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Vertex-to-self trajectories on the platonic solids
[SNAP-2020-003-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-15)
We consider the problem of walking in a straight line
on the surface of a Platonic solid. While the tetrahedron,
octahedron, cube, and icosahedron all exhibit
the same behavior, we find a remarkable difference
with the ...
From Betti numbers to ℓ²-Betti numbers
[SNAP-2020-001-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-15)
We provide a leisurely introduction to ℓ²-Betti numbers,
which are topological invariants, by relating
them to their much older cousins, Betti numbers. In
the end we present an open research problem about
ℓ²-Betti numbers.
Determinacy versus indeterminacy
[SNAP-2020-004-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-22)
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 ...
Shape space – a paradigm for character animation in computer graphics
[SNAP-2020-007-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-10-07)
Nowadays 3D computer animation is increasingly realistic
as the models used for the characters become
more and more complex. These models are typically
represented by meshes of hundreds of thousands or
even millions ...
Rotating needles, vibrating strings, and Fourier summation
[SNAP-2020-006-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-09-21)
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
[SNAP-2020-005-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-06-04)
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 ...
Quantum symmetry
[SNAP-2020-009-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-12-31)
The symmetry of objects plays a crucial role in many
branches of mathematics and physics. It allowed, for
example, the early prediction of the existence of new
small particles. “Quantum symmetry” concerns a
generalized ...
Random matrix theory: Dyson Brownian motion
[SNAP-2020-002-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-15)
The theory of random matrices was introduced by
John Wishart (1898–1956) in 1928. The theory was
then developed within the field of nuclear physics
from 1955 by Eugene Paul Wigner (1902–1995) and
later by Freeman John ...
Higgs bundles without geometry
[SNAP-2020-008-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-09-29)
Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal stroll ...