arXiv research

The paper explains how microlocal analysis solves geometric inverse problems.

Geometric programming approach for traffic equilibrium problems.

Open problems in billiards and optics from a workshop.

New geometric mechanism solves four envelope problems.

Study of control problems on Carnot groups with SO(3) symmetry using geometric algebra.

Unified approach to geometric structure equivalence problem.

New framework tackles geometric structure existence and classification.

In this note, we extend our previous work on the inverse $σ_k$ problem. Inverse $σ_{k}$ problem is a fully nonlinear geometric PDE on compact Kähler manifolds. Given a proper geometric condition, we prove that a large family of nonlinear geometric flows converges to the desired solution of the given PDE.

Enhances quantum circuit synthesis using deep learning and geometric methods.

Study on geometric variational problems for existence, regularity, and uniqueness of solutions.

This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition series of projective embeddings. The concept of "geometric composition series" is…

Equivariant networks improve geometric prediction without scalar approximations.

Solves optimal liquidation problem for stock price following geometric Brownian motion.

Researchers solved the multiple fibration problem for Seifert 3-orbifolds.

Notes on relative algebroids for geometric problems.

Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then the class of fundamental groups of closed, geometric 3-manifolds is algorithmically recognizable. In our terminology, the …

Uniform small energy regularity for fractional geometric problems proved.

Geometric Algebra Transformer (GATr) handles various geometric data types efficiently.

Solves geometric Cauchy problem for submanifolds with constant rank.

Explains Bernstein theorems for various geometric PDEs.

We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propos…

Solves geometric problems using fully nonlinear equations and Morse theory.

Geometric framework for inverse problems using foliations and dual connections.

Solves initial boundary value problem for vacuum Einstein equations and proves geometric uniqueness.

Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.

Develops Hodge theory for boundary-value problems on general geometric structures.

High-dimensional ConvNets detect patterns in 32+ dimensions for geometric registration.

Study geometric manifolds in arbitrary dimensions, focusing on maps and diffeomorphisms.

Unified geometric description of Kepler flow across all energies.

New theorems prove uniqueness of solutions to geometric PDEs.

We give a more geometric approach to an algorithm for deciding whether two hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.

In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.

Geometric Hydrodynamics tackles open problems in fluid dynamics.

A new method for group invariant machine learning using geometric projections.

How to understand deep learning systems remains an open problem. In this paper we propose that the answer may lie in the geometrization of deep networks. Geometrization is a bridge to connect physics, geometry, deep network and quantum computation and this may result in a new scheme to reveal the rule of the physical w…

Researchers tackle the globalization problem of locally cosymplectic Hamiltonian dynamics.

In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method (GeoPG), converges with a linear rate $(1-1/\sqrtκ)$ and thus achieves the optimal …

We construct the solution to the periodic Cauchy problem of the Schrödinger flow on the sphere. Such construction of solutions is formulated explicitly and therefore a geometric algorithm of solving this periodic Cauchy problem follows. Theoretical and experimental results will be discussed.

New proofs for curvature problems using a viscosity approach.

GNPs learn operators on non-Euclidean geometries using neural networks.

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell$ on a combinatorial surface of complexity $n$ we describe simple algo…

Unified treatment of stability problems in geometry and analysis.

Paper shows how to use geometric median for robust SGD in high dimensions.

We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the geometric intersection in the Johnson filtration. Also, we further consider the problem…

We study the tracking of a trajectory for a nonholonomic system by recasting the problem as a constrained optimal control problem. The cost function is chosen to minimize the error in positions and velocities between the trajectory of a nonholonomic system and the desired reference trajectory, both evolving on the dist…

Smooth even solutions found for a generalized convex geometry problem.

The problem of evaluating heat invariants can be computerized. Geometric symbol calculus of pseudodifferential operators is the main tool of such computerization.

Geometric vector perceptrons improve protein structure learning.