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In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new…
In this paper we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of…
Learning-based 3D reconstruction methods have shown impressive results. However, most methods require 3D supervision which is often hard to obtain for real-world datasets. Recently, several works have proposed differentiable rendering…
We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of $d$-dimensional manifolds, structured hierarchically so that each $d$-dimensional manifold is contained in the boundary of…
Differentiable rendering has received increasing interest for image-based inverse problems. It can benefit traditional optimization-based solutions to inverse problems, but also allows for self-supervision of learning-based approaches for…
We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of…
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical…
Implicit neural representations have emerged as a powerful tool in learning 3D geometry, offering unparalleled advantages over conventional representations like mesh-based methods. A common type of INR implicitly encodes a shape's boundary…
Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself…
We introduce a constructive method that provides the local solution of general implicit systems in arbitrary dimension via Hamiltonian type equations. A variant of this approach constructs parametrizations of the manifold, extending the…
Even for the gradient descent (GD) method applied to neural network training, understanding its optimization dynamics, including convergence rate, iterate trajectories, function value oscillations, and especially its implicit acceleration,…
We introduce the Visual Implicit Geometry Transformer (ViGT), an autonomous driving geometric model that estimates continuous 3D occupancy fields from surround-view camera rigs. ViGT represents a step towards foundational geometric models…
We integrate in closed implicit form the Navier-Stokes equations for an incompressible fluid and the kinematical dynamo equation, in smooth manifolds and Euclidean space. This integration is carried out by applying Stochastic Differential…
Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the {\it extrinsic} curvature (instead of the intrinsic curvature). Such an…
The paper introduces a new differential-geometric system which originates from the theory of $m$-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise…
Realizations of four dimensional Lie algebras as vector fields in the plane are explicitly constructed. Fourth order ordinary differential equations which admit such Lie symmetry algebras are derived. The route to their integration is…
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm…
In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic…
Tools of the intrinsic analysis on manifolds, helpful in solving the invariant inverse problem of the calculus of variations are being presented comprising a combined approach which consists in the simultaneous imposition of symmetry…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…