I am Junior Professor in the Algebra and Number Theory group at the Department of Mathematics, of Universität Hamburg.

I am PI at the new SFB (Collaborative Research Center) on "Higher Structures, Moduli and Integrable Systems"". I am also a member of the "Cluster of Excellence" Quantum Universe.

Before that I was a College Associate Lecturer and Fellow of St John's College, affiliated with the University of Cambridge, a Senior Research Associate at Lancaster University, a Postdoctoral Fellow at the Max Planck Institute for Mathematics in Bonn and a College Lecturer and Fellow of Christ's College.

Contact

Room 336, Geomatikum

Email: julian.holstein, uni-hamburg.de

Research

My research interests lie at the intersection of algebraic geometry and algebraic topology with higher category theory and homotopical algebra.

Here is a list of my papers and preprints in reverese chronological order:

1. Gluing invariants of Donaldson-Thomas type -- Part II: Matrix factorizations uses our results on the Darboux stack of a (-1)-shifted symplectic stack to glue sheaves of matrix factorizations up to isotopy in the presence of orientation data. This is joint work with Benjamin Hennion and Marco Robalo.
2. Koszul duality and Calabi-Yau structures shows that under Koszul duality smooth (respectively proper) Calabi-Yau structures on dg categories correspond to proper (respectively smooth) Calabi-Yau structures on pointed curved coalgebras. This is joint work with Manuel Rivera .
3. Gluing invariants of Donaldson-Thomas type -- Part I: the Darboux stack studies the Darboux stack of a (-1)-shifted symplectic stack, the moduli of all local presentations as a derived critical locus. This is the first part of a project on gluing categorified Donaldson-Thomas invariants, in particular sheaves of matrix factorizations. This is joint work with Benjamin Hennion and Marco Robalo.
4. Hochschild cohomology of the second kind: Koszul duality and Morita invariance defines a Hochschild cohomology for dg or curved algebras that is Morita invariant compatible with global Koszul duality. We also discuss applications to topology and algebraic geometery. This is joint work with Ai Guan and Andrey Lazarev.
5. Enriched Koszul duality for dg categories establishes a monoidal equivalence betweend dg categories and a category ptdco* of curved coalgebras. Moreover it shows that dg cat is a ptdco*-enriched model category. The upshot is that important but somewhat subtle constructions like the internal hom of dg categories can be easily and conceptually computed with the help of coalgebras. This is joint work with Andrey Lazarev. To appear in Documenta Mathematica.
6. Categorical Koszul duality establishes Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. This may be seen as a linearization of the Quillen equivalence between quasicategories and simplicial categories and provides a conceptual interpretation of the dg nerve of a dg category. This is joint work with Andrey Lazarev. Published in Advances in Mathematics, Volume 409, Part B (2022). The latest arxiv version corrects a statement in the published version.
7. Analytification of mapping stacks examines the interplay between the analytification functor and the mapping stack functor in derived geometry. In particular we show that in good cases the mapping stack between two analytifications is the analytification of the mapping stack. Along the way we prove a number of very useful technical results about analytic perfect complexes, analytic Tannaka duality and working with the derived analytification functor. This is joint work with Mauro Porta. Published in Algebraic Geometry (2025).
8. Homotopy theory of monoids and derived localization derives a number of interesting consequences from the close relationship between the algebraic bar construction and the nerve construction on monoids. We prove that Adams' cobar construction is the analogue of the loop space even for non-simply connected spaces, and we model topological spaces by an infinity category of discrete monoids. This is joint work with Joe Chuang and Andrey Lazarev. Published in Journal of Homotopy and Related Structures (2021).
9. Maurer-Cartan moduli and theorems of Riemann-Hilbert type considers different notions of equivalence for Maurer-Cartan elements in dg algebras. As an application we compute some categories of twisted modules and give new descriptions of infinity local systems. This is joint work with Joe Chuang and Andrey Lazarev. Published in Applied Categorical Structures (2021). Unfortunately the published version contains a mistake that is fixed in the latest arXiv version, thanks to Zhaoting Wei for pointing out the problem and the solution!
10. The global derived period map constructs an analogue of Griffiths' period map in derived geometry. This is joint work with Carmelo di Natale. Published in Advances in Mathematics, Volume 353 (2019).
11. Explicity homotopy limits of dg-categories and twisted complexes constructs some explicit homotopy limits of dg-categories and thus gives a description of the twisted complexes of Toledo and Tong. This is joint work with Zhaoting Wei and Jonathan Block. Published in Homology, Homotopy ond Applications, Volume 19 (2017), Number 2.
12. Morita Cohomology constructs cohomology of a topological space X with coefficients in the dg-category of perfect complexes in two different ways. The results agree and may be characterized as the dg category of infinity local systems on X. Published in Mathematical Proceedings of the Cambridge Philosophical Society, Volume 158, Issue 01.
13. Morita Cohomology and homotopy locally constant presheaves shows that Morita cohomology is moreover equivalent to the dg-category of homotopy locally constant sheaves of perfect complexes on X. Published in Mathematical Proceedings of the Cambridge Philosophical Society, Volume 158, Issue 01.
14. Properness and simplicial resolutions for the model category dgCat proves two technical results about the category of dg-categories. When defined over a ring of flat dimension 0 dgCat is a left proper model category. Simplicial resolutions in dgCat are explicitly given by categories of Maurer-Cartan elements. Note that the newest arXiv version contains an erratum, fixing some errors in the proof of the main theorem. Published in Homology, Homotopy ond Applications, Volume 16 (2014), Number 2.

The paper title links to the arXiv, the journal title links to the published versions; there are no differences in content unless noted otherwise.

The last three papers are based on my PhD thesis Morita Cohomology. My advisor was Ian Grojnowski.

Seminars

Together with Birgit Richter I run a research seminar on algebraic topology (paused for summer semester 2023).

Together with Craig Lawrie and Ingo Runkel I organize the ZMP Colloquium on mathematical physics and the ZMP Seminar on Quantum Physics and Geometry.

Group

My current PhD students are Kristoffer Rasmussen, Patrick Antweiler and Yannick Hoyer.

Dasha Poliakova is working with me as a postdoc

Teaching

In Sommersemseter 2025 I will teach a course on Sheaves in Topology (Master).

In Wintersemester 2024/25 I am teaching a Proseminar zum Computerbeweisassistenten LEAN (Bachelor),

In Wintersemester 2023/24 I taught Linear Algebra I (Bachelor) and in Sommersemester 2024 I taught Linear Algebra II.

In Sommersemseter 2023 I taught Algebraic Topology (Master).

In Wintersemester 2022/23 I taught a Proseminar "Mathematik und Alltag".

In Sommersemester 2022 I taught Seminar über Topologie (Bachelor) on topology and groups.

In Wintersemester 2021/22 I taught Topologie (Bachelor).

In Sommersemester 2021 I taught a Proseminar über Zahlentheorie (LPSI).

In Wintersemester 2020/21 I taught a course on Rational Homotopy Theory (Master).

In Sommersemester 2020 I taught a Seminar über Topologie (Bachelor) on de Rham cohomology.

In Wintersemester 2019/20 I taught Topologie (Bachelor) at Universität Hamburg.

In Lent Term 2015 I lectured a Part III course, Homological and Homotopical Algebra at University of Cambridge.

Family

My mother is Susannne Holstein, she is an artist.