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.Fri, 19 Apr 2019 17:56:15 GMT2019-04-19T17:56:15ZOn radial basis functions
http://publications.mfo.de/handle/mfo/1410
On radial basis functions
Buhmann, Martin; Jäger, Janin
Many sciences and other areas of research and applications
from engineering to economics require the approximation
of functions that depend on many variables.
This can be for a variety of reasons. Sometimes
we have a discrete set of data points and we
want to find an approximating function that completes
this data; another possibility is that precise
functions are either not known or it would take too
long to compute them explicitly. In this snapshot
we want to introduce a particular method of approximation
which uses functions called radial basis functions.
This method is particularly useful when approximating
functions that depend on very many variables.
We describe the basic approach to approximation
with radial basis functions, including their computation,
give several examples of such functions and
show some applications.
Wed, 13 Mar 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14102019-03-13T00:00:00ZBuhmann, MartinJäger, JaninMany sciences and other areas of research and applications
from engineering to economics require the approximation
of functions that depend on many variables.
This can be for a variety of reasons. Sometimes
we have a discrete set of data points and we
want to find an approximating function that completes
this data; another possibility is that precise
functions are either not known or it would take too
long to compute them explicitly. In this snapshot
we want to introduce a particular method of approximation
which uses functions called radial basis functions.
This method is particularly useful when approximating
functions that depend on very many variables.
We describe the basic approach to approximation
with radial basis functions, including their computation,
give several examples of such functions and
show some applications.Snake graphs, perfect matchings and continued fractions
http://publications.mfo.de/handle/mfo/1405
Snake graphs, perfect matchings and continued fractions
Schiffler, Ralf
A continued fraction is a way of representing a real
number by a sequence of integers. We present a new
way to think about these continued fractions using
snake graphs, which are sequences of squares in the
plane. You start with one square, add another to
the right or to the top, then another to the right or
the top of the previous one, and so on. Each continued
fraction corresponds to a snake graph and vice
versa, via “perfect matchings” of the snake graph. We
explain what this means and why a mathematician
would call this a combinatorial realization of continued
fractions.
Wed, 13 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14052019-02-13T00:00:00ZSchiffler, RalfA continued fraction is a way of representing a real
number by a sequence of integers. We present a new
way to think about these continued fractions using
snake graphs, which are sequences of squares in the
plane. You start with one square, add another to
the right or to the top, then another to the right or
the top of the previous one, and so on. Each continued
fraction corresponds to a snake graph and vice
versa, via “perfect matchings” of the snake graph. We
explain what this means and why a mathematician
would call this a combinatorial realization of continued
fractions.Mixed volumes and mixed integrals
http://publications.mfo.de/handle/mfo/1400
Mixed volumes and mixed integrals
Rotem, Liran
In recent years, mathematicians have developed new
approaches to study convex sets: instead of considering
convex sets themselves, they explore certain functions
or measures that are related to them. Problems
from convex geometry become thereby accessible to
analytic and probabilistic tools, and we can use these
tools to make progress on very difficult open problems.
We discuss in this Snapshot such a functional extension
of some “volumes” which measure how “big”
a set is. We recall the construction of “intrinsic volumes”,
discuss the fundamental inequalities between
them, and explain the functional extensions of these
results.
Sat, 29 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14002018-12-29T00:00:00ZRotem, LiranIn recent years, mathematicians have developed new
approaches to study convex sets: instead of considering
convex sets themselves, they explore certain functions
or measures that are related to them. Problems
from convex geometry become thereby accessible to
analytic and probabilistic tools, and we can use these
tools to make progress on very difficult open problems.
We discuss in this Snapshot such a functional extension
of some “volumes” which measure how “big”
a set is. We recall the construction of “intrinsic volumes”,
discuss the fundamental inequalities between
them, and explain the functional extensions of these
results.Estimating the volume of a convex body
http://publications.mfo.de/handle/mfo/1396
Estimating the volume of a convex body
Baldin, Nicolai
Sometimes the volume of a convex body needs to
be estimated, if we cannot calculate it analytically.
We explain how statistics can be used not only to
approximate the volume of the convex body, but also
its shape.
Sun, 30 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13962018-12-30T00:00:00ZBaldin, NicolaiSometimes the volume of a convex body needs to
be estimated, if we cannot calculate it analytically.
We explain how statistics can be used not only to
approximate the volume of the convex body, but also
its shape.Topological Complexity, Robotics and Social Choice
http://publications.mfo.de/handle/mfo/1384
Topological Complexity, Robotics and Social Choice
Carrasquel, José; Lupton, Gregory; Oprea, John
Topological complexity is a number that measures
how hard it is to plan motions (for robots, say) in
terms of a particular space associated to the kind of
motion to be planned. This is a burgeoning subject
within the wider area of Applied Algebraic Topology.
Surprisingly, the same mathematics gives insight into
the question of creating social choice functions, which
may be viewed as algorithms for making decisions by
artificial intelligences.
Fri, 10 Aug 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13842018-08-10T00:00:00ZCarrasquel, JoséLupton, GregoryOprea, JohnTopological complexity is a number that measures
how hard it is to plan motions (for robots, say) in
terms of a particular space associated to the kind of
motion to be planned. This is a burgeoning subject
within the wider area of Applied Algebraic Topology.
Surprisingly, the same mathematics gives insight into
the question of creating social choice functions, which
may be viewed as algorithms for making decisions by
artificial intelligences.A short story on optimal transport and its many applications
http://publications.mfo.de/handle/mfo/1381
A short story on optimal transport and its many applications
Santambrogio, Filippo
We present some examples of optimal transport problems
and of applications to different sciences (logistics,
economics, image processing, and a little bit of
evolution equations) through the crazy story of an
industrial dynasty regularly asking advice from an
exotic mathematician.
Wed, 08 Aug 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13812018-08-08T00:00:00ZSantambrogio, FilippoWe present some examples of optimal transport problems
and of applications to different sciences (logistics,
economics, image processing, and a little bit of
evolution equations) through the crazy story of an
industrial dynasty regularly asking advice from an
exotic mathematician.Number theory in quantum computing
http://publications.mfo.de/handle/mfo/1380
Number theory in quantum computing
Schönnenbeck, Sebastian
Algorithms are mathematical procedures developed
to solve a problem. When encoded on a computer,
algorithms must be "translated" to a series of simple
steps, each of which the computer knows how
to do. This task is relatively easy to do on a classical
computer and we witness the benefits of this
success in our everyday life. Quantum mechanics,
the physical theory of the very small, promises to enable
completely novel architectures of our machines,
which will provide specific tasks with higher computing
power. Translating and implementing algorithms
on quantum computers is hard. However, we will
show that solutions to this problem can be found and
yield surprising applications to number theory.
Tue, 07 Aug 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13802018-08-07T00:00:00ZSchönnenbeck, SebastianAlgorithms are mathematical procedures developed
to solve a problem. When encoded on a computer,
algorithms must be "translated" to a series of simple
steps, each of which the computer knows how
to do. This task is relatively easy to do on a classical
computer and we witness the benefits of this
success in our everyday life. Quantum mechanics,
the physical theory of the very small, promises to enable
completely novel architectures of our machines,
which will provide specific tasks with higher computing
power. Translating and implementing algorithms
on quantum computers is hard. However, we will
show that solutions to this problem can be found and
yield surprising applications to number theory.Tropical geometry
http://publications.mfo.de/handle/mfo/1378
Tropical geometry
Brugallé, Erwan; Itenberg, Ilia; Shaw, Kristin; Viro, Oleg
What kind of strange spaces hide behind the enigmatic
name of tropical geometry? In the tropics, just
as in other geometries, one of the simplest objects is
a line. Therefore, we begin our exploration by considering
tropical lines. Afterwards, we take a look at
tropical arithmetic and algebra, and describe how to
define tropical curves using tropical polynomials.
Thu, 19 Jul 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13782018-07-19T00:00:00ZBrugallé, ErwanItenberg, IliaShaw, KristinViro, OlegWhat kind of strange spaces hide behind the enigmatic
name of tropical geometry? In the tropics, just
as in other geometries, one of the simplest objects is
a line. Therefore, we begin our exploration by considering
tropical lines. Afterwards, we take a look at
tropical arithmetic and algebra, and describe how to
define tropical curves using tropical polynomials.Data assimilation: mathematics for merging models and data
http://publications.mfo.de/handle/mfo/1375
Data assimilation: mathematics for merging models and data
Morzfeld, Matthias; Reich, Sebastian
When you describe a physical process, for example,
the weather on Earth, or an engineered system, such
as a self-driving car, you typically have two sources of
information. The first is a mathematical model, and
the second is information obtained by collecting data.
To make the best predictions for the weather, or most
effectively operate the self-driving car, you want to
use both sources of information. Data assimilation
describes the mathematical, numerical and computational
framework for doing just that.
Tue, 10 Jul 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13752018-07-10T00:00:00ZMorzfeld, MatthiasReich, SebastianWhen you describe a physical process, for example,
the weather on Earth, or an engineered system, such
as a self-driving car, you typically have two sources of
information. The first is a mathematical model, and
the second is information obtained by collecting data.
To make the best predictions for the weather, or most
effectively operate the self-driving car, you want to
use both sources of information. Data assimilation
describes the mathematical, numerical and computational
framework for doing just that.Fast Solvers for Highly Oscillatory Problems
http://publications.mfo.de/handle/mfo/1370
Fast Solvers for Highly Oscillatory Problems
Barnett, Alex
Waves of diverse types surround us. Sound, light
and other waves, such as microwaves, are crucial for
speech, mobile phones, and other communication technologies.
Elastic waves propagating through the Earth
bounce through the Earth’s crust and enable us to
“see” thousands of kilometres in depth. These propagating
waves are highly oscillatory in time and space,
and may scatter off obstacles or get “trapped” in
cavities. Simulating these phenomena on computers
is extremely important. However, the achievable
speeds for accurate numerical modelling are low even
on large modern computers. Our snapshot will introduce
the reader to recent progress in designing
algorithms that allow for much more rapid solutions.
Tue, 26 Jun 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13702018-06-26T00:00:00ZBarnett, AlexWaves of diverse types surround us. Sound, light
and other waves, such as microwaves, are crucial for
speech, mobile phones, and other communication technologies.
Elastic waves propagating through the Earth
bounce through the Earth’s crust and enable us to
“see” thousands of kilometres in depth. These propagating
waves are highly oscillatory in time and space,
and may scatter off obstacles or get “trapped” in
cavities. Simulating these phenomena on computers
is extremely important. However, the achievable
speeds for accurate numerical modelling are low even
on large modern computers. Our snapshot will introduce
the reader to recent progress in designing
algorithms that allow for much more rapid solutions.