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. Manuscripta Mathematica 173.3 (2024): 1273-1304, DOI 10.1007/s00229-023-01492-5. 

2. with T. Brzeziński and B. Rybołowicz - From pre-trusses to skew-braces. Publicacions Matemàtiques, volume 66(2), 2022, 683-714.  

Preprints

3. with A. Vargas -  On the polyhedral geometry of tropical solution sets associated to ODE's. In preparation.

4. with G.L. Marchetti, V. Shahverdi, M. Trager, K. Kohn - An invitation to neuroalgebraic geometry. https://www.arxiv.org/abs/2501.18915. Submitted.

5. with F. 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. Submitted.

6. with M. Beyer, É. Roldán, P. Voran - Cubical sliding puzzles. https://arxiv.org/abs/2307.14143. Submitted.

7. with S. Falkensteiner - The fundamental theorem of tropical differential algebra for formal Puiseux series. https://arxiv.org/abs/2312.10036. 

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 explicit computations can also be found in the appendix. 

Talks

31/10/2024 - "The space of valuated (pre)orders as the spectrum of a ring", Seminario di algebra e geometria, Università di Bologna, Bologna, Italy

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.