(click here for less standard but outdated pic)
I am an Assistant Professor in the Math Department at the University of Washington, Seattle.
I can be reached via [first 8 letters of last name]@uw.edu
My main field of research is Analysis. I like questions combining PDEs, Spectral Theory, Harmonic Analysis
and all sorts of applications. My research is supported by NSF DMS-1763179 and the Sloan Foundation.
Spring 2021: MATH 300
Office Hours: tbd
Some Open Problems
- (with Greengard, Liu, Tsyvinski) Factor Clustering with t-SNE, ssrn
- Free Convolution of Measures via Roots of Polynomials, arxiv
- Surrounding the solution of a Linear System of Equations from all sides, arxiv
- Using Expander Graphs to test whether samples are iid, arxiv
- (with Noah Kravitz) The Smoothest Average: Dirichlet, Fejer and Chebychev, arxiv
- On the Regularization Effect of Stochastic Gradient Descent applied to Least Squares, arxiv
- A Weighted Randomized Kaczmarz Method for Solving Linear Systems, arxiv
- Randomized Kaczmarz converges along small singular vectors, arxiv
- Polynomials with Roots on the Unit Circle: Regularity of Leja sequences, arxiv
- (with Jeremy Hoskins) A Semicircle Law for Derivatives of Random Polynomials, arxiv
- Fourier Uncertainty Principles, Scale Space Theory and the Smoothest Average, arxiv
- On the Stability of Fourier Phase Retrieval, arxiv
- (with Aleh Tsyvinski) On Vickreys Income Averaging, arxiv
- A Spectral Approach to the Shortest Path Problem, arxiv
- (with Adela DePavia) Spectral Clustering Revisited: Information Hidden in the Fiedler Vector, arxiv
- (with Ofir Lindenbaum) Randomly Aggregated Least Squares for Support Recovery, arxiv, Signal Processing , accepted.
- Regularized Potentials of Schrodinger Operators and a Local Landscape Function, arxiv
- (with Jaffe, Kluger, Lindenbaum, Patsenker, Peterfreund), The Spectral Underpinning of word2vec, arxiv, Front. Appl. Math. Stat., accepted.
- (with Kovalsky, Aigerman, Daubechies, Khazdan, Lu), Non-Convex Planar Harmonic Maps, arxiv
- (with Jeremy Hoskins), Towards Optimal Gradient Bounds of the Torsion Function in the Plane, arxiv, Journal of Geometric Analysis , accepted.
- (with Roy Lederman), Extreme Values of the Fiedler Vector on Trees, arxiv
- (with Jianfeng Lu), Synchronization of Kuramoto Oscillators in Dense Networks, arxiv, Nonlinearity , accepted.
- (with Sean O'Rourke) A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation
arxiv, Proc. Amer. Math. Soc., accepted.
- (with Louis Brown) On the Wasserstein Distance between Classical Sequences and the Lebesgue Measure arxiv, Trans. Amer. Math. Soc. , accepted.
- (with Louis Brown) Positive-definite Functions, Exponential Sums and the Greedy Algorithm: a curious Phenomenon arxiv, Journal of Complexity, accepted.
- (with Jianfeng Lu), Optimal Trapping of Brownian Motion: A Nonlinear Analogue of the Torsion Function, arxiv, Potential Analysis, accepted.
- Hot Spots in Convex Domains are in the Tips (up to an Inradius), arxiv, Comm. PDE, accepted.
- A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds, arxiv
- (with Beck, Brandolini, Burdzy, Henrot, Langford, Larson, Smits), Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions, arxiv, Journal of Geometric Analysis, accepted.
- On Sublevel Set Estimates and the Laplacian, arxiv, Potential Analysis, accepted.
- (with Amir Sagiv), Transport and Interface: an Uncertainty Principle for the Wasserstein distance, arxiv, SIAM J. Math. Anal., accepted.
- (with Jianfeng Lu), A Dimension-Free Hermite-Hadamard Inequality via Gradient Estimates for the Torsion Function, arxiv, Proc. Amer. Math. Soc., accepted.
- (with Aleh Tsyvinski) Tax Mechanisms and Gradient Flows arxiv
- (with Rima Al-Aifari, Xiuyuan Cheng, Lillian Pierce),
On Matrix Rearrangement Inequalities, arxiv, Proc. Amer. Math. Soc., accepted.
- Roots of trigonometric polynomials and the Erdos-Turan theorem, arxiv, Mathematika, accepted.
- (with Rick Barnard) Three Convolution Inequalities on the Real Line with Connections to Additive Combinatorics
arxiv, Journal of Number Theory, accepted.
- (with Markus Faulhuber) An Extremal Property of the Hexagonal Lattice arxiv, Journal of Statistical Physics, accepted.
- (with Felipe Goncalves and Diogo Oliveira e Silva) A Universality Law For Sign Correlations of Eigenfunctions of Differential Operators arxiv, Journal of Spectral Theory, accepted.
- (with Dmitry Kobak, George Linderman, Yuval Kluger, Philipp Berens) Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations,
arxiv, ECML PKDD 2019
- Dynamically Defined Sequences with Small Discrepancy, arxiv, Monatshefte Math., accepted.
- A Nonlocal Functional Promoting Low-Discrepancy Point Sets, arxiv, Journal of Complexity, accepted.
- A forgotten Theorem of Schoenberg on one-sided integral averages, arxiv, American Mathematical Monthly, accepted.
- Poissonian Pair Correlation in Higher Dimensions, arxiv, Journal of Number Theory, accepted.
- (with Jianfeng Lu and Chris Sogge) Approximating Pointwise Products of Laplacian Eigenfunctions, arxiv, Journal of Functional Analysis , 373, 3271-3282 (2019)..
- A Nonlocal Transport Equation Describing Roots of Polynomials Under Differentiation, arxiv, Proc. Amer. Math. Soc., accepted.
- Quantitative Homogenization and Convergence of Moving Averages, arxiv
- (with Raphy Coifman) A Remark on the Arcsine Distribution and the Hilbert Transform arxiv, Journal of Fourier Analysis and Applications , accepted, see also.
- (with Trevor Richards) Leaky Roots and Stable Gauss-Lucas Theorems, arxiv, Complex Variables and Elliptic Equations, 64, 1898-1904 (2019).
- (with Peter W. Jones) Localization of Neumann Eigenfunctions near Irregular Boundaries, arxiv, Nonlinearity, 32, 768-776 (2019).
- A metric Sturm-Liouville theory in two dimensions, arxiv, Calculus of Variations and Partial Differential Equations, accepted.
- The Hermite-Hadamard inequality in higher dimensions, arxiv, Journal of Geometric Analysis, accepted.
- (with Hau-tieng Wu) On Zeroes of Random Polynomials and Applications to Unwinding, arxiv, IMRN, accepted.
- (with Eric Chi) Recovering Trees with Convex Clustering, arxiv, SIAM J. Math. Data Science, 1, 383-407 (2019).
- A Stability Version of the Gauss-Lucas Theorem and Applications, arxiv, J. Aust. Math. Soc., accepted.
- A Compactness Principle for Maximizing Smooth Functions over Toroidal Geodesics, arxiv, Bull. Aust. Math. Soc., 100, 148-154 (2019).
- (with Jakob Kapeller and Matthias Aistleitner) Citation Patterns in Economics and Beyond, Science in Context, accepted.
- (with Alex Cloninger) On the Dual Geometry of Laplacian Eigenfunctions, arxiv, Experimental Mathematics, accepted.
- Electrostatic Interpretation of Roots of Orthogonal Polynomials, arxiv, Proc. Amer. Math. Soc., 146, 5323-5331 (2018).
- An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane, arxiv, Canad. Math. Bull., 62, 643-651 (2019).
- Quantitative Projections in the Sturm Oscillation Theorem, arxiv, J. Math. Pure Appl., accepted.
- (with Jianfeng Lu and Matthias Sachs) Quadrature Points via Heat Kernel Repulsion, arxiv, Constructive Approximation, accepted.
- Wasserstein Distance, Fourier Series and Applications, arxiv
- (with George Linderman) Numerical Integration on Graphs: where to sample and how to weigh, arxiv, Mathematics of Computation, accepted.
- Generalized Designs on Graphs: Sampling, Spectra, Symmetries, arxiv, Journal of Graph Theory, accepted.
- Refined Heinz-Kato-Loewner inequalities, arxiv, Journal of Spectral Theory, accepted.
- (with Dmitriy Bilyk and Feng Dai) General and Refined Montgomery Lemmata, arxiv, Math. Annalen, 373, 1283-1297 (2019).
- A Sharp Estimate for Probability Distributions, arxiv, Stat. Prob. Lett., accepted.
- (with George Linderman, Manas Rachh, Jeremy Hoskins, Yuval Kluger) Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding, arxiv, Nature Methods, 16, 243-245 (2019).
- On the Spectral Resolution of Products of Laplacian Eigenfunctions, arxiv, Journal of Spectral Theory, 9, p. 1367–1384 (2019)
- (with Sam Johnson) Intuitions about mathematical beauty: A case study in the aesthetic experience of ideas, Cognition, 189, 242-259 (2019).
- (with Sam Johnson) The Aesthetic Psychology of Mathematics, arXiv, Math. Intelligencer, 41, 67-70 (2019)
- (with Janna Lierl) A Local Faber-Krahn inequality and Applications to Schrodinger's Equation, arxiv, Comm. PDE, 43, 66-81 (2018).
- Varadhan Asymptotics for the Heat Kernel on Finite Graphs, arxiv
- (with George Linderman, Gal Mishne, Ariel Jaffe, Yuval Kluger) Randomized Near Neighbor Graphs, Giant Components, and Applications in Data Science, arxiv, Advances in Applied Probability, accepted.
- Poissonian Pair Correlation and Discrepancy, arxiv, Indag. Math. 29, 1167-1178 (2018).
- (with Jianfeng Lu) Detecting Localized Eigenstates of Linear Operators, arxiv, Res. Math. Sci. 5, no. 34 (2018)
- Spectral Limitations of Quadrature Rules and Generalized Spherical Designs, arxiv, IMRN, accepted.
- Oscillatory functions vanish on a large set, arxiv, Asian J. Math., 24, 177-190 (2020).
- Exponential Sums and Riesz energies, arxiv, Journal of Number Theory, 182, 37-56 (2018).
- (with Nick Marshall) Triangles capturing many lattice points, arxiv, Mathematika, 64, 551-582 (2018).
- (with George Linderman) Clustering with t-SNE, provably, arxiv, SIAM J. Math. Data Science, 1, 313-332 (2019).
- (with Xiuyuan Cheng and Gal Mishne) The Geometry of Nodal Sets and Outlier Detection, arxiv, Journal of Number Theory, 185 , 48-64 (2018).
- Topological Bounds on Fourier Coefficients and Applications to Torsion, arxiv, Journal of Functional Analysis, 274, 1611-1630 (2018).
- (with Noah Kravitz) Ulam Sequences and Ulam sets, arxiv, Integers, 18 , A80 (2018).
- (with Florian Pausinger and Manas Rachh) Optimal Jittered Sampling for Two Points in the Unit Square, arxiv, Statist. Probab. Lett., 132, 55-61 (2018).
- (with Uri Shaham) Stochastic Neighbor Embedding separates well-separated clusters, arxiv
- (with Bogdan Georgiev and Mayukh Mukherjee) A Spectral Gap Estimate and Applications, arxiv, Potential Analysis, 49, 635-645 (2018).
- (with Xiuyuan Cheng and Manas Rachh) On the Diffusion Geometry of Graph Laplacians and Applications, arxiv, Appl. Comp. Harm. Anal., 46, p. 674-688 (2019).
- (with SMALL 2016) On algorithms to calculate integer complexity, arxiv, Integers 19, A12, (2019).
- Well-Distributed Great Circles on S^2, arxiv, Discrete & Computational Geometry, 60, 40-56 (2018).
- Fast Escape in Incompressible Vector Fields, arxiv, Monatshefte Math., 186, 525-537 (2018).
- (with Jakob Kapeller and Matthias Aistleitner) The Power of Scientometrics and the Development of Economics, Journal of Economics Issues, 52, 816-834 (2018).
- (with Jianfeng Lu) A Variation on the Donsker-Varadhan Inequality for the Principal Eigenvalue, arxiv, Proc. Royal Soc. A, 473 (2017).
- (with Jakob Kapeller) Stability, Fairness and Random Walks in the Bargaining Problem, arxiv, Physica A, 488, 60-71 (2017).
- Localized Quantitative Criteria for Equidistribution, arxiv, Acta Arithmetica, 180, 183-199 (2017).
- (with Manas Rachh) On the location of maxima of solutions of Schroedinger's equation, arxiv, Comm. Pure. Appl. Math., 71, 1109-1122 (2018).
- (with Raphy Coifman and Hau-tieng Wu) Carrier frequencies, holomorphy and unwinding, arxiv, SIAM J. Math. Anal., 49 , 4838-4864 (2017).
- (with Roy Lederman) Stability Estimates for Truncated Fourier and Laplace Transforms, arxiv, Integral Equations and Operator Theory, 87, 529-543 (2017).
- An amusing sequence of functions, arxiv, Mathematics Magazine, 91, 262-266 (2018). pictures more fun
- (with Alex Cloninger) On Suprema of Autoconvolutions with an Application to Sidon sets, arxiv, Proc. Amer. Math. Soc.,145, 3191-3200 (2017).
- (with Alex Cloninger) Spectral Echolocation via the Wave Embedding, arxiv, Appl. Comp. Harm. Anal., 43, 577-590 (2017).
- (with Florian Pausinger, Heating a Room with Number Theory Mathematics Magazine, accepted.
- Localization of Quantum States and Landscape Functions, arxiv, Proc. Amer. Math. Soc., 145, 2895-2907 (2017).
- A Hidden Signal in the Ulam sequence, arxiv, Experimental Mathematics, 23, 460-467 (2017).
- (with Raphy Coifman) Nonlinear phase unwinding of functions arxiv, Journal of Fourier Analysis and Applications, 23, 778-809 (2017).
- (with Felipe Goncalves and Diogo Oliveira e Silva) Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots, arxiv, J. Math. Anal. Appl.,451, 678-711 (2017).
- (with Markus Faulhuber) Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions, arxiv, J. Math. Anal. Appl., 445 , 407-422 (2017).
- (with Yuke Li, Tianhao Wu and Nicholas Marshall) Extracting Geography from Trade Data, arxiv, Physica A, 473 205-212 (2017)
- (with Jakob Kapeller) Emergent phenomena in scientific publishing: a simulation exercise, Research Policy 45, 1945-1952 (2016).
- (with Alberto Enciso and Daniel Peralta-Salas) Prescribing the nodal set of the first eigenfunction in each conformal class, arxiv, IMRN, 54, 555-569.
- Directional Poincare Inequalities along Mixing Flows, Arkiv för Matematik 54 , 555-569, 2016 arxiv .
- A Filtering Technique for Markov Chains with Applications to Spectral Embedding, Applied and Computational Harmonic Analysis , 40 (2016), 575-587, arxiv.
- (with Florian Pausinger) On the Discrepancy of Jittered Sampling, Journal of Complexity, 33 (2016), 199-216, arxiv
- (with Rima Al-Aifari and Lillian Pierce) Lower bounds for the truncated Hilbert transform, Revista Matematica Iberoamericana, 32 (2016), 23-56, arxiv.
- A Rigidity Phenomenon for the Hardy-Littlewood maximal function, Studia Mathematica, 229 , 263-278 (2015), arxiv
- An Uncertainty Principle on compact manifolds, Journal of Fourier Analysis and Applications, 21 (2015), 575-599 arxiv.
- (with Herbert Koch) Convolution Estimates for Singular Measures and Some Global Nonlinear Brascamp-Lieb
Inequalities, arxiv, Proc. Roy. Soc. Edinburgh, 145 (2015), 1223-1237.
- Sharp L^1 Poincare inequalities correspond to optimal hypersurface cuts, Archiv der Mathematik, 105 (2015), 179-188 arxiv.
- On the Number of Positions in Chess without Promotion, International Journal of Game Theory 44, 761-767 (2015).
- (with Florian Pausinger) Local Extrema in Quantum Chaos, Physics Letters A, 379 (2015), 535-541 arxiv.
- Lower bounds on nodal sets of eigenfunctions via the heat flow, Communications in Partial Differential Equations, 39 (2014), arxiv.
- New Bounds for the Traveling Salesman Constant, Advances in Applied Probability, 47 (2015), arxiv.
- A Remark on Disk Packings and Numerical Integration of Harmonic Functions, Journal of Complexity: Oberwolfach Special Issue 31 (2015), 486-493, arxiv.
- Dispersion dynamics for the generalized Korteweg-de Vries equation, Proceedings of the American Mathematical Society 143 (2015), 789-800, arxiv.
- A Geometric Uncertainty Principle with an Application to Pleijel's Estimate, Annales Henri Poincare, 15 (2014), 2299 - 2319, arxiv.
- (with Michaela Nieuwenhuis and James Robinson ), Minimal Periods for Ordinary Differential Equations in
Strictly Convex Banach Spaces and Explicit Bounds for some L^p-Spaces, Journal of Differential Equations 256 (2014), 2846 - 2857, arxiv.
- (with Jakob Kapeller) Modeling the Evolution of Preferences: An Answer to Schubert and Cordes, Journal of Institutional Economics 10 (2014), 337- 347 pdf.
- (with Jakob Kapeller and Bernhard Schuetz ) The Impossibility of Rational Consumer Choice - A Problem and its Solution.
Journal of Evolutionary Economics 23 (2013) 29 - 60, link.
- (with Jakob Kapeller) How Formalism shapes Perception: An Experiment on Mathematics as a Language, International Journal of Pluralism and Economics Education
4 (2013): 138 - 156.
- Random restricted matching and lower bounds for combinatorial optimization.
Journal of Combinatorial Optimization 24 (2012) no.3, 280 - 298, link.
- (with Erhard Aichinger) A proof of a Theorem by Fried and MacRae and applications to the composition of polynomial functions,
Archiv der Mathematik 97 (2011) no.2, 115-124, link.
- On the optimal interpoint distance sum inequality,
Archiv der Mathematik 97 (2011) no.3, 289-298., link.
- Extremal uniform distribution and random chord lengths,
Acta Mathematica Hungarica 130 (2011) no.4, 321-339, link.
- A note on implicitly defined sets in uniform distribution theory,
Uniform Distribution Theory 6 (2011) no.2, 85-94, link.
- The asymptotic behavior of the average $L^p-$discrepancies and a randomized discrepancy,
The Electronic Journal of Combinatorics 17 (2010) R106, link.
- A New Lower Bound for the Geometric Traveling Salesman Problem in Terms of Discrepancy,
Operations Research Letters 38 (2010) no. 4, 318-319, link.
- A note on the number of different inner products generated by a finite set of vectors,
Discrete Mathematics 310 (2010) no. 5, 1112-1117, link.
- Uniform distribution preserving mappings and variational problems Uniform Distribution Theory 4 (2009) no. 1, 117-145, link.
- (with Friedrich Pillichshammer) Average distance between consecutive points of uniformly distributed sequences,
Uniform Distribution Theory 4 (2009) no. 1, 51-67, link.
Optimal Transport and Point Distributions on Manifolds (2020)
Optimal Transport and Point Distributions on the Torus (2020)
Solving Linear Systems of Equations via Random Kaczmarz/Stochastic Gradient Descent (2020)
Interpolation Inequalities for the Wasserstein Distance and Applications (2020)
Nodal Sets of Eigenfunctions: Progress via Optimal Transport (2020)
Designs on Graphs (2020)
Dimension Reduction via t-SNE: Theory and Challenges (2020)
Dynamics of Roots of Polynomials (2019)
Nonlinear Phase Unwinding of Functions (2019)
Tax Mechanisms and Gradient Flows (2019)
Parabolic Techniques for Elliptic Problems (2017)
New Interactions between Analysis and Number Theory (2017)
Toronto (Jan. 13-14)
U Minnesota (Jan. 21-22)
Columbia (Feb. 18)
Localization of Waves Annual Meeting, Simons Foundation (Feb. 20-21)
Spectral Geometry in the Clouds (June 22)
Corona Seminar: Inequalities on Function Spaces of Smooth Functions (July 9)
Applied Math Seminar, UC Berkeley (Aug 26)
2nd Mid-Atlantic Analysis Seminar (Sep 4)
Number Theory Seminar, Rutgers (Sep 15)
Analysis Seminar, Northwestern (Sep 21)
Point Distribution Webinar (Sep 30 and Oct 1)
Joint UCLA/Caltech Analysis Seminar (Oct 6)
Colloquium, Syracuse (Oct 16)
Math Seminar, IIT Delhi (Oct 27)
Seminar, Fernuni Hagen (Nov 10)
Rainwater Seminar, UW (Nov 10)
Analysis Seminar, Erlangen (Jan 13)
Uniform Distribution Theory, JKU Linz (Feb 22-26)
Weil-Chern Symposium, U Chicago