About me
I am a researcher currently working as an Advanced Fellow at the ETH-ITS in Zürich. In February 2025, I will join the Perimeter Institute as a junior faculty member. My field of research is mathematical physics.
Much of my research explores the interplay between moduli spaces and quantum field theory. Moduli spaces are pivotal objects in mathematics, while quantum field theory is the language of physics at the elementary particle level. I am fascinated by the connections between these two domains due to their foundational significance in their corresponding fields.
This article in Quanta Magazine gives a popular scientific account of parts of my research.
Download my CV, view my or browse my articles on arXiv.
Contact
michael.borinsky@eth-its.ethz.ch
Publications
PhD and MSc student contributions are marked with grad.
Preprints
Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, preprint August 2024
with Gerald Dunne and Karen Yeats
On the Euler characteristic of the commutative graph complex and the top weight cohomology of Mg, preprint May 2024
with an appendix by Don Zagier
Journal articles
Bivariate exponential integrals and edge-bicolored graphs, preprint September 2024, to appear in Le Matematiche
with Chiara Meroni and Maximilian Wiesmann (further illustrations and computer code)
Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group, preprint August 2023, to appear in Selecta Mathematica
with Benjamin Brück and Thomas Willwacher
The Sn-equivariant Euler characteristic of the moduli space of graphs, preprint June 2023, to appear in Algebraic & Geometric Topology
with Jos Vermaseren
The Euler characteristic of the moduli space of graphs, Advances in Mathematics, 432 (2023) 109290 (arXiv:2301.01121)
with Karen Vogtmann. This work is featured in a popular science piece by Leila Sloman in Quanta Magazine.
Tropical Feynman integration in the Minkowski regime, Computer Physics Communications, 292 (2023) 108874 (arXiv:2302.08955)
with Henrik Munchgrad and Felix Tellandergrad
Try out the Feynman integration program feyntrop that we describe in the paper! It is fast and easy to use.
Flow-oriented perturbation theory, Journal of High Energy Physics, 172 (2023) (arXiv:2210.05532)
with Zeno Capattigrad, Eric Laenen, and Alexandre Salas-Bernárdezgrad
Computing Euler characteristics using quantum field theory, preprint February 2022, to appear in Geometric methods in group theory: papers dedicated to Ruth Charney
with Karen Vogtmann
Recursive computation of Feynman periods, Journal of High Energy Physics, 291 (2022) (arXiv:2206.10460)
with Oliver Schnetz
Bayesian Integrals on Toric Varieties, SIAM Journal on Applied Algebra and Geometry 7 (2023) 77-103, (arXiv:2204.06414)
with Anna-Laura Sattelberger, Bernd Sturmfels, and Simon Telen (further illustrations and computer code)
Resonant resurgent asymptotics from quantum field theory, Nuclear Physics B, 981 (2022) 115861 (arXiv:2202.01513)
with David Broadhurst
Graphical functions in even dimensions, Communications in Number Theory and Physics, 16 (2022) 515-614 (arXiv:2105.05015)
with Oliver Schnetz
Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: ϕ3 QFT in 6 Dimensions, SIGMA 17 (2021), 087 (arXiv:2104.00593)
with Gerald Dunne and Max Meyniggrad
Five loop renormalization of ϕ3 theory with applications to the Lee-Yang edge singularity and percolation theory, Physical Review D, Volume 103, June 2021, 116024, (distinguished as Editors' suggestion) (arXiv:2103.16224)
with John Gracey, Mikhail Kompaniets and Oliver Schnetz
Tropical Monte Carlo quadrature for Feynman integrals, Ann. Inst. Henri Poincaré Comb. Phys. Interact., Volume 10 (2023), no. 4, pp. 635–685, (arXiv:2008.12310)
the proof-of-concept implementation of the algorithm for numerical Feynman integration is available on this page
Non-Perturbative Completion of Hopf-Algebraic Dyson-Schwinger Equations, Nuclear Physics B, Volume 957, August 2020, 115096 (arXiv:2005.04265)
with Gerald Dunne
The Hopf algebra structure of the R∗-operation, Journal of High Energy Physics, Article 61, July 2020 (arXiv:2003.04301)
with Robert Beekveldtgrad and Franz Herzog
The Euler characteristic of Out(Fn), Commentarii Mathematici Helvetici, 95 (2020), 703-748 (arXiv:1907.03543)
with Karen Vogtmann
Renormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, Volume 385, October 2017, Pages 95-135 (arXiv:1703.00840)
Generating asymptotics for factorially divergent sequences, Electronic Journal of Combinatorics, Volume 25, Issue 4, October 2018, P4.1 (arXiv:1603.01236)
The results of this article are summarized in this colorful poster for FPSAC 2017.
Algebraic lattices in QFT renormalization, Letters in Mathematical Physics, Volume 106, Issue 7, July 2016, Pages 879-911 (arXiv:1509.01862).
Featured in Edward Tufte's new book Seeing with fresh eyes.
Feynman graph generation and calculations in the Hopf algebra of Feynman graphs, Computer Physics Communications, Volume 185, Issue 12, December 2014, Pages 3317-3330 (arXiv:1402.2613):
Information and downloads to the two programs feyngen and feyncop is available on this page.
Proceedings
Tropical Feynman integration in the physical region, Proceedings contribution to EPS-HEP2023, PoS 449, (arXiv:2310.19890)
with Henrik Munch and Felix Tellander
Flow Oriented Perturbation Theory, Proceedings contribution to RADCOR2023, PoS 432, (arXiv:2310.09708)
with Alexandre Salas-Bernárdez, Zeno Capatti and Eric Laenen
Taming a resurgent ultra-violet renormalon, Proceedings contribution to Loops and Legs in Quantum Field Theory 2022, PoS 416, (arXiv:2209.10586)
with David Broadhurst
Generating Asymptotics for factorially divergent sequences, Séminaire Lotharingien de Combinatoire, 78B.28 (2017), 12 pp.
Proceedings contribution for FPSAC 2017.
Feynman diagrams and their algebraic lattices, Resurgence, Physics and Numbers CRM Series - Edizioni della Normale, Volume 20, 2017, Pages 91-107 (arXiv:1512.05237)
with Dirk Kreimer: Proceedings contribution for Resurgence, Physics and Numbers.
Computations and generation of elements on the Hopf algebra of Feynman graphs, Proceedings of 16th International workshop on Advanced Computing and Analysis Techniques in physics research (ACAT), J. Phys.: Conf. Ser. 608, 2015 (arXiv:1410.7324)
Proceedings contribution for ACAT 2014.
Other
Graphs in perturbation theory: Algebraic structure and asymptotics, May 2018 (arXiv:1807.02046):
My PhD thesis - available as hardcover in the Springer Theses series (amazon)
Algorithmization of the Hopf algebra of Feynman graphs, August 2013:
My master's thesis
Organization of scientific events
9-13 September 2024: With Karen Vogtmann, I organized a workshop on Moduli spaces of curves, moduli spaces of graphs and graph complexes at the ETH-ITS in Zürich, Switzerland.
19-23 August 2024: With Christian Blohmann, Ralph Kaufmann, Dirk Kreimer, and Karen Yeats, I organized a workshop on Combinatorics, Resurgence and Algebraic Geometry in Quantum Field Theory at the Max Planck Institute for Mathematics in Bonn, Germany.
05-09 September 2022: With Erik Panzer, I organized a workshop on Tropical and Convex Geometry and Feynman integrals at the ETH-ITS in Zürich, Switzerland.
Supervision of MSc and BSc theses
MSc thesis, 2023, by Andrea Favorito, ETH Zurich, on 'Tropical Feynman period integration'.
BSc thesis, 2022, by Jean-Luc Portner, ETH Zurich, on 'The forested graph complex'.
MSc thesis, 2019, by Robert Beekvelt, Universiteit van Amsterdam, on the 'Hopf algebra structure of the R* operation' (co-supervision with Franz Herzog).
Feel free to contact me if you are interested in a BSc, MSc or PhD project along the lines of my research.
Teaching activities
13-17 May 2024, Max Planck Institute for Mathematics in the Sciences, Leipzig - Germany
Invited 5-part lecture series on 'Counting graphs using quantum field theory'
17-21 June 2024, Institute for Advanced Study (IAS), Princeton - US
Invited 4-part lecture series on 'The tropical and discrete geometry of Feynman integrals'
26-27 October 2021, ETH Zürich, Switzerland
Mini-course From Feynman graphs to Calabi-Yau manifolds with Johannes Brödel and Albrecht Klemm
Summer 2019, Universiteit van Amsterdam, The Netherlands
Field Theory in Particle Physics, exercise sessions for course on particle physics
Winter 2016/2017, Humboldt Universität zu Berlin, Germany
Winter 2015/2016, Humboldt Universität zu Berlin, Germany
Proseminar on Infrared and collinear divergences in perturbative QFT
Invited talks
2024/2025
Research Seminar - Centre for Quantum Mathematics, Odense - Denmark, 04 December 2024
Talk 'On the Euler characteristic of the commutative graph complex'
MITP Workshop on Combinatorics in Fundamental Physics, Mainz - Germany, 28 November 2024
Talk on 'Bivariate exponential integrals and edge-bicolored graphs' (virtual)
IAS - Combinatorics of Fundamental Physics Workshop, Princeton - US, 18 November 2024
Talk on 'Probing Moduli Space Cohomology Using Quantum Field Theory' (video)
Hamburg University - ZMP Colloquium, Hamburg - Germany, 24 October 2024
Talk on 'On the Dark-Matter Problem of the Moduli Space of Curves'
DESY - Theory Seminar, Hamburg - Germany, 21 October 2024
Talk on 'Tropical (Feynman) Sampling'
2023/2024
Max Planck Institute for Mathematics, Bonn - Germany, 7 August 2024
Talk 'On the Euler characteristic of the commutative graph complex'
Center Of Mathematical Sciences And Applications (CMSA), Cambridge - US, 21 May 2024
Talk 'On the Euler characteristic of the commutative graph complex and the top weight cohomology of Mg'
A panorama of moduli spaces, Frankfurt - Germany, 28 February 2024
Talk 'On the top-weight cohomology of the moduli space of curves' (video)
Perimeter Institute Colloquium, Waterloo - Canada, 20 February 2024
Talk on 'Algebraic geometry at the limits of perturbative QFT and dark matter in moduli spaces' (video)
Particle Physics Seminar, Torino - Italy, 22 November 2023
Talk on 'Tropical Feynman integration'
Purdue Topology Seminar, Lafayette - US, 15 November 2023
Talk 'On the top-weight cohomology of the moduli space of curves' (online)
IAS HEP Seminar, Princeton - US, 6 October 2023
Talk on 'The Dark-Matter Problem of the Moduli Space of Curves and its (partial) Tropical Solution' (video)
2022/2023
Amplitudes 2023, CERN - Switzerland, 8 August 2023
Talk on 'Tropical Feynman integration in Minkowski space' (slides/video)
Max-Planck-Institute for Physics, Munich - Germany, 18 July 2023
Talk on 'Tropical Feynman integration in the Minkowski regime'
SIAM Conference on Applied Algebraic Geometry, Eindhoven - The Netherlands, 13 July 2023
Talk on 'Generalized permutahedra tropical sampling' (slides)
Algebra/Topology Seminar, Copenhagen - Denmark, 23 June 2023
Talk on 'On the amount of top-weight cohomology in the moduli space of curves'
From perturbative to non-perturbative QFT, Muenster - Germany, 15 June 2023
Talk on 'Probing the non-perturbative regime with tropical Feynman integration' (video)
International Symposium on Radiative Corrections - RADCOR, Crieff - UK, 29 May 2023
Talk on 'Feynman amplitudes and the moduli space of graphs'
Geometry and Topology seminar, Warwick - UK, 25 May 2023
Talk on 'The commutative graph complex and the amount of top-weight cohomology in the moduli space of curves'
Institute of Advanced Studies - Special Topology Seminar, Princeton - US, 19 May 2023
Talk on 'Graphs, Curves, and their Moduli Spaces'
Institute of Advanced Studies - Amplitudes Pizza Seminar, Princeton - US, 12 May 2023
Talk on 'Greatest hits on tropical geometry and Feynman integrals'
Geometry seminar, ETH Zürich - Switzerland, 19 April 2023
Talk on 'On the amount of top-weight cohomology in the moduli space of curves'
Non-Linear Algebra Seminar, Leipzig - Germany, 07 April 2023
Talk on 'Graphs, Curves, and their Moduli Spaces'
Theory Seminar DESY Zeuthen, Zeuthen - Germany, 06 April 2023
Talk on 'Tropical Feynman integration in the Minkowski regime'
Nico van Kampen Colloquium in Theoretical Physics, Utrecht - The Netherlands, 15 March 2023
Colloquium talk on 'Feynman diagrams at the limits of perturbation theory'
Feynman Integrals - UZH workshop, Zürich - Switzerland, 22 February 2023
Talk on 'Tropical Feynman integration in the Minkowski regime'
Applicable resurgent asymptotics: summary workshop at the Newton Institute, Cambridge - UK, 14 December 2022
Talk on 'Tropical sampling and its application to large-loop Feynman integration' (video)
Algebraic and Enumerative Combinatorics Seminar, Waterloo - Canada, 24 November 2022
Talk on 'Asymptotics of the Euler characteristic of Kontsevich's commutative graph complex'
Perimeter Institute: Quantum Fields and Strings Seminar, Waterloo - Canada, 22 November 2022
Talk on 'Computability in QFT and tropical QFT' (video and slides)
Geometry and Topology Seminar, McMaster University - Canada, 17 November 2022
Talk on 'Euler characteristics of Out(Fn)'
SLAC Theoretical Physics Seminar, Stanford - US, 8 November 2022
Talk on 'Tropical sampling and fast Feynman integration'
Bethe Center for Theoretical Physics, Bonn - Germany, 18 October 2022
Talk on 'Tropical sampling and its applications to Feynman integrals'
Physical resurgence: On quantum, gauge, and stringy, Cambridge - UK, 13 September 2022
Talk on 'Resonant resurgent asymptotics from quantum field theory' (online talk) (video)
2021/2022
Algebra, Topology and the Grothendieck-Teichmüller group, Les Diablerets - Switzerland, 30 August 2022
Talk on 'Asymptotics for graph complex Euler characteristics' (online talk) (video)
Joint Mathematical Congress AMS-EMS-SMF 2022, Grenoble - France, 20 July 2022
Talk on 'Euler characteristics of Out(Fn) and renormalized topological field theory' (slides)
Graph Complexes and Quantum Field Theory, Berlin - Germany, 28 March 2022
Talk on 'Asymptotics for graph complex Euler characteristics' (video)
Feynman integrals and nonlinear algebra, Leipzig - Germany, 10 March 2022
Talk on 'Tropical Geometries in Feynman Integrals'
Structure of Local Quantum Field Theories, Berlin - Germany, 21 February 2022
Talk on 'Resonant resurgent asymptotics from quantum field theory' (online talk)
QCD Seminar, CERN - Switzerland, 3 December 2021
Talk on 'Fast Feynman integration with tropical geometry' (slides)
Talks in mathematical physics, ETH Zurich - Switzerland, 4 November 2021
Talk on 'The Euler characteristic of Out(Fn) and renormalized topological field theory'
Algebra/Topology Seminar, Copenhagen - Denmark, 15 October 2021
Talk on 'The Euler characteristic of Out(Fn) and renormalized topological field theory'
Niels Bohr Institute - High Energy Physics Seminar, Copenhagen - Denmark, 14 October 2021
Talk on 'Tropical Feynman integration'
Humboldt University - Kolleg Mathematics and Physics Berlin Seminar (KMPB day), Berlin - Germany, 27 September 2021
Talk on 'Graphical functions in even dimensions' (online talk)
2020/2021
Theory Seminar of the Max-Planck-Institute, Munich - Germany, 30 July 2021
Talk on 'Numerical Feynman integration: The tropical way' (online talk)
Geomplitudes III, UC Davis - US, 4 June 2021
Talk on 'Tropical Feynman integration' (online talk)
Nikhef Theory Day, Amsterdam - The Netherlands, 28 May 2021
Talk on 'Numerical Feynman integration: The tropical way' (online talk)
Radcor & Loopfest 2021, Tallahassee - US, 17 May 2021
Talk on 'The tropical approach to numerical Feynman integration' (online talk: video/slides)
IAS Informal Amplitudes Meeting, Princeton - US, 24 March 2021
Talk on 'Tropical QFT and its applications' (online talk)
Amplitudes Seminar - Higgs Centre, Edinburgh - UK, 10 February 2021
Talk on 'Tropical quantum field theory and its applications' (online talk: video/slides)
Physics@Veldhoven: The Magic of Physics, Veldhoven - The Netherlands, 20 January 2021
Talk on 'Magic numbers at the heart of physics' (online talk for a broad audience)
Structure of Local Quantum Field Theories, Berlin - Germany, 18 January 2021
Talk on 'Numerical Feynman integration: The tropical way' (online talk)
Algebraic Structures in Perturbative Quantum Field Theory, IHES - France, 16-20 November 2020
Talk on 'The Euler characteristic of Out(F_n) and the Hopf algebra of graphs' (online talk: video/slides)
ACPMS, Intereuropean virtual seminar, 18 September 2020
Talk on 'The Euler characteristic of Out(F_n) and the Hopf algebra of graphs' (online talk: video/slides)
Graph Complexes and Quantum Field Theory, Berlin - Germany, 21-25 September 2020Talk on 'Out(F_n), M_gn and renormalized topological field theory'(postponed due to SARS-CoV-2)
2019/2020
Topology Seminar, Karlsruhe - Germany, 02 July 2020
Talk on 'The Euler characteristic of Out(Fn) and renormalized topological field theory' (online talk)
Structure of Local Quantum Field Theories, Berlin - Germany, 04 May 2020
Talk on 'Fast Feynman parametric integral evaluation via tropical sampling' (online talk)
Resurgence in string and gauge theory, AIM - San Jose - US, 05 Mai 2020Talk on 'The ring of factorially divergent power series and its applications'(canceled due to SARS-CoV-2)Algebraic Structures in Perturbative Quantum Field Theory, IHES - Paris - France, 30 March 2020Talk on 'The Hopf algebra structure of the R* operation'(postponed due to SARS-CoV-2)Theoretical Physics Seminar, Liverpool - UK, 26 February 2020
Talk on 'Graphical functions applied to phi^3 in D = 6' (slides)
Emmy-Noether-Seminar, Erlangen - Germany, 7 February 2020
Talk on 'The Euler characteristic of Out(F_n)' (slides)
Geometry Seminar, Nijmegen - Netherlands, 28 January 2020
Talk on 'Renormalized topological field theory and the Euler characteristic of Out(F_n)' (slides)
Algebraic Combinatorics Seminar, Waterloo - Canada, 12 December 2019
Talk on 'Out(F_n), M_gn and renormalized topological field theory'
Canadian Mathematical Society Winter Meeting, Toronto - Canada, 9 December 2019
Talk on 'Constrained Graph Counting' (slides)
Theory Seminar of the Max-Planck-Institute, Munich - Germany, 4 December 2019
Talk on 'Efficient Monte-Carlo on Feynman parametric integrals'
Combinatorics Seminar, Vienna - Austria, 21 November 2019
Talk on 'Renormalized topological field theory'
Mathematical Physics Seminar, Vienna - Austria, 19 November 2019
Talk on 'The graphical function method applied to phi^3 in D=6' (slides)
Topology Seminar, Oxford - UK, 4 November 2019
Talk on 'The Euler characteristic of Out(Fn) and renormalized topological field theory'
Emmy-Noether-Seminar, Erlangen - Germany, 18 September 2019
Talk on 'The graphical function method in 2n-dimensions' (slides)
RADCOR 2019, Avignon - France, 12 September 2019
Talk on 'Systematic integration of higher loop gauge theory amplitudes in position space' (slides)
2018/2019
Structure of Local Quantum Field Theories, Berlin - Germany, 28 Januar 2019
Talk on 'The Euler characteristic of OutFn'
Emmy-Noether-Seminar, Erlangen - Germany, 16 November 2018
Talk on 'Deriving asymptotics for factorially divergent power series'
Higgs as Probe and Portal Meeting, Amsterdam - Netherlands, 21 September 2018
Talk on 'Asymptotics and algebraic structure in perturbation theory'
2017/2018
Summer school on structures in local quantum field theory, Les Houches - France, 4 - 15 June 2018
Talk on 'Graphs in Perturbation Theory' (slides)
Combinatorial structures in perturbative QFT, Waterloo - Canada, 25 May 2018
Talk on 'Counting graphs without given edge-induced subgraphs' (slides)
University of Waterloo: Algebraic Combinatorics Seminar, Waterloo - Canada, 10 May 2018
Talk on 'Hopf algebra of graphs for restricted graphical enumeration'
Workshop on Enumerative Combinatorics, Vienna - Austria, 16 - 20 October, 2017
Talk on 'Hopf algebras and factorial divergent power series: Algebraic tools for graphical enumeration' (slides)
ALEA in Europe, Vienna - Austria, 9 - 13 October, 2017
Talk on 'Generating Asymptotics for factorially divergent sequences' (slides)
2016/2017
Formal Power Series and Algebraic Combinatorics (FPSAC), London - United Kingdom, 9 - 13 July, 2017
Poster about 'Generating Asymptotics for factorially divergent sequences'
Proceedings contribution: Generating Asymptotics for factorially divergent sequences, Séminaire Lotharingien de Combinatoire, 78B.28 (2017), 12 pp.
Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA), Luminy - France, 26 - 30 June, 2017
Talk on 'Flag decompositions of graphs and their Hopf algebraic structure'
International school of subnuclear physics, Erice - Italy, 14 - 23 June 2017
Talk on 'Bounds and estimates for Feynman-perturbative expansions' (slides)
University of Waterloo: Algebraic Combinatorics Seminar, Waterloo - Canada, 30 March 2017
Talk on 'Zero-dimensional quantum field theory or counting classes of graphs using Gaussian integrals and Hopf algebras'
2015/2016
Resurgence in Gauge and String Theories, Lisbon - Portugal, 18 - 22 July 2016
Talk on 'Asymptotic Calculus for Combinatorial Dyson-Schwinger Equations' (slides)
Kolleg Mathematik Physik Berlin (KMPB-Day), Berlin - Germany, 20 June 2016
Talk on 'Factorially divergent expansions from a QFT perspective'
Combinatorial Structures in Perturbative Quantum Field Theory, Vancouver - Canada, 21 March 2016
Talk on 'Generating functions' asymptotics' generating functions' (slides)
Paths to, from and in renormalisation, Potsdam - Germany, 8 - 12 February 2016
Talk on 'Asymptotic expansions and combinatorial Dyson-Schwinger equations' (slides)
2014/2015
Resurgence, Physics and Numbers, Pisa - Italy, 18 - 22 May 2015
Talk on 'Combinatorics of Feynman diagrams and the algebraic lattice structure in QFT' (slides)
Proceedings contribution with Dirk Kreimer: Feynman diagrams and their algebraic lattices, Resurgence, Physics and Numbers CRM Series - Edizioni della Normale, Volume 20, 2017, Pages 91-107 (arXiv:1512.05237)
Dyson-Schwinger Equations in Modern Mathematics & Physics, Trento - Italy, 22 - 26 September 2014
Talk on 'The Hopf-Algebra of Feynman graphs as the foundation of Dyson-Schwinger equations'
Parallel session - Track 3, 16th International workshop on Advanced Computing and Analysis Techniques in physics research (ACAT), Prague - Czech Republic, 1 - 5 September 2014
Talk on 'Computations and generation of elements on the Hopf algebra of Feynman graphs'
Proceedings contribution: Computations and generation of elements on the Hopf algebra of Feynman graphs, Proceedings of 16th International workshop on Advanced Computing and Analysis Techniques in physics research (ACAT), J. Phys.: Conf. Ser. 608, 2015 (arXiv:1410.7324).
2013/2014
Summer school on structures in local quantum field theory, Les Houches - France, 9 - 20 June 2014
Talk on 'The Hopf algebra of Feynman graphs as foundation for Dyson Schwinger equations'
2012/2013
CERN Summer Student Lecture Programme, CERN - Switzerland, Summer 2012