# Dr. Nikolas Tapia

**Address:**Mohrenstraße 39, 10117 Berlin, Germany.**Office:**Hausvogteiplatz 11, room 0.02**Telephone:**+49 30 20372-448.**e-mail**: tapia (at) wias (dash) berlin (dot) de.- 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, and the Work Group on Stochastics and Financial Mathematics at the Technical University of Berlin.

I work as a Postdoctoral Researcher in the MATH+ EF1-13 project "Stochastic and rough aspects in deep neural networks", led by P. K. Friz and C. Bayer.

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, 2021.

**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 and Peter Friz*. - Discrete-time signatures,
*with Joscha Diehl and Kurusch Ebrahimi-Fard*. - Numerical methods for rough SPDEs,
*with Ana Djurdjevac and Christian Bayer*.

## Preprints and Publications

### Articles in Refereed Journals

- J. Diehl, K. Ebrahimi-Fard,
__N. Tapia__.*Tropical time series, iterated-sums signatures and quasisymmetric functions*. (2022) To appear in SIAM J. Appl. Algebra Geom. (SIAGA). arXiv:2009.08443 [math.RA] - 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. (2021) - 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

- K. Ebrahimi-Fard, F. Patras,
__N. Tapia__, L. Zambotti.*Shifted substitution in non-commutative multivariate power series with a view toward free probability*. (2022) arXiv:2204.01445 [math.PR] - M. Ghani Varzaneh, S. Riedel, A. Schmeding,
__N. Tapia__.*The geometry of controlled rough paths*. (2022) arXiv:2203.05946 [math.PR] - C. Bayer, P. K. Friz,
__N. Tapia__.*Stability of deep Neural Networks via discrete rough paths*. (2022) WIAS Preprint no. 2732, arXiv:2201.07566 [cs.LG] - J. Diehl, K. Ebrahimi-Fard,
__N. Tapia__.*Generalized iterated-sums signatures*. (2020) arXiv:2012.04597 [math.RA]

### 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

## 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.