Granja) Intersections of complex structures, 2024.
To appear in Homology, Homotopy and Applications.
arxiv

Published:

(with Michael Albanese, Spiro Karigiannis, and Lucía Martín-Merchán) Hodge theoretic results for nearly Kähler manifolds in all dimensions, 2026.
journal; arxiv
(with Jonas Stelzig and Leopold Zoller) Poincaré dualization and Massey products,
Journal of Homotopy and Related Structures, 2026.
journal; arxiv
(with Michael Albanese) Obstructions to almost complex structures following Massey,
L’Enseignement Mathématique, 2026.
journal; arxiv
Universal covers of non-negatively curved manifolds and formality,
Annals of Global Analysis and Geometry, 2024.
journal; arxiv
(with Jonas Stelzig) Bigraded notions of formality and Aeppli-Bott-Chern-Massey products,
Communications in Analysis and Geometry, 2024.
journal; arxiv
On the behavior of Massey products under field extension,
Journal of Pure and Applied Algebra, 2024.
journal; arxiv
(with Gustavo Granja) Topology of almost complex structures on six-manifolds,
SIGMA, 2022.
journal; arxiv
(with Jiahao Hu) Infinite symmetric products of rational algebras and spaces,
Comptes Rendus Mathématique, 2022.
journal; arxiv
The weak form of Hirzebruch’s prize question via rational surgery,
Journal of Topology and Analysis, 2022.
journal; pdf
On the characterization of rational homotopy types and Chern classes of closed almost complex manifolds,
Complex Manifolds, 2022.
journal; pdf
based on a large part of my thesis:link extended abstract from Oberwolfach workshop 2020: link
(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.
journal; arxiv
(with Michael Albanese) Spinʰ and further generalisations of spin,
Journal of Geometry and Physics, 2021.
journal; corrigendum; arxiv
On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds,
Proceedings of the American Mathematical Society, 2021.
journal; arxiv
Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence,
Complex Manifolds, 2020.
journal
(with Spencer Cattalani) Verifying the Hilali conjecture up to formal dimension twenty,
Journal of Homotopy and Related Structures, 2020.
journal; arxiv
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 spinᶜ Dirac operator,
Topology and its Applications, 2019.
journal; arxiv
(with Michael Albanese) On the minimal sum of Betti numbers of an almost complex manifold,
Differential Geometry and its Applications, 2019.
journal; arxiv

Errata and comments on publications.
(The two of some 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.)

Coauthors:

Michael Albanese
Frederik Benirschke
Spencer Cattalani
Bora Ferlengez
Gustavo Granja
Jiahao Hu
Spiro Karigiannis
Lucía Martín-Merchán
Jonas Stelzig
Leopold Zoller

Teaching

Lectures and assignments from my course PMATH965: ``Topics in Geometry and Topology - Rational homotopy theory in geometry'' are on the course overleaf.

Recorded talks

Geometria em Lisboa seminar, IST Lisbon, June 2022
link; here is a related Oberwolfach report: https://publications.mfo.de/handle/mfo/3920
Institut Mittag-Leffler program, February 2022
link
Stony Brook University capsule talks, May 2021
link

Some notes

Remark on the Deligne-Griffiths-Morgan-Sullivan formality criterion, 2023
(with Scott Wilson)
Invariant Dolbeault cohomology for homogeneous almost complex manifolds, 2022
(with Bora Ferlengez) A trichotomy of consequences of the existence of holomorphic charts on the six sphere, 2020; 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; 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
Some calculations of the rational homotopy type of the classifying space for fibrations up to fiber homotopy equivalence, 2018
A note on the difference between the sum of the Hodge numbers and Betti numbers on a non-Kähler complex manifold, 2018

Student mentorship

Bachelor 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.

Master students:

(unofficial) Assisted Peter Teichner in supervising Anton Ablov's Masters thesis at the University of Bonn, "Formality and coformality in rational homotopy theory"

Very old notes

Slides for a series of five lectures I gave virtually at IISER Kolkata in 2021, on topological aspects of (almost) complex manifolds.
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.

I am occasionally on MathOverflow: link

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