Nikhil Srivastava

Nikhil Srivastava

Professor of Mathematics (1035 Evans Hall)
Senior Scientist, Simons Institute (204 Calvin Lab)
UC Berkeley.

email: firstname at math.berkeley.edu.

Research Interests: theory of computing, spectral graph theory, random matrices, convex geometry, geometry of polynomials, numerical algorithms.

Spring 2026 Office Hours: W 3:30-4:50pm, F 3:30-5:00pm

Current Teaching: Math 55

Seminars: Student Discrete Analysis Seminar: schedule. Mondays 1:30pm-3:00pm in 939 Evans.
Spectral Theory Seminar (w/ Jitomirskaya and Zworski): schedule. Wednesdays 2-3pm in 736 Evans.

Papers:

Tight Bounds on Plurality (with A. Taylor), IPL 96 (2005). [PDF]
Voting with Rubber Bands, Weights, and Strings (with D. Cervone, R. Dai, D. Gnoutcheff,
G. Lanterman, A. Mackenzie, A. Morse, and W. Zwicker), Mathematical Social Sciences. [LINK]
On the Longest Path Algorithm for Reconstructing Trees from Distance Matrices (with L. Reyzin), IPL 101 (2007). [PDF]
Learning and Verifying Graphs Using Queries, with a Focus on Edge Counting (with L. Reyzin), ALT 2007. [PDF] [slides]
Graph Sparsification by Effective Resistances (with D. Spielman), STOC 2008, SICOMP special issue (2011). [arXiv] [slides]
Twice-Ramanujan Sparsifiers (with J. Batson and D. Spielman), STOC 2009, SICOMP special issue + SIAM Review (2012),. [arXiv] [slides] [video] and [bourbaki notes] by Assaf Naor.
An Elementary Proof of the Restricted Invertibility Theorem (with D. Spielman), Israel J. Math 190 (2012). [arXiv] [video]
On Contact Points of Convex Bodies, Geometric Aspects of Functional Analysis, Springer Lec. Notes in Math. (2012) [PDF]
Covariance Estimation for Distributions with 2+epsilon Moments (with R. Vershynin), Annals of Probability. [arXiv] [slides]
Graph Densification (with M. Hardt and M. Tulsiani), ITCS 2012. [PDF]
Zero-One Rounding of Singular Vectors (with A. Deshpande and R. Kannan), ICALP 2012. [PDF]
A New Approach to Computing Maximum Flows using Electrical Flows (with Y. Lee and S. Rao), STOC 2013. [PDF] [slides]
Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees (with A. Marcus and D. Spielman), Annals of Mathematics & FOCS 2013 [arXiv] [slides]
Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem (with A. Marcus and D. Spielman), Annals of Mathematics [arXiv] and [blog post]
Spectral Sparsification of Graphs: Theory and Algorithms (with J. Batson, D. Spielman, and S-H. Teng), Communications of the ACM 2013. [PDF] and [technical perspective] by Assaf Naor.
Ramanujan Graphs and the Solution of the Kadison-Singer Problem. (with A. Marcus and D. Spielman), Proc. ICM 2014. [arxiv]
Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes (with A. Marcus and D. Spielman), FOCS 2015, SICOMP special issue. [arxiv]
Real Stability Testing (with P. Raghavendra and N. Ryder), ITCS 2017. [arxiv]
Aproximating the Largest Root and Applications to Interlacing Families (with N. Anari, S. Oveis Gharan, and A. Saberi), SODA 2018. [arxiv]
Asymptotically Optimal Multi-Paving (with M. Ravichandran), IMRN. [arxiv]
Group Synchronization on Grids (with E. Abbe, L. Massoulie, A. Montanari, and A. Sly), Mathematical Statistics and Learning. [arxiv]
An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification (with L. Trevisan), SODA 2018. [arxiv]
Localization of Electrical Flows (with A. Schild and S. Rao), SODA 2018. [arxiv]
A Matrix Expander Chernoff Bound (with A. Garg, Y. Lee, and Z. Song), STOC 2018. [arxiv] [slides]
Exponential Lower Bounds on Spectrahedral Representations of Hyperbolicity Cones (with P. Raghavendra, N. Ryder, B. Weitz), SODA 2019. [arxiv] [video]
Interlacing Families III: Sharper Restricted Invertibility Estimates (with A. Marcus and D. Spielman), Israel J. Math. [arxiv]
The Solution of the Kadison-Singer Problem (with A. Marcus), Proc. CDM 2016. [arxiv]
Optimal Lower Bounds for Sketching Graph Cuts (with C. Carlson, A. Kolla, and L. Trevisan), SODA 2019. [arxiv]
On Non-Localization of Eigenvectors of High Girth Graphs (with S. Ganguly), IMRN. [arxiv]
Finite Free Convolutions of Polynomials (with A. Marcus and D. Spielman), Probability Theory and Related Fields. [arxiv]
Gaussian Regularization of the Pseudospectrum and Davies' Conjecture (with J. Banks, S. Mukherjee, A. Kulkarni), Commun. Pure Appl. Math. [arxiv] [video]
High-girth near-Ramanujan Graphs with Localized Eigenvectors (with N. Alon and S. Ganguly), Israel J. Math. [arxiv] [slides]
Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time (with J. Banks, J. Garza Vargas, A. Kulkarni), FOCS 2020, Found. Comp. Math. [arxiv] [video]
On Concentration Inequalities for Random Matrix Products (with T. Kathuria and S. Mukherjee), unpublished note. [arxiv]
Overlaps, Eigenvalue Gaps, and Pseudospectrum under Real Ginibre and Absolutely Continuous Perturbations (with J. Banks, J. Garza Vargas, A. Kulkarni), Ann. l'Institute Henri Poincare B. [arxiv]
Scalar Poincare Implies Matrix Poincare (with A. Garg and T. Kathuria) Electron. Comm. Probab. [arxiv]
Support of Closed Walks and Second Eigenvalue Multiplicity of Graphs (with T. McKenzie and P. Rasmussen) STOC 2021. [arxiv]
A Spectral Approach to Polytope Diameter (with H. Narayanan and R. Shah) ITCS 2022, Discrete Comput. Geom. [arxiv].
Many Nodal Domains in Random Regular Graphs (with S. Ganguly, T. McKenzie, S. Mohanty) Comm. Math. Phys.. [arxiv] [video].
Bit Complexity of Jordan Normal Form and Spectral Factorization (with P. Dey, R. Kannan, N. Ryder) ITCS 2023. [arxiv] [video].
Global Convergence of Hessenberg Shifted QR I: Exact Arithmetic (with J. Banks, J. Garza-Vargas) Found. Comp. Math. [arxiv] [slides].
Global Convergence of Hessenberg Shifted QR II: Finite Arithmetic (with J. Banks, J. Garza-Vargas) SIAM J. Matrix Analysis. [arxiv].
Global Convergence of Hessenberg Shifted QR III: Approximate Ritz Values via Shifted Inverse Iteration (with J. Banks, J. Garza-Vargas) SIAM J. Matrix Analysis. [arxiv].
On Eigenvalue Gaps of Integer Matrices (with A. Abrams, J. Pommersheim, Z. Landau). Mathematics of Computation. [arxiv].
The Complexity of Diagonalization. ISSAC 2023.[arxiv].
On the Ground State Energies of Discrete and Semiclassical Schrödinger Operators (with I. Detherage and Z. Stier). Pure and Applied Analysis. [arxiv].
Ramanujan Graphs and Interlacing Families. Proc. ICBS 2024. [arxiv].
Sparse Pseudospectral Shattering (with R. Shah and E. Zeng). Submitted. [arxiv].
On quantum to classical comparison for Davies generators (with J. Basso, S. Ganguly, A. Sinclair, Z. Stier, T-D. Vuong). QIP 2026. [arxiv].
Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop (with D. Kressner et al.). [arxiv].
First Proof (with M. Abouzaid et al.). [arxiv] [website].
Finite Free Information Inequalities (with J. Garza-Vargas and Z. Stier). Submitted. [arxiv].

Theory at the Institute and Beyond: Jan'24, Oct'24, Feb'25, Sep'25, Feb'26.

CV: [PDF]

Thesis: Spectral Sparsification and Restricted Invertibility [PDF] [slides]
Advisor: Dan Spielman.

Current Graduate Students:Zack Stier, Isabel Detherage, Joao Basso (co-advised with Lin Lin).
Past Graduate Students: Rikhav Shah, Jorge Garza Vargas (co-advised with Dan Voiculescu), Theo McKenzie, Jess Banks, Satyaki Mukherjee, Archit Kulkarni, Nick Ryder, Aaron Schild (EECS, co-advised with Satish Rao)

Program Committees: ITCS 2023, ITCS 2021, STOC 2019, FOCS 2018, ITCS 2017, ICALP 2016, STOC 2015, FSTTCS 2012.

Organizational Activities:

IPAM lectures on Expected Characteristic Polynomials: one, two, three.

Simons Institute Tutorial on Graph Sparsification: one, two, three.

General Audience Talk on Graph Sparsification: [vimeo] (Kavli Frontiers of Science, Agra 2013)

Previous Teaching

Math 55, Discrete Mathematics
Fall 2021: Math 55, Discrete Mathematics
Spring 2021: Berkeley Connect
Spring 2021: Math 54
Fall 2020: Math 224A: Mathematical Methods for Physical Sciences.
Spring 2020: Math 54
Fall 2019: Math 224A: Mathematical Methods for Physical Sciences.
Fall 2018: Math 54
Spring 2018: Math 53, Multivariable Calculus.
Spring 2017/18: Berkeley Connect
Fall 2016: Math 54, Linear Algebra and Differential Equations.
Spring 2016: Math 55, Discrete Mathematics
Fall 2015: Math 185, Complex Analysis
Fall 2015: Math 270, Hot Topics: The Geometry of Polynomials in Algorithms, Combinatorics, and Probability
Spring 2015: Math 121A, Mathematical Tools for the Physical Sciences
Spring 2012: COS 521 Advanced Algorithms (Princeton)

Funding: I am grateful to the NSF and to the Sloan Foundation for generously supporting my research.

Previously, I was at Microsoft Research India, Princeton, MSRI, IAS, Yale, and Union College.