2 - Snapshots of modern mathematics from Oberwolfach
http://publications.mfo.de/handle/mfo/20
The snapshot project is designed to promote the understanding and appreciation of modern mathematics and mathematical research in the general public world-wide. It is part of the project "Oberwolfach meets IMAGINARY“, supported by the Klaus Tschira Foundation.Sun, 18 Mar 2018 17:06:48 GMT2018-03-18T17:06:48Z2 - Snapshots of modern mathematics from Oberwolfachhttp://publications.mfo.de:8080/bitstream/id/23bdbb71-2942-4102-9945-4cc8f6d60131/
http://publications.mfo.de/handle/mfo/20
Computing the long term evolution of the solar system with geometric numerical integrators
http://publications.mfo.de/handle/mfo/1355
Computing the long term evolution of the solar system with geometric numerical integrators
Fiorelli Vilmart, Shaula; Vilmart, Gilles
Simulating the dynamics of the Sun–Earth–Moon system
with a standard algorithm yields a dramatically
wrong solution, predicting that the Moon is ejected
from its orbit. In contrast, a well chosen algorithm
with the same initial data yields the correct behavior.
We explain the main ideas of how the evolution of
the solar system can be computed over long times
by taking advantage of so-called geometric numerical
methods. Short sample codes are provided for the
Sun–Earth–Moon system.
Wed, 27 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13552017-12-27T00:00:00ZFiorelli Vilmart, ShaulaVilmart, GillesSimulating the dynamics of the Sun–Earth–Moon system
with a standard algorithm yields a dramatically
wrong solution, predicting that the Moon is ejected
from its orbit. In contrast, a well chosen algorithm
with the same initial data yields the correct behavior.
We explain the main ideas of how the evolution of
the solar system can be computed over long times
by taking advantage of so-called geometric numerical
methods. Short sample codes are provided for the
Sun–Earth–Moon system.Computing with symmetries
http://publications.mfo.de/handle/mfo/1354
Computing with symmetries
Roney-Dougal, Colva M.
Group theory is the study of symmetry, and has many
applications both within and outside mathematics.
In this snapshot, we give a brief introduction to symmetries,
and how to compute with them.
Tue, 06 Mar 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13542018-03-06T00:00:00ZRoney-Dougal, Colva M.Group theory is the study of symmetry, and has many
applications both within and outside mathematics.
In this snapshot, we give a brief introduction to symmetries,
and how to compute with them.Topological recursion
http://publications.mfo.de/handle/mfo/1353
Topological recursion
Sułkowski, Piotr
In this snapshot we present the concept of topological
recursion – a new, surprisingly powerful formalism
at the border of mathematics and physics, which has
been actively developed within the last decade. After
introducing necessary ingredients – expectation values,
random matrices, quantum theories, recursion
relations, and topology – we explain how they get
combined together in one unifying picture.
Mon, 05 Mar 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13532018-03-05T00:00:00ZSułkowski, PiotrIn this snapshot we present the concept of topological
recursion – a new, surprisingly powerful formalism
at the border of mathematics and physics, which has
been actively developed within the last decade. After
introducing necessary ingredients – expectation values,
random matrices, quantum theories, recursion
relations, and topology – we explain how they get
combined together in one unifying picture.Spaces of Riemannian metrics
http://publications.mfo.de/handle/mfo/1352
Spaces of Riemannian metrics
Bustamante, Mauricio; Kordaß, Jan-Bernhard
Riemannian metrics endow smooth manifolds such as
surfaces with intrinsic geometric properties, for example
with curvature. They also allow us to measure
quantities like distances, angles and volumes. These
are the notions we use to characterize the “shape” of
a manifold. The space of Riemannian metrics is a
mathematical object that encodes the many possible
ways in which we can geometrically deform the shape
of a manifold.
Thu, 28 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13522017-12-28T00:00:00ZBustamante, MauricioKordaß, Jan-BernhardRiemannian metrics endow smooth manifolds such as
surfaces with intrinsic geometric properties, for example
with curvature. They also allow us to measure
quantities like distances, angles and volumes. These
are the notions we use to characterize the “shape” of
a manifold. The space of Riemannian metrics is a
mathematical object that encodes the many possible
ways in which we can geometrically deform the shape
of a manifold.Mathematics plays a key role in scientific computing
http://publications.mfo.de/handle/mfo/1351
Mathematics plays a key role in scientific computing
Shu, Chi-Wang
I attended a very interesting workshop at the research
center MFO in Oberwolfach on “Recent Developments
in the Numerics of Nonlinear Hyperbolic Conservation
Laws”. The title sounds a bit technical,
but in plain language we could say: The theme is
to survey recent research concerning how mathematics
is used to study numerical algorithms involving
a special class of equations. These equations arise
from computer simulations to solve application problems
including those in aerospace engineering, automobile
design, and electromagnetic waves in communications
as examples. This topic belongs to the general
research area called “scientific computing”.
Fri, 29 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13512017-12-29T00:00:00ZShu, Chi-WangI attended a very interesting workshop at the research
center MFO in Oberwolfach on “Recent Developments
in the Numerics of Nonlinear Hyperbolic Conservation
Laws”. The title sounds a bit technical,
but in plain language we could say: The theme is
to survey recent research concerning how mathematics
is used to study numerical algorithms involving
a special class of equations. These equations arise
from computer simulations to solve application problems
including those in aerospace engineering, automobile
design, and electromagnetic waves in communications
as examples. This topic belongs to the general
research area called “scientific computing”.Prony’s method: an old trick for new problems
http://publications.mfo.de/handle/mfo/1338
Prony’s method: an old trick for new problems
Sauer, Tomas
In 1795, French mathematician Gaspard de Prony invented
an ingenious trick to solve a recovery problem,
aiming at reconstructing functions from their values
at given points, which arose from a specific application
in physical chemistry. His technique became
later useful in many different areas, such as signal
processing, and it relates to the concept of sparsity
that gained a lot of well-deserved attention recently.
Prony’s contribution, therefore, has developed into a
very modern mathematical concept.
Tue, 06 Mar 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13382018-03-06T00:00:00ZSauer, TomasIn 1795, French mathematician Gaspard de Prony invented
an ingenious trick to solve a recovery problem,
aiming at reconstructing functions from their values
at given points, which arose from a specific application
in physical chemistry. His technique became
later useful in many different areas, such as signal
processing, and it relates to the concept of sparsity
that gained a lot of well-deserved attention recently.
Prony’s contribution, therefore, has developed into a
very modern mathematical concept.The Algebraic Statistics of an Oberwolfach Workshop
http://publications.mfo.de/handle/mfo/1337
The Algebraic Statistics of an Oberwolfach Workshop
Seigal, Anna
Algebraic Statistics builds on the idea that statistical
models can be understood via polynomials. Many
statistical models are parameterized by polynomials
in the model parameters; others are described implicitly
by polynomial equalities and inequalities. We explore
the connection between algebra and statistics
for some small statistical models.
Tue, 27 Feb 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13372018-02-27T00:00:00ZSeigal, AnnaAlgebraic Statistics builds on the idea that statistical
models can be understood via polynomials. Many
statistical models are parameterized by polynomials
in the model parameters; others are described implicitly
by polynomial equalities and inequalities. We explore
the connection between algebra and statistics
for some small statistical models.Solving quadratic equations in many variables
http://publications.mfo.de/handle/mfo/1335
Solving quadratic equations in many variables
Tignol, Jean-Pierre
Fields are number systems in which every linear equation
has a solution, such as the set of all rational
numbers $\mathbb{Q}$ or the set of all real numbers $\mathbb{R}$. All fields
have the same properties in relation with systems of
linear equations, but quadratic equations behave differently
from field to field. Is there a field in which
every quadratic equation in five variables has a solution,
but some quadratic equation in four variables
has no solution? The answer is in this snapshot.
Sat, 30 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13352017-12-30T00:00:00ZTignol, Jean-PierreFields are number systems in which every linear equation
has a solution, such as the set of all rational
numbers $\mathbb{Q}$ or the set of all real numbers $\mathbb{R}$. All fields
have the same properties in relation with systems of
linear equations, but quadratic equations behave differently
from field to field. Is there a field in which
every quadratic equation in five variables has a solution,
but some quadratic equation in four variables
has no solution? The answer is in this snapshot.Computational Optimal Transport
http://publications.mfo.de/handle/mfo/1332
Computational Optimal Transport
Solomon, Justin
Optimal transport is the mathematical discipline of
matching supply to demand while minimizing shipping
costs. This matching problem becomes extremely
challenging as the quantity of supply and demand
points increases; modern applications must cope with
thousands or millions of these at a time. Here, we
introduce the computational optimal transport problem
and summarize recent ideas for achieving new
heights in efficiency and scalability.
Thu, 21 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13322017-12-21T00:00:00ZSolomon, JustinOptimal transport is the mathematical discipline of
matching supply to demand while minimizing shipping
costs. This matching problem becomes extremely
challenging as the quantity of supply and demand
points increases; modern applications must cope with
thousands or millions of these at a time. Here, we
introduce the computational optimal transport problem
and summarize recent ideas for achieving new
heights in efficiency and scalability.A few shades of interpolation
http://publications.mfo.de/handle/mfo/1329
A few shades of interpolation
Szpond, Justyna
The topic of this snapshot is interpolation. In the
ordinary sense, interpolation means to insert something
of a different nature into something else. In
mathematics, interpolation means constructing new
data points from given data points. The new points
usually lie in between the already-known points. The
purpose of this snapshot is to introduce a particular
type of interpolation, namely, polynomial interpolation.
This will be explained starting from basic ideas
that go back to the ancient Babylonians and Greeks,
and will arrive at subjects of current research activity.
Thu, 07 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13292017-12-07T00:00:00ZSzpond, JustynaThe topic of this snapshot is interpolation. In the
ordinary sense, interpolation means to insert something
of a different nature into something else. In
mathematics, interpolation means constructing new
data points from given data points. The new points
usually lie in between the already-known points. The
purpose of this snapshot is to introduce a particular
type of interpolation, namely, polynomial interpolation.
This will be explained starting from basic ideas
that go back to the ancient Babylonians and Greeks,
and will arrive at subjects of current research activity.