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.Mon, 14 Oct 2019 22:54:55 GMT2019-10-14T22:54:55ZConfiguration spaces and braid groups
http://publications.mfo.de/handle/mfo/2519
Configuration spaces and braid groups
Jiménez Rolland, Rita; Xicoténcatl, Miguel A.
In this snapshot we introduce configuration spaces
and explain how a mathematician studies their ‘shape’.
This will lead us to consider paths of configurations
and braid groups, and to explore how algebraic properties
of these groups determine features of the spaces.
Tue, 08 Oct 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25192019-10-08T00:00:00ZJiménez Rolland, RitaXicoténcatl, Miguel A.In this snapshot we introduce configuration spaces
and explain how a mathematician studies their ‘shape’.
This will lead us to consider paths of configurations
and braid groups, and to explore how algebraic properties
of these groups determine features of the spaces.Limits of graph sequences
http://publications.mfo.de/handle/mfo/2516
Limits of graph sequences
Klimošová, Tereza
Graphs are simple mathematical structures used to
model a wide variety of real-life objects. With the
rise of computers, the size of the graphs used for
these models has grown enormously. The need to efficiently
represent and study properties of extremely
large graphs led to the development of the theory of
graph limits.
Wed, 04 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25162019-09-04T00:00:00ZKlimošová, TerezaGraphs are simple mathematical structures used to
model a wide variety of real-life objects. With the
rise of computers, the size of the graphs used for
these models has grown enormously. The need to efficiently
represent and study properties of extremely
large graphs led to the development of the theory of
graph limits.On Logic, Choices and Games
http://publications.mfo.de/handle/mfo/2515
On Logic, Choices and Games
Oliva, Paulo
Can we always mathematically formalise our taste
and preferences? We discuss how this has been done
historically in the field of game theory, and how recent
ideas from logic and computer science have brought
an interesting twist to this beautiful theory.
Wed, 04 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25152019-09-04T00:00:00ZOliva, PauloCan we always mathematically formalise our taste
and preferences? We discuss how this has been done
historically in the field of game theory, and how recent
ideas from logic and computer science have brought
an interesting twist to this beautiful theory.Nonlinear Acoustics
http://publications.mfo.de/handle/mfo/2514
Nonlinear Acoustics
Kaltenbacher, Barbara; Brunnhuber, Rainer
Nonlinear acoustics has been a topic of research for
more than 250 years. Driven by a wide range and a
large number of highly relevant industrial and medical
applications, this area has expanded enormously
in the last few decades. Here, we would like to give
a glimpse of the mathematical modeling techniques
that are commonly employed to tackle problems in
this area of research, with a selection of references
for the interested reader to further their knowledge
into this mathematically interesting field.
Wed, 04 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25142019-09-04T00:00:00ZKaltenbacher, BarbaraBrunnhuber, RainerNonlinear acoustics has been a topic of research for
more than 250 years. Driven by a wide range and a
large number of highly relevant industrial and medical
applications, this area has expanded enormously
in the last few decades. Here, we would like to give
a glimpse of the mathematical modeling techniques
that are commonly employed to tackle problems in
this area of research, with a selection of references
for the interested reader to further their knowledge
into this mathematically interesting field.Random permutations
http://publications.mfo.de/handle/mfo/2510
Random permutations
Betz, Volker
100 people leave their hats at the door at a party and
pick up a completely random hat when they leave.
How likely is it that at least one of them will get
back their own hat? If the hats carry name tags,
how difficult is it to arrange for all hats to be returned
to their owner? These classical questions of
probability theory can be answered relatively easily.
But if a geometric component is added, answering
the same questions immediately becomes very hard,
and little is known about them. We present some
of the open questions and give an overview of what
current research can say about them.
Fri, 12 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25102019-07-12T00:00:00ZBetz, Volker100 people leave their hats at the door at a party and
pick up a completely random hat when they leave.
How likely is it that at least one of them will get
back their own hat? If the hats carry name tags,
how difficult is it to arrange for all hats to be returned
to their owner? These classical questions of
probability theory can be answered relatively easily.
But if a geometric component is added, answering
the same questions immediately becomes very hard,
and little is known about them. We present some
of the open questions and give an overview of what
current research can say about them.Counting self-avoiding walks on the hexagonal lattice
http://publications.mfo.de/handle/mfo/1424
Counting self-avoiding walks on the hexagonal lattice
Duminil-Copin, Hugo
In how many ways can you go for a walk along a
lattice grid in such a way that you never meet your
own trail? In this snapshot, we describe some combinatorial
and statistical aspects of these so-called
self-avoiding walks. In particular, we discuss a recent
result concerning the number of self-avoiding walks
on the hexagonal (“honeycomb”) lattice. In the last
part, we briefly hint at the connection to the geometry
of long random self-avoiding walks.
Tue, 04 Jun 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14242019-06-04T00:00:00ZDuminil-Copin, HugoIn how many ways can you go for a walk along a
lattice grid in such a way that you never meet your
own trail? In this snapshot, we describe some combinatorial
and statistical aspects of these so-called
self-avoiding walks. In particular, we discuss a recent
result concerning the number of self-avoiding walks
on the hexagonal (“honeycomb”) lattice. In the last
part, we briefly hint at the connection to the geometry
of long random self-avoiding walks.Algebra, matrices, and computers
http://publications.mfo.de/handle/mfo/1415
Algebra, matrices, and computers
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
What part does algebra play in representing the real
world abstractly? How can algebra be used to solve
hard mathematical problems with the aid of modern
computing technology? We provide answers to these
questions that rely on the theory of matrix groups
and new methods for handling matrix groups in a
computer.
Fri, 03 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14152019-05-03T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWhat part does algebra play in representing the real
world abstractly? How can algebra be used to solve
hard mathematical problems with the aid of modern
computing technology? We provide answers to these
questions that rely on the theory of matrix groups
and new methods for handling matrix groups in a
computer.Positive Scalar Curvature and Applications
http://publications.mfo.de/handle/mfo/1414
Positive Scalar Curvature and Applications
Rosenberg, Jonathan; Wraith, David
We introduce the idea of curvature, including how it
developed historically, and focus on the scalar curvature
of a manifold. A major current research topic
involves understanding positive scalar curvature. We
discuss why this is interesting and how it relates to
general relativity.
Thu, 25 Apr 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14142019-04-25T00:00:00ZRosenberg, JonathanWraith, DavidWe introduce the idea of curvature, including how it
developed historically, and focus on the scalar curvature
of a manifold. A major current research topic
involves understanding positive scalar curvature. We
discuss why this is interesting and how it relates to
general relativity.Diophantine equations and why they are hard
http://publications.mfo.de/handle/mfo/1413
Diophantine equations and why they are hard
Pasten, Hector
Diophantine equations are polynomial equations whose
solutions are required to be integer numbers. They
have captured the attention of mathematicians during
millennia and are at the center of much of contemporary
research. Some Diophantine equations are easy,
while some others are truly difficult. After some time
spent with these equations, it might seem that no
matter what powerful methods we learn or develop,
there will always be a Diophantine equation immune
to them, which requires a new trick, a better idea, or
a refined technique. In this snapshot we explain why.
Wed, 24 Apr 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14132019-04-24T00:00:00ZPasten, HectorDiophantine equations are polynomial equations whose
solutions are required to be integer numbers. They
have captured the attention of mathematicians during
millennia and are at the center of much of contemporary
research. Some Diophantine equations are easy,
while some others are truly difficult. After some time
spent with these equations, it might seem that no
matter what powerful methods we learn or develop,
there will always be a Diophantine equation immune
to them, which requires a new trick, a better idea, or
a refined technique. In this snapshot we explain why.On 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.