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, 12 Aug 2019 14:55:17 GMT2019-08-12T14:55:17ZRandom 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.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.