Research
My current interest lies in the development of differential algebra for semirings, in particular a notion of D-module compatible with tropical differential equations, and applications of tropical geometry to the mathematical foundations of neural networks (in particular ReLU NN).
My PhD work focused on building links between p-adic differential equations and their radii of convergence as functions on the Berkovich analytification and the recent development of the theory of tropical differential equations. Since I am at KTH, I started to apply my knowledge of tropical geometry to the geometry of neural networks, in particular to understand what is the space of functions of a ReLU network for a given architecture.
The theory p-adic differential equations tracks back several decades and it is strictly linked with non-archimedean geometry and Berkovich spaces. p-adic differential equations are fundamental objects of study in arithmetic geometry and in number theory: they are connected to p-adic representations and thus with the Langlands program. The subject has a solid background but most of its results remain limited to dimension one and the linear case.
Tropical differential equations were introduced by D. Grigoriev few years ago and a small group of experts is working in the development of this theory, which looks very promising for applications in the context of p-adic differential equations and elsewhere.
Publications
1. with J. Giansiracusa, A general framework for tropical differential equations. To appear in Manuscripta Mathematica, 2023, DOI 10.1007/s00229-023-01492-5. Available here.
2. with T. Brzeziński and B. Rybołowicz, From pre-trusses to skew-braces. Publicacions Matemàtiques, volume 66(2), 2022, 683-714.
Preprints
3. with Alejandro Vargas, On the polyhedral geometry of tropical solution sets associated to ODE's. In preparation.
4. with Sebastian Falkensteiner, The fundamental theorem of tropical differential algebra for formal Puiseux series. https://arxiv.org/abs/2312.10036. Submitted.
5. with Moritz Beyer, Érika Roldán, Peter Voran, Cubical sliding puzzles. https://arxiv.org/abs/2307.14143. Submitted.
6. with Francesco Gallinaro, The fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations. https://arxiv.org/abs/2303.12124.
PhD Thesis
Here you can find my PhD thesis, entitled "A general framework for tropical differential equations", containing a more general and broader point of view on the result of the paper with the same title that I coauthored with J. Giansiracusa. Some explicite computations can also be found in the appendix.
Talks
30/7/2024 - "The fundamental theorem of tropical differential algebra with nontrivially valued coefficients", MEGA, MPI MiS & Lepzig University, Leipzig, Germany
12/6/2024 - "Tropical differential algebra and its fundamental theorem", GeUmetric deep learning, Umeå University, Umeå, Sweden
19/6/2023 - "The fundamental theorem of tropical differential algebra with nontrivially valued coefficients", Algebraic and Tropical Methods for Solving Differential Equations, UNAM, Unidad Oaxaca, Oaxaca de Juárez, Mexico
16/5/2023 - "The fundamental theorem of tropical differential algebra with nontrivially valued coefficients", Applied CATS Seminar, KTH, Stockholm, Sweden
21/4/2023 - "The fundamental theorem of tropical differential algebra with nontrivially valued coefficients", LAGARTOS online seminar
14/4/2023 - "The fundamental theorem of tropical differential algebra with nontrivially valued coefficients", Kolchin seminar in Differential Algebra (online)
3/8/2022 - "A general framework for tropical differential equations", Nonlinear algebra seminar, MPI MiS, Leipzig, Germany
25/6/2021 -"Tropical differential equations", TGiF Tropical Geometry in Frankfurt, Goethe-Universität Frankfurt, Frankfurt am Main, Germany (On Zoom due to the pandemic)
23/4/2021 - "Tropical differential equations", Mathematics PhD Seminar, Swansea University, Swansea, Wales, UK (On Zoom due to the pandemic)
8-10/12/2020 - Poster "An inverse limit theorem in tropical differential algebra", Tropical Geometry, Berkovich Spaces, Arithmetic D-Modules and p-adic Local Systems, Imperial College of London, London, UK (On Zoom due to the pandemic)
26/10/2020 - "Scheme theoretic tropicalization", Online reading course in F_1-geometry organized by Oliver Lorscheid
3/3/2020 - "An introduction to tropical geometry", Journal Club, Physics Department, Swansea University, Swansea, Wales, UK
4/12/2019 - "A scheme theoretic approach to differential tropical geometry", Tropical Differential Algebra Workshop, Queen Mary University, London, England, UK.
15/1/2020 - "A scheme theoretic approach to differential tropical geometry", LMS Tropical Meeting, University of Manchester, Manchester, England, UK.
22/3/2019 - "The fundamental theorem of tropical geometry", Mathematics PhD Seminar, Swansea University, Swansea, Wales, UK
Research visits
3-7/10/2022 - Commutative algebra group, Università degli Studi di Genova, Genova, IT
Conferences, meetings and workshops
Ongoing - Reading group in tropical geometry, Swansea University/Durham University, UK
29/7-2/8/2024 - MEGA, MPI MiS & Lepzig University, Leipzig, Germany
11-13/6/2024 - GeUmetric deep learning, Umeå University, Umeå, Sweden
12-23/6/2023 - Algebraic and Tropical Methods for Solving Differential Equations, UNAM, Unidad Oaxaca, Oaxaca de Juárez, Mexico
3/4/2023 - Combinatoire et équations différentielles p-adiques, Institut Henri Poincaré, Paris, France.
14-18/11/2022 - Newton-Okounkov bodies and tropical geometry, Goethe Universität, Frankfurt am Main, Germany.
20-24/6/2022 - Nordfjordeid summer school 2022, Topics in real and tropical geometry, Sophus Lie conference center, Nordfjordeid, Norway.
10-12/2020 - Online reading course on F_1-geometry, organised by Oliver Lorscheid http://oliver.impa.br/2020-F1/F1.html
15/1/2020 - LMS Tropical Meeting, University of Manchester, Manchester, England, UK.
3-7/12/2019 - Tropical Differential Algebra Workshop, Queen Mary University, London, England, UK.
29/7 - 2/8/2019 - WARTHOG 2019 - Foundations of Tropical Geometry, University of Oregon, Eugene, OR, USA.
15-19/7/2019 - Recent advances in the arithmetic of Galois representations, Università degli Studi di Genova, Genova, IT.
20-22/5/2019 - Welsh Mathematical Colloquium, Gregynog Hall, Newtown, Wales, UK.
9/5/2019 - COW meeting, Cardiff University, Cardiff, Wales, UK.
24/1/2019 - Tropical Mathematics and its Applications, University of Birmingham, Birmingham, England, UK.
11-14/12/2018 - A3G 2018, University of Bristol, Bristol, England, UK.
20/11/2018 - CALF meeting, University of Oxford, Oxford, England, UK.