C∗-algebras: structure and classification
Submitted by IMAGINARY on
Analysis
mathemathical subject image:
Quantum symmetry
Submitted by IMAGINARY on
Determinacy versus indeterminacy
Submitted by IMAGINARY on
Can a continuous function on an interval be uniquely determined if we know all the integrals of the function against the natural powers of the variable? Following Weierstrass and Stieltjes, we show that the answer is yes if the interval is finite, and no if the interval is infinite.
Rotating needles, vibrating strings, and Fourier summation
Submitted by IMAGINARY on
We give a brief survey of the connection between seemingly unrelated problems such as sets in the plane containing lines pointing in many directions, vibrating strings and drum heads, and a classical problem from Fourier analysis.
Quantum symmetry
Submitted by IMAGINARY on
In mathematics, symmetry is usually captured using the formalism of groups. However, the developments of the past few decades revealed the need to go beyond groups: to “quantum groups”. We explain the passage from spaces to quantum spaces, from groups to quantum groups, and from symmetry to quantum symmetry, following an analytical appr
The Mathematics of Fluids and Solids
Submitted by IMAGINARY on
Fluid-structure interaction is a rich and active field of mathematics that studies the interaction between fluids and solid objects. In this short article, we give a glimpse into this exciting field, as well as a sample of the most significant questions that mathematicians try to answer.
A surprising connection between quantum mechanics and shallow water waves
Submitted by IMAGINARY on
We describe a connection between quantum mechanics and nonlinear wave equations and highlight a few problems at the forefront of modern research in the intersection of these areas.
Analogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964)
Submitted by IMAGINARY on
Throughout the history of dynamical systems, instruments have been used to calculate and visualize (approximate) solutions of differential equations. Here we describe the approach of a group of physicists and engineers in the period 1948–1964, and we give ex- amples of the specific (analogue) mathematical in- struments they conceived and used. These examples also illustrate how their analogue culture and practices faced the advent of the digital computer, which appeared at that time as a new instrument, full of promises.
Expander graphs and where to find them
Submitted by IMAGINARY on
Graphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical properties of graphs can provide an insight into real-life phenomena. One interesting property is how connected a graph is, in the sense of how easy it is to move between the vertices along the edges. The topic dealt with here is the construction of particularly well-connected graphs, and whether or not such graphs can happily exist in worlds similar to ours.
Deep Learning and Inverse Problems
Submitted by IMAGINARY on
Big data and deep learning are modern buzz words which presently infiltrate all fields of science and technology. These new concepts are impressive in terms of the stunning results they achieve for a large variety of applications. However, the theoretical justification for their success is still very limited. In this snapshot, we highlight some of the very recent mathematical results that are the beginnings of a solid theoretical foundation for the subject.