I am an Assistant Professor of Applied Mathematics in the School of Engineering and Applied Sciences at Harvard, and an Affiliate in the Department of Statistics.
My research lies in the areas of applied algebra, tensors and multilinear algebra, applied algebraic geometry, and algebraic statistics.
I study data analysis questions and use algebraic approaches to answer them. This involves factorising data using matrix and tensor decompositions, solving systems of equations, and finding group structure and symmetries in models. I work on problems arising in parameter estimation, dimensionality reduction, causal inference, and optimization, as well as applications to study complex physical and biological systems.
Office hours: by appointment.
If you are a postdoc, graduate student, or undergraduate student interested in working on a research project with me, feel free to get in touch.
Here is my google scholar profile and CV [last updated Aug '23].
I am an Assistant Professor of Applied Mathematics in the School of Engineering and Applied Sciences at Harvard, and an Affiliate in the Department of Statistics.
Previously, I was a Junior Fellow at the Society of Fellows, Harvard University. Before that, I was a Hooke Research Fellowship in the Mathematical Institute at the University of Oxford and a Junior Research Fellowship at The Queen's College. I received my PhD from UC Berkeley in 2019, where my advisor was Bernd Sturmfels. My thesis was awarded the SIAM Richard C. DiPrima Prize and the Friedman Memorial Prize.
27. Identifiability of overcomplete independent component analysis (with Kexin Wang), arXiv:2401.14709.
26. Complete collineations for maximum likelihood estimation (with Gergely Bérczi, Eloise Hamilton, and Philipp Reichenbach), arXiv:2311.03329.
25. Rectifiable paths with polynomial log-signature are straight lines (with Peter Friz and Terry Lyons), arXiv:2305.19210.
24. Supermodular Rank: Set Function Decomposition and Optimization (with Rishi Sonthalia and Guido Montúfar), arXiv:2305.14632.
23. Multilinear Hyperquiver Representations (with Tommi Muller and Vidit Nanda), arXiv:2305.05622.
22. Differential equations for Gaussian statistical models with rational maximum likelihood estimator (with Carlos Améndola, Lukas Gustafsson, Kathlén Kohn, and Orlando Marigliano), arXiv:2304.12054. [code]
21. Deciphering causal genomic templates of complex molecular phenotypes (with Salil Bhate and Juan Caicedo), bioRxiv:2022.08.15.503769.
20. Symmetries in Directed Gaussian Graphical Models (with Visu Makam and Philipp Reichenbach), Electronic Journal of Statistics 2023, Vol. 17, No. 2, 3969-4010. [arXiv] [pdf] [code]
19. Linear Causal Disentanglement via Interventions (with Salil Bhate, Chandler Squires, and Caroline Uhler), Proceedings of the 40th International Conference on Machine Learning (ICML 2023), PMLR 202, 2023. [arXiv] [pdf] [code]
18. Lower bounds on the rank and symmetric rank of real tensors (with Kexin Wang), Journal of Symbolic Computation 118 (2023) 69–92. [arXiv] [pdf] [code]
17. Principal Components along Quiver Representations (with Heather Harrington and Vidit Nanda), Foundations of Computational Mathematics (2022). [arXiv] [pdf]
16. A blood atlas of COVID-19 defines hallmarks of disease severity and specificity (as part of 100+ author Covid-19 Multi-omics Blood Atlas (COMBAT) Consortium), Cell 185 (2022) 916–938. [medRxiv] [pdf]
15. Toric Invariant Theory for Maximum Likelihood Estimation in Log-linear Models (with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach), Algebraic Statistics 12 (2021) no. 2, 187-211. [arXiv] [pdf]
14. The Maximum Likelihood Degree of Linear Spaces of Symmetric Matrices (with Carlos Améndola, Lukas Gustafsson, Kathlén Kohn, and Orlando Marigliano), Le Matematiche 76 (2021) no. 2, 535-557. [arXiv]
13. Invariant Theory and Scaling Algorithms for Maximum Likelihood Estimation (with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach), SIAM Journal on Applied Algebra and Geometry 5 (2021) no. 2, 304-337. [arXiv] [pdf] [case study]
12. Ranks and Singularities of Cubic Surfaces (with Eunice Sukarto), Le Matematiche 75 (2020) no. 2, 575-594. [arXiv] [pdf]
11. Ranks and Symmetric Ranks of Cubic Surfaces, Journal of Symbolic Computation 101 (2020) 304-317. [arXiv]
10. Learning Paths from Signature Tensors (with Max Pfeffer and Bernd Sturmfels), SIAM Journal on Matrix Analysis and Applications 40 (2019) no. 2, 394-416. [arXiv] [pdf] [code]
9. Tensor clustering with algebraic constraints gives interpretable groups of crosstalk mechanisms in breast cancer (with Mariano Beguerisse-Díaz, Birgit Schoeberl, Mario Niepel and Heather Harrington), Journal of the Royal Society Interface 16 (2019) no. 151, 20180661. [arXiv] [pdf] [News story]
8. Duality of Graphical Models and Tensor Networks (with Elina Robeva), Information and Inference: A Journal of the IMA, iay009. [arXiv] [pdf]
7. Mixtures and products in two graphical models (with Guido Montúfar), Journal of Algebraic Statistics 9 (2018) no. 1, 1-20. [arXiv] [pdf]
6. Gram Determinants of Real Binary Tensors, Linear Algebra and its Applications 544 (2018) 350-369. [arXiv] [pdf]
5. Real Rank Two Geometry (with Bernd Sturmfels), Journal of Algebra 484 (2017), 310-333. [arXiv] [pdf]
4. Does Antibiotic Resistance Evolve in Hospitals? (with Portia Mira, Bernd Sturmfels and Miriam Barlow), Bulletin of Mathematical Biology 79 (2017) 191-208. [arXiv] [pdf] [R package] [News story]
3. Singular Vectors of Orthogonally Decomposable Tensors (with Elina Robeva), Linear and Multilinear Algebra 65 (2017) no. 12, 2457-2471. [arXiv]
2. Eigenconfigurations of Tensors (with Hirotachi Abo and Bernd Sturmfels), Algebraic and Geometric Methods in Discrete Mathematics, Contemporary Mathematics 685, Amer. Math. Soc., Providence, RI (2017) 1-25.
1. Local dynamics of intersections: V. I. Arnold's theorem revisited (with Sergei Yakovenko) Israel Journal of Mathematics 201 (2014) no. 2 813-833. [arXiv] [pdf]
Structured Tensors and the Geometry of Data, 2019.
Ideas for a complex world, a public lecture I gave as part of the Oxford Mathematical Institute public lecture series, Oxford, November 2020. Featured in the Oxford Mathematics Annual Newsletter 2021.
The Algebraic Statistics of an Oberwolfach Workshop, Snapshots of modern mathematics from Oberwolfach (2018) no. 1.
Applied Algebra and Geometry: A SIAGA of Seven Pictures, also featured as a series of three articles in SIAM news: I, II, III, Spring 2016.
Picture this Maths, a joint blog with Rachael Boyd to present mathematical research centered around pictures. Featured here in the AMS Blog on math blogs.
Iwasawa Theory of Elliptic Curves with Complex Multiplication, Part III essay (master's thesis) supervised by John Coates, Cambridge, Spring 2014.
Imagining things: What do you think about, when you think about maths?, a mini documentary about mathematical thinking, directed by Richard Braham and me, Cambridge, 2013.
Spring 2023: I am organizing, with Melanie Weber, a seminar on Applied Algebra and Geometry in Harvard Applied Math.
Spring 2022: I organized, with Xiao-Li Meng, a mini-workshop on Algebraic Statistics at Harvard University.
Summer 2021: I organized, with Michael Walter, a mini-symposium at the SIAM Conference on Applied Algebraic Geometry called "Applied Invariant Theory: Statistics and Algorithms".
2020-2021: I organized, with Vidit Nanda, the Data Science seminar at the Mathematical Institute in Oxford.
Spring 2021: I gave two introductory lectures at the IPAM long program on "Tensor Methods and Emerging Applications to the Physical and Data Sciences". They were (i) From linear algebra to multi-linear algebra (ii) Tensors in statistics and data analysis.
Summer 2019: I organized, with Carlos Améndola, Joscha Diehl, and Francesco Galuppi, a mini-symposium at the SIAM Conference on Applied Algebraic Geometry, called "Signature tensors of paths".
Spring 2019: I organized a learning seminar on Applied Invariant Theory.
Summer 2018: I organized, with Elina Robeva, a mini-symposium at the SIAM Annual meeting "Theoretical Challenges of Tensor Decomposition".
Summer 2018: I organized, with Andre Uschmajew and Bart Vandereycken, a mini-symposium in the SIAM Conference on Applied Linear Algebra, called "Tensors and Multilinear Algebra".
Summer 2017: I organized, with Elina Robeva, a mini-symposium in the SIAM Conference on Applied Algebraic Geometry, called "Tensors: From Algebra to Applications".
Spring 2017: I organized the workshop Interactions between Algebra and the Sciences with Jacinta Torres.
Fall 2016: I co-organized a tensors reading group Foundational Tensor Theory with Rockford Sison.
Fall 2015: I organized a Commutative Algebra reading group.
Summer 2013: I did a research project at Imperial College, London under the supervision of Mike Levin. We analysed biological pathways associated with susceptibility to inflammatory diseases.
Summer 2012: I attended the Kupcinet-Getz Program at the Weizmann Institute in Israel, working with Sergei Yakovenko on algebraic properties of local dynamics of intersections, see here.
Fall 2023: Algebraic Fundamentals of Representing Data (Applied Math 210).
Trinity 2020: Tutorials for Statistics and Data Analysis, The Queen's College, Oxford.
Spring 2018: GSI for Math 10B with Prof. Kelli Talaska, UC Berkeley.
Spring 2017: Head GSI for Math 10B with Prof. Craig Evans. A fun worksheet I made for the last class of the semester, UC Berkeley.
Spring 2016: I organized a Mathematical Biology reading group for a group of Berkeley undergraduate students.
Fall 2015: GSI for Math 10A with Prof. Richard Bamler, UC Berkeley.