Using elementary methods from differential equations and analysis we will consider the existence and multiplicity of solutions to semilinear partial differential equations with boundary conditions.
In this talk we discuss the Hilbert space approach, or the variational approach, in the study of questions of existence and multiplicity for some two-point boundary-value problems for nonlinear, second order, ordinary differential equations (ODEs). We illustrate the use of the Hilbert space approach in obtaining some old existence results for periodic solutions of a […]
We comment on the main steps to take when studying some variational problems. This includes optimization problems arising in geometry, machine learning, non linear elasticity, fluid mechanics, etc... For the sake of illustration, in this talk, we keep our focus on a minimization problem obtained after a time-discretization of the incompressible Navier-Stokes equations. Elementary geometric […]
Given a unital AF (approximately finite-dimensional) algebra A equipped with a faithful tracial state, we equip each (norm-closed two-sided) ideal of A with a metrized quantum vector bundle structure, when canonically viewed as a module over A, in the sense of Latrémolière using previous work of Aguilar and Latrémolière. Moreover, we show that convergence of […]
Graph products of groups were introduced in E. Green’s thesis in the 90’s as generalizations of Right-Angled Artin Groups. These have become objects of intense study due to their key roles in topology and group theory. Recently, Caspers and Fima introduced graph products of von Neumann algebras. Since their inception, several structural aspects such as […]
We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) to include both linear multifractional Brownian sheets and linear multifractional stable sheets. The purpose of the framework is to study the existence and joint continuity of the local times of LMSS, and also the local Holder condition of the local times […]
Let x_1,...,x_n be an overdetermined spanning set for the Euclidean space R^k, where n > k. Let L be the integer span of these vectors. Then L is an additive subgroup of R^n. When is it discrete in R^n? Naturally, this depends on the choice of the spanning set, but in which way? We will […]
We introduce a family of norms on the space of self-adjoint trace class symmetric tensor power of linear operators acting on an infinite-dimensional Hilbert space. Our technique is to extend to infinite dimension an original and nice idea of a very recent result by K. Aguilar, Á. Chávez, S. R. Garcia and J. Volčič, in […]
Quantum metrics in the sense of Rieffel were introduced to prove some statements arising in the high-energy physics literature. Since then, the area of quantum metric geometry has been used to answer questions stemming from within mathematics as well. To prove such results, it is often the case that certain properties of a quantum metric […]
Let \mathcal{L}(X,Y) denote the normed vector space of all continuous operators from \(X\) to \(Y\), \(X^*\) be the dual space of \(X\), and \(\mathcal{K}(X,Y)\) denote the collection of all compact operators from \(X\) to \(Y\). Denote by \(T^{*} \in \mathcal{L}(Y^{*}, X^{*} )\) the adjoint operator of \(T\in \mathcal{L} (X, Y)\). The well known theorem of […]
Noncommutative metric geometry is the study of certain noncommuative algebras in the context of metric geometry. For instance, the Lipschitz constant (which measures the maximum slope obtained by a real-valued continuous function on a metric space (allowed to be infinite)) is a vital tool in metric geometry, and a main feature of noncommutative metric geometry […]
The theory of compact linear operators between Banach spaces has a classical core and is familiar to many. Perhaps lesser known is the factorization of compact maps through a closed subspace of c_0 . This factorization theorem has a number of important connections and consequences analogous to how the ideals of continuous linear operators factoring […]
