Research

My general area of research is Geometric Analysis. To be slightly more specific, the problems I generally work on encompass elements in spectral theory, PDEs, differential geometry, harmonic analysis, probability and their applications. 

Presently, my main research interests are :

spectral theory (particularly in the setting of non-positively curved Riemannian manifolds)

Preprints:

6. Finitely generated groups and harmonic functions of slow growth (with Soumyadeb Samanta and Soumyadip Thandar), preprint (arXiv:2405.07688).

5. On the impact of stable ranks in deep nets (with Bogdan Georgiev, Lukas Franken and Georgios Arvanitidis), preprint (arXiv:2110.02333).

4. Heat profile, level sets and hot spots of Laplace eigenfunctions (with Soumyajit Saha), preprint (arXiv:2109.06531).

3. On the effects of small perturbation on low energy Laplace eigenfunctions (with Soumyajit Saha), preprint (arXiv:2108.13874).

2. Mass non-concentration at the nodal set and a sharp Wasserstein uncertainty principle, preprint (arXiv:2103.11633).

1. Compactifications and maximal ideal spaces arising from holomorphic functions (with Bedanta Bose), preprint.

Publications (reverse chronological order):

15.  On the spectrum of certain Hadamard manifolds (with Werner Ballmann and Panagiotis Polymerakis),  SIGMA  19 (2023), 050, 19 pages (special issue in honor of Jean-Pierre Bourguignon for his 75th birthday) (arXiv:2301.10986, online edition).

14. Some applications of heat flow to Laplace eigenfunctions (with Bogdan Georgiev), Communications in PDE 47 (2022), no. 4, 677 - 700 (arXiv:2109.00710, journal link).

13. Heating up decision boundaries: isocapacitory saturation, adversarial scenarios and generalization bounds (with Lukas Franken and Bogdan Georgiev), ICLR 2021 (arXiv:2101.06061, openreview).

12. Nodal sets of Laplace eigenfunctions under small perturbations (with Soumyajit Saha), Mathematische Annalen 383 (2022), no. 1-2, 475–491  (arXiv:2003.04833, journal link).

11. Polyhedral billiards, eigenfunction concentration and almost periodic control (with Mihajlo Cekic and Bogdan Georgiev), Communications in Mathematical Physics 377 (2020), no. 3, 2451–2487 (arXiv:1903.09857, journal link).

10. Some remarks on nodal geometry in the smooth setting (with Bogdan Georgiev),  Calculus of Variations & Partial Differential Equations 58 (2019), no. 3, Art. 93, 25 pp. (arXiv:1608.05344, journal link).

9. A Spectral Gap Estimate and Applications (with Bogdan Georgiev and Stefan Steinerberger), Potential Analysis 49 (2018), no. 4, 635–645 (arXiv:1612.08565, journal link).

8. On maximizing the fundamental frequency of the complement of an obstacle (with Bogdan Georgiev). C. R. Math. Acad. Sci. Paris 356 (2018), no. 4, 406–411 (arXiv:1706.02138, journal link).

7. Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary, Mathematische Zeitschrift, 289 (2018), no. 3 - 4, 1011 - 1031 (arXiv:1410.6041, journal link).

6. Nodal geometry, heat diffusion and Brownian motion (with Bogdan Georgiev),  Analysis & PDE, 11 (2018), no. 1, 133–148 (arXiv:1602.07110, journal link).

5. Nonlinear travelling waves on complete Riemannian manifolds, Advances in Differential Equations, 23 (2018), no. 1-2, 65 - 88 (journal link).  

4. A special class of nonlinear hypoelliptic equations on spheres, NoDEA Nonlinear Differential Equations and Applications, 24 (2017), no. 2, Art. 15, 25 pp (journal link).

3. Variation of Laplace spectra of compact "nearly" hyperbolic surfaces, C. R. Math. Acad. Sci. Paris, 355 (2017), no. 2, 216 - 221 (journal link). 

2. Gelfand-Kolmogoroff theorem for rings of analytic functions (with Bedanta Bose),  Topology and its Applications, 218 (2017), 53 - 65 (journal link).

1. Extremal values of the (fractional) Weinstein functional on the hyperbolic space, Forum Mathematicum, 29 (2017), no. 4, 959 - 970 (journal link).