13 citations · 20 across the 15 of their papers we have counts for
22 papers
A sharp covering theorem and Solyanik estimates for Euclidean balls
Mayukh Mukherjee
For every finite family of Euclidean balls in , , and every , we select a subfamily whose -dilations cover the original union and whose undilat…
Spectral Dehn functions and a characterisation of word-hyperbolicity
Mayukh Mukherjee
We introduce a \emph{spectral Dehn function} \[ Λ_{\mathcal{P}}(n):=\inf λ_1(Δ), \] where is the first Dirichlet eigenvalue of the random-walk Laplacian on a van Kampen di…
An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature
Mayukh Mukherjee
A classical theorem of Colin de Verdière shows that on a closed manifold of fixed topology one can prescribe an arbitrary finite portion of the Laplace-Beltrami spectrum (including…
Nodal Domains on Surfaces under Perturbation: Upper Semicontinuity, Courant-Sharpness, and Boundary Intersections
Saikat Maji, Mayukh Mukherjee, Soumyajit Saha
We study how the number of nodal domains of eigenfunctions of Schrödinger operators on closed surfaces changes under smooth perturbations of along conver…
On the Spectrum of Certain Hadamard Manifolds
Werner Ballmann, Mayukh Mukherjee, Panagiotis Polymerakis
We show the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds and for specific domains and quotients of one of the class…
On the Impact of Stable Ranks in Deep Nets
Bogdan Georgiev, Lukas Franken, Mayukh Mukherjee +1
A recent line of work has established intriguing connections between the generalization/compression properties of a deep neural network (DNN) model and the so-called layer weights'…
Some applications of heat flow to Laplace eigenfunctions
Bogdan Georgiev, Mayukh Mukherjee
We consider mass concentration properties of Laplace eigenfunctions , that is, smooth functions satisfying the equation , on a smooth closed Riemannian manifold.…
Heat profile, level sets and hot spots of Laplace eigenfunctions
Mayukh Mukherjee, Soumyajit Saha
We use probabilistic tools based on Brownian motion and Feynman-Kac formulae to investigate the heat profile for the ground state Dirichlet and second Neumann eigenfunctions. Among…
On the effects of small perturbation on low energy Laplace eigenfunctions
Mayukh Mukherjee, Soumyajit Saha
We investigate several aspects of the nodal geometry and topology of Laplace eigenfunctions, with particular emphasis on the low frequency regime. This includes investigations in a…
Mass non-concentration at the nodal set and a sharp Wasserstein uncertainty principle
Mayukh Mukherjee
We prove -mass concentration properties of Laplace eigenfunctions away from their nodal sets, extending a recent result in \cite{GM3} to all dimensions, and giving a slight re…
Heating up decision boundaries: isocapacitory saturation, adversarial scenarios and generalization bounds
Bogdan Georgiev, Lukas Franken, Mayukh Mukherjee
In the present work we study classifiers' decision boundaries via Brownian motion processes in ambient data space and associated probabilistic techniques. Intuitively, our ideas co…
Nodal sets of Laplace eigenfunctions under small perturbations
Mayukh Mukherjee, Soumyajit Saha
We study the stability properties of nodal sets of Laplace eigenfunctions on compact manifolds under specific small perturbations. We prove that nodal sets are fairly stable if sai…
Polyhedral billiards, eigenfunction concentration and almost periodic control
Mihajlo Cekić, Bogdan Georgiev, Mayukh Mukherjee
We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only…
On maximizing the fundamental frequency of the complement of an obstacle
Bogdan Georgiev, Mayukh Mukherjee
Let be a bounded domain satisfying a Hayman-type asymmetry condition, and let be an arbitrary bounded domain referred to as "obstacle". We are interes…
A Spectral Gap Estimate and Applications
Bogdan Georgiev, Mayukh Mukherjee, Stefan Steinerberger
We consider the Schrödinger operator $$-\frac{d^2}{d x^2} + V \qquad \mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions},$$ where is bounded from below and p…
Some remarks on nodal geometry in the smooth setting
Bogdan Georgiev, Mayukh Mukherjee
We consider a Laplace eigenfunction on a smooth closed Riemannian manifold, that is, satisfying . We introduce several observations about the geometry of its van…
Nodal Geometry, Heat Diffusion and Brownian Motion
Bogdan Georgiev, Mayukh Mukherjee
We use tools from -dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold . On…
Variation of Laplace spectra of compact "nearly" hyperbolic surfaces
Mayukh Mukherjee
We use the time real analyticity of Ricci flow proved by Kotschwar to extend a result in ~\cite{B}, namely, we prove that the Laplace spectra of negatively curved compact surfaces…
Discrete -filters and rings of analytic functions
Bedanta Bose, Mayukh Mukherjee
Consider rings of single variable real analytic or complex entire functions, denoted by . We study "discrete -filters" on and their conn…
Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary
Mayukh Mukherjee
Consider a second order, strongly elliptic negative semidefinite differential operator (maybe a system) on a compact Riemannian manifold with smooth boundary, wh…
Extremal values of the (fractional) Weinstein functional on the hyperbolic space
Mayukh Mukherjee
We make a study of Weinstein functionals, first defined in ~\cite{W}, on the hyperbolic space . We are primarily interested in the existence of Weinstein functional m…
Nonlinear travelling waves on non-Euclidean spaces
Mayukh Mukherjee
We study travelling wave solutions, that is, solutions of the form , to nonlinear Schrödinger and Klein-Gordon equations on Riemannian manifolds, both co…