### News

I am a postdoctoral fellow at MSRI/SLMath from January–May 2024 for the spring programme on Noncommutative Algebraic Geometry

### Research interests

My main research centres around noncommutative algebraic geometry, representation theory and mathematical physics. I am developing new techniques in the deformation theory of algebraic structures to describe and study

deformations of associative and A∞ algebras via reduction systems

(strict) deformation quantization

deformations of Abelian or derived categories of coherent sheaves and noncommutative algebraic geometry

noncommutative deformations of singularities and their singularity categories

geometry of noncommutative varieties

deformations of Fukaya categories and mirror symmetry

A∞ structures and deformations of graded algebras

noncommutative deformation theory (i.e. deformations over a noncommutative base).

Together with Zhengfang Wang I developed a combinatorial approach to the deformation theory of associative algebras (in this paper). Jointly with Daniel Labardini Fragoso I am developing a visual applet implementing this method to help with the combinatorics when studying particular examples: QUIVERed.

I am also working on a (for now) separate line of research concerning

categorical techniques in theoretical physics, in particular the categorification of scattering amplitudes.

### CV

I am currently a postdoc at the University of Cologne in the group of Sibylle Schroll.

### Short biography

born in Germany, 10 years of school education in Germany

one year of high school in the UK

MA(Hons) Philosophy & Mathematics (undergraduate) at the University of Edinburgh

one year of volunteer work ("Zivildienst") in Takasaki, Japan

MSc Mathematical Sciences at the University of Tokyo (with Toshiyuki Kobayashi)

PhD Mathematics (defended December 2018) from the University of Münster, Germany — my advisers were Elizabeth Gasparim and Lutz Hille

9-month postdoc at the MPIM in Bonn where my mentor was Catharina Stroppel

2-year postdoc at the University of Freiburg with Katrin Wendland

currently postdoc at the University of Cologne with Sibylle Schroll

My CV is available as PDF.

### Articles

### In progress

A∞ deformations of partially wrapped Fukaya categories of surfaces, with S. Schroll, Z. Wang

Deformation quantization and singularity categories, with M. Kalck, Z. Wang

A new structure sheaf on the spectrum of a module category, with A. Siqveland

Singular Hochschild cohomology of affine toric varieties, with M. Filip, Z. Wang

### Preprints

A∞ deformations of extended Khovanov arc algebras and Stroppel's Conjecture, with Z. Wang [arXiv]

Towards a categorification of scattering amplitudes, with P. Oak, A. Pal, K. Ray, H. Treffinger [arXiv]

Deformations of path algebras of quivers with relations, with Z. Wang [arXiv]

### Journal publications

Deformations of categories of coherent sheaves via quivers with relations, with Z. Wang, Algebr. Geom. 11 (1), 1–36 (2024) [doi] [arXiv]

Strict quantization of polynomial Poisson structures, with P. Schmitt, Comm. Math. Phys. 398 (3), 1085–1127 (2023) [doi] [arXiv]

Learning scattering amplitudes by heart, with K. Ray, Phys. Lett. B 820, 136594 (2021) [doi] [arXiv]

PhD project

Quantizations of local surfaces and rebel instantons, with E. Gasparim, J. Noncommut. Geom. 16 (1), 311–351 (2022) [doi] [arXiv]

Deformation–obstruction theory for diagrams of algebras and applications to geometry, with Y. Frégier, J. Noncommut. Geom. 14 (3), 1019–1047 (2020) [doi] [arXiv]

Classical deformations of noncompact surfaces and their moduli of instantons, with E. Gasparim, J. Pure Appl. Algebra 223 (6), 2543–2561 (2019) [doi] [arXiv]

A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors, with E. Ballico, E. Gasparim, L. Grama, L. A. B. San Martin, Manuscripta Math. 158 (1–2), 85–101 (2019) [doi] [arXiv]

Master's project

Deformations of the discrete Heisenberg group, Proc. Japan Acad. Ser. A Math. Sci. 89 (4), 55–59 (2013) [doi] [arXiv]

Graduate research

Isomorphisms of moduli spaces, with C. C. Amilburu, B. Callander and E. Gasparim, in Proceedings Second Latin Congress on Symmetries in Geometry and Physics, Mat. Contemp. 41, 1–16 (2012) [doi] [arXiv]

### Theses

Moduli of vector bundles on non-compact surfaces and their deformations, PhD thesis, University of Münster (2018)

Deformations of the discrete Heisenberg group, Master's thesis, University of Tokyo (2012)

### Videos

Extended Khovanov arc algebras: Quivers, Hochschild cohomology and A∞ deformations

Informal Categorification Seminar, Columbia UniversityMinicourse Deformation theory via L∞ algebras

Higher Structures in Deformation Theory, University of FreiburgDeformations of categories of coherent sheaves via quivers with relations

LAGOON SeminarMinicourse Deformations of path algebras of quivers with relations (slides first, second, fourth talk)

New Trends in Representation Theory, Hausdorff Research Institute for Mathematics in Bonn)

### Conference and seminar organization

### Conferences, workshops and research schools

Cologne Representation Theory Day

17 October 2023

University of Cologneτ-Research School

5–8 September 2023

University of CologneQuivers, Representation Theory and Physics

7–9 September 2022

University of CologneHigher Structures in Deformation Theory

31 August – 2 September 2022

University of FreiburgGEOQUANT 2021 — Geometry & Quantization

International school "Deformation quantization and convergence" and international conference (hybrid events)

2–13 August 2021

University of Freiburg

### Research seminars

Longitudinal Algebra and Geometry Open ONline Seminar (LAGOON)

(since 2023)Oberseminar Algebra und Geometrie

University of Münster (2017/2018)Simetrias & Representações

State University of Campinas, Brazil (2013–2015)

### Diagrams

A vanishing cycle and its two thimbles in a symplectic Lefschetz fibration with two critical points (from my PhD thesis)

14 intermediate t-structures on the derived category of the A₃ quiver 1 → 2 → 3 and their hearts (shaded) and projective objects (circled) (from this paper on scattering amplitudes)

A picture of the quiver for the extended Khovanov arc algebra K₂² (from this paper on Stroppel's Conjecture)

### Miscellaneous

### Talks

### 2024

[21 February] Combinatorial deformation quantization via the Koszul complex

Seminar Noncommutative Projective Schemes, MSRI/SLMathNoncommutative geometry via quivers with relations

Introductory Workshop: Noncommutative Algebraic Geometry, MSRI/SLMath

### 2023

A∞ deformations of extended Khovanov arc algebras

Seminar on Quantum groups, Hopf algebras and monoidal categories, UCLouvainIntroduction to representation and character varieties

Seminar Factorization Homology, University of BonnThe Diamond Lemma

Seminar Operads, University of BonnHochschild cohomology and A∞ deformations of extended Khovanov arc algebras

Algebras and Representation Theory in Germany (ARTIG 2), University of StuttgartKoszul duality for operads

Seminar Operads, University of BonnA categorical perspective on scattering amplitudes

Oberseminar Algebra, Zahlentheorie und algebraische Geometrie, University of FreiburgA∞ deformations of extended Khovanov arc algebras and Stroppel's Conjecture

Algebraic Geometry Seminar, University of LuxembourgFrom cluster categories to scattering amplitudes

Algebra Group Seminar, University of CologneExtended Khovanov arc algebras: Quivers, Hochschild cohomology and A∞ deformations (video first talk, second talk by Zhengfang Wang)

Informal Categorification Seminar, Columbia UniversityA∞ deformations of extended Khovanov arc algebras and Stroppel’s Conjecture

FD SeminarStrict deformation quantization of polynomial Poisson structures

Seminar semiklassische Analysis und Darstellungstheorie, University of Cologne

### 2022

Deformations of path algebras of quivers with relations

Oberseminar Geometry, Algebra and Topology, University of MünsterThe Bardzell–Chouhy–Solotar resolution

Algebra Group Seminar, University of CologneDeformations and spectra in noncommutative algebraic geometry

Spectral Methods in Algebra, Geometry and Topology, Hausdorff Research Institute for Mathematics, BonnFrom formal to actual deformations of associative algebras I, II

Algebra Group Seminar, University of CologneMinicourse Deformation theory via L∞ algebras (videos)

Higher Structures in Deformation Theory, University of FreiburgVarieties of reduction systems and PBW deformations

Aachen–Bochum–Cologne Darstellungstheorie Seminar, RWTH Aachen UniversityIntroduction to deformation theory

Algebra Group Seminar, University of CologneDeformations of categories of coherent sheaves via quivers with relations (video)

Longitudinal Algebra and Geometry Open ONline Seminar (LAGOON)Combinatorial deformations of associative algebras and applications to geometry

Forschungsseminar Algebra/Algebraische Geometrie, Freie Universität BerlinA combinatorial approach to deformation quantization

Séminaire de Mathématiques, Université Haute-Alsace, Mulhouse

### 2021

Deformations of diagrams of algebras and applications to geometry

Deformation Theory Seminar, University of PennsylvaniaStrict deformation quantizations of polynomial Poisson structures

Deformation Quantization Seminar, University of WürzburgChern classes of tautological bundles and Nakajima operators

Seminar Heisenberg algebras and Hilbert schemes of points on surfaces, University of FreiburgHilbert schemes of points on surfaces: the moduli problem

Seminar Heisenberg algebras and Hilbert schemes of points on surfaces, University of FreiburgFrom quiver representations to scattering amplitudes and back

Oberseminar Differentialgeometrie, University of FreiburgScattering amplitudes from derived categories and cluster categories

Longitudinal Algebra and Geometry Open ONline Seminar (LAGOON)Introduction to deformation theory

Basic Notions Seminar, University of Freiburg

### 2020

Minicourse Deformations of path algebras of quivers with relations (videos)

New Trends in Representation Theory, Hausdorff Research Institute for Mathematics, BonnCovariant quantization of strings

Seminar String theory, University of FreiburgPerspectives on GIT

Seminar Higgs bundles, University of FreiburgIntroduction to homological mirror symmetry

Basic Notions Seminar, University of FreiburgDeformations of path algebras of quivers with relations

Oberseminar Algebra und Zahlentheorie, University of Freiburg

### 2019

The derived category of projective space

Seminar Derived categories, University of FreiburgDeformations of path algebras of quivers with relations from a geometric perspective

Oberseminar Differentialgeometrie, University of FreiburgPoisson manifolds, star products and deformation quantization after Kontsevich

Valeurs zêta multiples en quantification par déformation, Institut Montpelliérain A. Grothendieck, MontpellierMinicourse An introduction to algebraic deformation theories

Working Group in Deformation Theory, ICPT, TriesteDeformations of Abelian categories in algebra, geometry & physics

Algebra, Geometry & Physics Seminar, Max Planck Institute for Mathematics, BonnL∞ algebras for deformations of Abelian categories and applications to geometry

Higher Differential Geometry Seminar, Max Planck Institute for Mathematics, BonnDeformations of Abelian categories and applications to geometry

Oberseminar Geometrie und Topologie, University of MainzLocal holomorphic Euler characteristic and instanton charge

Oberseminar Algebra, Geometry & Topology, University of MünsterBrauer groups, Azumaya algebras and twisted sheaves

Seminar Derived categories of singular surfaces, University of HannoverCategorical resolutions of singularities

Seminar Derived categories of singular surfaces, Free University of BerlinL∞ algebra for deformations of categories of quasi-coherent sheaves

Interactions between commutative algebra, representation theory, and algebraic geometry — A conference in memoriam Ragnar-Olaf Buchweitz, University of MünsterDiagrams of algebras, categories of coherent sheaves and deformations

Bielefeld Representation Theory Seminar (BIREP), University of BielefeldDiagrams of algebras, categories of coherent sheaves and their deformation theory

Forschungsseminar Algebra/Algebraische Geometrie, Free University of Berlin

### Teaching

### Supervision

Anahita Pouralijanki (Master's)

Simon Rottmann (Master's)

### Courses

University of Cologne

Seminar "Introduction to homological algebra"

Algebra

Linear algebra II (lecturer)

Linear algebra I

Seminar "Topics in representation theory: From matrix problems to quiver representations and back"

Hyperbolic geometry

Elementary differential geometry

University of Freiburg

Differential geometry II — The simple Lie algebras and singularities

Undergraduate seminar "Big theorems and beautiful proofs"

Seminar "Crystallographic groups"

Complex analysis

Undergraduate seminar "Representation theory of quivers"

Algebraic topology

State University of Campinas

Calculus 2

Calculus 1

### Links

### Author profiles

### Contact

### Email

s.barmeier ⟨æт⟩ gmail.com

### Address

Mathematical Institute

University of Cologne

Weyertal 86-90

50931 Köln

Germany