# Dr. Nikolas Tapia

**Address:**Mohrenstraße 39, 10117 Berlin, Germany.**Office:**208**Telephone:**+49 30 20372-448.**e-mail**: lastname (at) FQDN of this page.- Here is my PGP public key. Fingerprint:
`fe:8b:40:ee:4d:7f:15:b9:a0:67:63:9f:7f:db:7a:f5:18:50:35:fc`

- My CV.

I am part of the Stochastic Algorithms and Nonparametric Statistics research group at the Weierstraß Institute for Applied Analysis and Stochastics.

I work as a Postdoctoral Researcher in the MATH+ AA4-2 project "Optimal Control in Energy Markets Using Rough Analysis and Deep Networks", led by C. Bayer, P. K. Friz, John Schoenmakers, and Vladimir Spokoiny.

Previously, I was part of the Differential Equations and Numerical Analysis research group at the Norwegian University of Science and Technology, where I worked with K. Ebrahimi-Fard.

Before that, I obtained a joint PhD in Mathematics from the University of Chile and Sorbonne Université under the direction of D. Remenik and L. Zambotti.

📷 Geraldine Cossio, 2023.

**Online Events:**

- I co-organize the Algebraic and Combinatorial Perspectives in the Mathematical Sciences seminar (Fridays @ 15:00 CET).

## Current Projects

- Robustness of Deep Neural Networks,
*with Christian Bayer, Simon Breneis, and Peter Friz*. - Translation of branched rough paths,
*with Joscha Diehl, Kurusch Ebrahimi-Fard, and Rosa Preiß*. - Higher-algebra structures in 2-parameter signatures,
*with Ilya Chevyrev, Joscha Diehl and Kurusch Ebrahimi-Fard*. - Reversible numerical schemes for rough differential equations,
*with James Foster and Cristopher Salvi*. - Orthogonal signature polynomials as regression models,
*with Emilio Ferrucci, Darrick Lee, Harald Oberhauser and Cristopher Salvi*.

## Preprints and Publications

### Articles in Refereed Journals

- J. Diehl, K. Ebrahimi-Fard,
__N. Tapia__.*Generalized iterated-sums signatures*. J. Algebra**623**(2023), pp. 801–824. - K. Ebrahimi-Fard, F. Patras,
__N. Tapia__, L. Zambotti.*Shifted substitution in non-commutative multivariate power series with a view toward free probability*. SIGMA Symmetry Integrability Geom. Methods Appl.**19**(2023), 038, 17 pp. - C. Bayer, P. K. Friz,
__N. Tapia__.*Stability of deep Neural Networks via discrete rough paths*. SIAM J. Math. Data Sci.**5**(2023), no. 1, pp. 50–76. - J. Diehl, K. Ebrahimi-Fard,
__N. Tapia__.*Tropical time series, iterated-sums signatures and quasisymmetric functions*. SIAM J. Appl. Algebra Geom.**6**(2022), no. 4, pp. 563–599. - P. Hager, P. K. Friz,
__N. Tapia__.*Unified signature cumulants and generalized Magnus expansions*. Forum Math. Sigma**10**(2022), no. e42. - J. Diehl, R. Preiß, M. Ruddy,
__N. Tapia__.*The moving frame method for iterated-integrals: orthogonal invariants*. Found. Comput. Math. (2022). - K. Ebrahimi-Fard, F. Patras,
__N. Tapia__, L. Zambotti.*Wick polynomials in non-commutative probability*. Canad. J. Math.**74**(2021), no. 6, pp. 1673–1699. - C. Bellingeri, A. Djurdjevac, P. K. Friz,
__N. Tapia__.*Transport and continuity equations with (very) rough noise*. Partial Differ. Equ. Appl.**2**(2021), no. 49. - J. Diehl, K. Ebrahimi-Fard,
__N. Tapia__.*Iterated-sums signature, quasisymmetric functions and time series analysis*. Sém. Lothar. Combin.**84B**(2020), Article 84B.86, 12 pp. - J. Diehl, K. Ebrahimi-Fard,
__N. Tapia__.*Time-warping invariants of multidimensional time series*. Acta Appl. Math.**171**(2020), no. 1, pp. 265–290. __N. Tapia__, L. Zambotti.*The geometry of the space of branched Rough Paths*. Proc. London Math. Soc.**121**(2020), no. 2, pp. 220–251.- K. Ebrahimi-Fard, F. Patras,
__N. Tapia__, L. Zambotti.*Hopf-algebraic deformations of products and Wick polynomials*. Int. Math. Res. Not.**2020**(2020), no. 24, pp. 10064–10099. - E. Celledoni, P.E. Lystad,
__N. Tapia__.*Signatures in Shape Analysis: an Efficient Approach to Motion Identification*. (2019) In*Geometric Science of Information. GSI 2019. Lect. Notes Comput. Sci. vol. 11712*

### Preprints

- L. Schmitz,
__N. Tapia__.*Free generators and Hoffman's isomorphism for the two-parameter shuffle algebra*. (2024) arXiv:2401.06691 [math.RA] - C. Bellingeri, E. Ferrucci,
__N. Tapia__.*Branched Itô forumla and natural Itô-Stratonovich isomorphism*. (2023) arXiv:2312.04523 [math.PR] - M. Ghani Varzaneh, S. Riedel, A. Schmeding,
__N. Tapia__.*The geometry of controlled rough paths*. (2022) arXiv:2203.05946 [math.PR]

### Other publications

*Transport and continuity equations with (very) rough noise*. Report for Oberwolfach Mini-Workshop 2207c "Regularization by noise: Theoretical foundations, numerical methods and applications". Oberwolfach Reports (2022). Published online: 2022-04-12*Unified signature cumulants and generalized Magnus expansions*. Report for Oberwolfach Workshop 2050a "New Directions in Rough Path Theory". Oberwolfach Reports**17**(2020), no. 4, pp. 1955–2019. Published online: 2021-09-13

## Editorial Work

- Guest editor for a special issue of "Symmetry, Integrability and Geometry: Methods and Applications" (SIGMA).
*Non-commutative Algebra, Probability and Analysis in Action*.

## Upcoming Events

- Mathematics, Data Science, and Education @ FernUniversität-Hagen, Germany. March 13–15, 2024.
- Mathematics of data streams: signatures, neural differential equations, and diffusion models @ AK WiKo Greifswald, Germany. April 8–12, 2024.
- Signatures of paths and images @ CAS Olso, Norway. June 10–14, 2024.
- 70 Years of Magnus Series @ Universitat Jaume I, Spain. July 1–5, 2024.

## Past Events

- Non-commutative Algebra, Probability and Analysis in Action @ Greifswald, Germany. September 20–25, 2021.
- LMMS Summer School: Mathematical Methods for Machine Learning @ Dagstuhl, Germany. August 23–26, 2021.
- 10th World Congress in Probability and Statistics, Online. July 19–23, 2021.
- TMS Summer School for young researchers between geometry and stochastic analysis, Online. June 16–18, 2021. Website.
- Mini-workshop: SPDEs & friends.
- CIRM Meeting: Pathwise Stochastic Analysis and Applications. March 8–12, 2021.
- Mini-workshop: Cumulants in Stochastic Analysis.
- 14th Berlin-Oxford meeting. February 10–12, 2021.
- Rough paths and SPDEs, Online. December 9–11, 2020.
- MFO Meeting: New directions in Rough Paths theory, Online. December 7–11, 2020.
- Higher structures emerging from renormalisation, Online. October 12–16, 2020.
- Bernoulli-IMS One World Symposium, Online. August 24–28, 2020.
- Geometry of curves in time series and shape analysis, Online. August 11–14, 2020
- 13th Oxford-Berlin meeting, Online. June 8–10, 2020.
- Young researchers between geometry and stochastic analysis @ Bergen, Norway. February 12–14, 2020.
- 12th Oxford-Berlin meeting @ Alan Turing Institute Oxford, UK. December 4–6, 2019.
- Visit: INI Program @ Oxford, UK. December 1–12, 2019.
- Analytic and Algebraic perspectives in the theory of rough paths and signatures @ Oslo, Norway. November 14–15, 2019.
- Recontre GDR Renormalisation @ Calais, France. September 30–October 4, 2019.
- GSI'19: Geometric Science of Information @ Toulouse, France. August 27–29, 2019.
- Summer School: Mathematics of Deep Learning @ Berlin, Germany. August 19–30, 2019.
- SciCADE @ University of Innsbruck, Austria. July 22–26, 2019.
- 11th Oxford-Berlin meeting @ WIAS Berlin, Germany. May 23–25, 2019.
- Noncommutative Stochastic Analysis 2019 @ NTNU Trondheim, Norway. May 8–10, 2019.
- New Directions in Stochastic Analysis: Rough Paths, SPDEs and Related Topics @ Zuse Institute. Berlin, Germany. March 18–22, 2019.
- MAGiC 2019 @ Ilsetra, Lillehammer, Norway. February 25–March 1st, 2019.
- Rencontres du GDR Renormalisation @ Laboratoire Blaise Pascal. Clermont-Ferrand, France. November 12–16, 2018.
- 10th Oxford-Berlin meeting @ Oxford Mathematical Institue. Oxford, UK. November 29th & 30th, December 1st, 2018.

## Scholarships and awards

- ERCIM Postdoctoral Fellowship. October 1, 2018 – May 31, 2019.
- CONICYT Doctorado Nacional 2013. March 2013 – September 2017.