Darrick Lee PhD Candidate, University of Pennsylvania applied algebraic topology
Contact Office: David Rittenhouse Labs 3C5 Email: ldarrick at sas.upenn.edu

I am currently a fifth year PhD student in the Applied Mathematics and Computational Sciences (AMCS) program at the University of Pennsylvania working under Professor Rob Ghrist. I received my Bachelor of Applied Sciences at the University of British Columbia in 2016, where I majored in Engineering Physics with an Electrical Engineering specialization and minored in Mathematics.

I received a Fulbright Canada student award for 2016-2017 and a NSERC PGS-D scholarship for 2018-2021.

My CV is here.

# Research Interests

I am broadly interested in algebraic topology and category theory, particularly in how these fields motivate a different point of view on approaching problems in data science.

More specifically, I study Chen’s iterated integral cochain model for path spaces and how these cochains may be considered as the semantics of multivariate time series data by providing interpretable and computable features. The 0-cochains in this model for the path space of $\mathbb{R}^n$ form a collection of functions called path signatures, which offer a reparametrization-invariant characterization of paths. I am interested in further developing tools derived from path signatures, understanding how higher cochains may be leveraged, and considering how generalizations of Chen’s cochain model to mapping spaces may provide methods to study more complex parametrized data.

A motivating example of path signatures is provided below, and a more complete survey can be found in my preprint with Chad Giusti.

### Path Signatures - A Motivating Example

For example, suppose we wish to detect leading/lagging relationships between two different time series, denoted $\gamma_1(t)$ (dark colors) and $\gamma_2(t)$ (light colors). A classical technique is the unbiased cross-correlation $r[\gamma_1, \gamma_2](t_d)$, where $t_d$ denotes the time delay. The cross-correlation can readily detect such behavior when the time series are of the same form (the blue curves on the left represent two sine waves off by a time delay) by locating the local maxima of the cross-correlation near the origin (circled point in blue on the right). The fact that the local maximum occurs at a negative time delay $t_d$ implies that $\gamma_1$ is leading $\gamma_2$. However, this indicator disappears once the time series are reparametrized (the animations depict a family of reparametrizions (left) and the corresponding cross-correlations (right) in red).

​ A reparametrization-invariant leading indicator is the signed area enclosed by the path defined by the time series (the ellipse on the right). The signed area is the area bounded by the curve, taking into account the multiplicity of the area (in this case, 4), and the orientation (in this case, counterclockwise, corresponding to a positive signed area). A positive signed area can be interpreted as an indicator that $\gamma_1$ is leading $\gamma_2$. Note that since reparametrization of the path does not change the image of the path, the signed area is reparametrization-invariant. Additionally, the signed area can be calculated as a linear combination of path signature terms.

​ For general multivariate time series, we compute the signed area for every pair of time series to obtain pairwise leading/lagging indicators. This was originally studied by Y. Baryshnikov and E. Schlafly.

# Publications and Preprints

1. Path signatures on Lie groups, D. Lee and R. Ghrist
(submitted) (2020)
2. Iterated integrals and population time series analysis, C. Giusti and D. Lee
In Nils A. Baas, Gunnar E. Carlsson, Gereon Quick, Markus Szymik, and Marius Thaule, editors, Topological Data Analysis, Abel Symposia, pages 219–246 (2020) [arXiv]
3. A methodology for morphological feature extraction and unsupervised cell classification. D. Bhaskar, D. Lee, H. Knútsdóttir, C. Tan, M. Zhang, P. Dean, C. Roskelley, L. Edelstein- Keshet
(under review) (2019)
4. Structure of vortex-bound states in spin singlet chiral superconductors, D. Lee and A. Schnyder
Physical Review B. 93: 064522 (2016) [arXiv]
5. Localization for transversally periodic random potentials on binary trees, R. Froese, D. Lee, C. Sadel, W. Spitzer, and G. Stolz
Journal of Spectral Theory. 6: 557-600 (2016) [arXiv]

# Teaching

I am one of the UPenn Math Department’s Master TAs, who help train, observe, and select TAs for the department.

• August 2020 Co-Instructor for Pre-Freshman Program
• Spring 2018 TA for MATH 241: Calculus IV (Partial Differential Equations)
• Fall 2017 TA for MATH 360: Advanced Calculus (Real Analysis)
Math Department Good Teaching Award