Related papers: On diffeologies from infinite dimensional geometry…

Bilevel optimization is defined as a mathematical program, where an optimization problem contains another optimization problem as a constraint. These problems have received significant attention from the mathematical programming community.…

Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize…

In this thesis we develop a functional analytic framework for shape optimization with elliptic partial differential equation (PDE) constraints in classical function spaces (H\"older spaces). This approach is motivated by shape optimization…

This research concerns design optimization problems involving numerous design parameters and large computational models. These problems generally consist in non-convex constrained optimization problems in large and sometimes complex search…

We consider a differential geometric setting on power sets and Borel algebras. Our chosen framework is based on diffeologies, and we make a link between the various diffeological structures that we propose, having in mind set-valued maps,…

In this paper, we use some basic quasi-topos theory to study two functors: one adding infinitesimals of Fermat reals to diffeological spaces (which generalize smooth manifolds including singular spaces and infinite dimensional spaces), and…

We study tangent spaces in the setting of diffeological spaces. Several distinct tangent functors have been introduced, each of which extends the classical tangent functor from smooth manifolds. In this paper, we construct infinitely many…

This paper provides necessary and sufficient optimality conditions for abstract constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of…

It has been noted that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different…

In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting.…

We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces,…

We study nonuniform Sobolev spaces, i.e., spaces of functions whose partial derivatives lie in possibly different Lebesgue spaces. Although standard proofs do not apply, we show that nonuniform Sobolev spaces share similar properties as the…

Ultrafunctions are a particular class of functions defined on a Non Archimedean field E. They have been introduced and studied in some previous works. In this paper we develop the notion of fine ultrafunctions which improves the older…

Derived geometry provides powerful tools to handle non-transverse intersections and singular moduli problems arising in geometry and theoretical physics. While derived algebraic geometry has been extensively developed, classical field…

We adapt concepts, methodology, and theory originally developed in the areas of multidimensional scaling and dimensionality reduction for multivariate data to the functional setting. We focus on classical scaling and Isomap -- prototypical…

The Dirichlet problem for a class of stochastic partial differential equations is studied in Sobolev spaces. The existence and uniqueness result is proved under certain compatibility conditions that ensure the finiteness of…

Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…

This paper is devoted to the study of some connections between coadjoint orbits in infinite dimensional Lie algebras, isospectral deformations and linearization of dynamical systems. We explain how results from deformation theory,…

In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural…

In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be…