250091 SE Seminar (Combinatorics) (2008W)
Continuous assessment of course work
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Vorbesprechung am 10. Oktober 2008, 15:00 Uhr; 2A180 (UZA 2)
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Information
Aims, contents and method of the course
Assessment and permitted materials
A seminar talk and participation in the discussions of the presentations of the other participants
Minimum requirements and assessment criteria
The topic of this seminar will be tilings of planar regions by dominos
(dimers) or rhombi. If we fix a bounded region in the plane and then choose
a tiling of this region by dominos (or by rhombi) at random: how does
it look like? What is the mathematics of these random tilings?While, at a first glance, these question do not even seem to make
sense, one notices quickly what could be meant if one looks at random
tilings of large regions (see http://www.mat.univie.ac.at/~kratt/limshap1.pdf.'>http://www.mat.univie.ac.at/~kratt/limshap1.pdf">http://www.mat.univie.ac.at/~kratt/limshap1.pdf.
Indeed, in the recent past, Richard Kenyon, Andrei Okounkov, Scott
Sheffield and co-authors have proved fascinating theorems, which show
that random tilings are not so "random" at all (as the pictures do
indeed suggest), but obey laws which can be described mathematically
in very precise ways.Starting from the "Lectures on dimers" by Richard Kenyon, which
provide an overview of the underlying mathematical theory of tilings,
we shall introduce ourselves to this area. This area draws upon
combinatorics, probability theory, and complex analysis.
It is therefore advantageous to be aquainted with the basics
in these areas.
(dimers) or rhombi. If we fix a bounded region in the plane and then choose
a tiling of this region by dominos (or by rhombi) at random: how does
it look like? What is the mathematics of these random tilings?While, at a first glance, these question do not even seem to make
sense, one notices quickly what could be meant if one looks at random
tilings of large regions (see http://www.mat.univie.ac.at/~kratt/limshap1.pdf.'>http://www.mat.univie.ac.at/~kratt/limshap1.pdf">http://www.mat.univie.ac.at/~kratt/limshap1.pdf.
Indeed, in the recent past, Richard Kenyon, Andrei Okounkov, Scott
Sheffield and co-authors have proved fascinating theorems, which show
that random tilings are not so "random" at all (as the pictures do
indeed suggest), but obey laws which can be described mathematically
in very precise ways.Starting from the "Lectures on dimers" by Richard Kenyon, which
provide an overview of the underlying mathematical theory of tilings,
we shall introduce ourselves to this area. This area draws upon
combinatorics, probability theory, and complex analysis.
It is therefore advantageous to be aquainted with the basics
in these areas.
Examination topics
The topic of this seminar will be tilings of planar regions by dominos
(dimers) or rhombi. If we fix a bounded region in the plane and then choose
a tiling of this region by dominos (or by rhombi) at random: how does
it look like? What is the mathematics of these random tilings?While, at a first glance, these question do not even seem to make
sense, one notices quickly what could be meant if one looks at random
tilings of large regions (see http://www.mat.univie.ac.at/~kratt/limshap1.pdf.'>http://www.mat.univie.ac.at/~kratt/limshap1.pdf">http://www.mat.univie.ac.at/~kratt/limshap1.pdf.
Indeed, in the recent past, Richard Kenyon, Andrei Okounkov, Scott
Sheffield and co-authors have proved fascinating theorems, which show
that random tilings are not so "random" at all (as the pictures do
indeed suggest), but obey laws which can be described mathematically
in very precise ways.Starting from the "Lectures on dimers" by Richard Kenyon, which
provide an overview of the underlying mathematical theory of tilings,
we shall introduce ourselves to this area. This area draws upon
combinatorics, probability theory, and complex analysis.
It is therefore advantageous to be aquainted with the basics
in these areas.
(dimers) or rhombi. If we fix a bounded region in the plane and then choose
a tiling of this region by dominos (or by rhombi) at random: how does
it look like? What is the mathematics of these random tilings?While, at a first glance, these question do not even seem to make
sense, one notices quickly what could be meant if one looks at random
tilings of large regions (see http://www.mat.univie.ac.at/~kratt/limshap1.pdf.'>http://www.mat.univie.ac.at/~kratt/limshap1.pdf">http://www.mat.univie.ac.at/~kratt/limshap1.pdf.
Indeed, in the recent past, Richard Kenyon, Andrei Okounkov, Scott
Sheffield and co-authors have proved fascinating theorems, which show
that random tilings are not so "random" at all (as the pictures do
indeed suggest), but obey laws which can be described mathematically
in very precise ways.Starting from the "Lectures on dimers" by Richard Kenyon, which
provide an overview of the underlying mathematical theory of tilings,
we shall introduce ourselves to this area. This area draws upon
combinatorics, probability theory, and complex analysis.
It is therefore advantageous to be aquainted with the basics
in these areas.
Reading list
Richard Kenyon: Lectures on dimers http://www.math.brown.edu/~rkenyon/papers/dimerlecturenotes.pdfHenry Cohn, Richard Kenyon, Jim Propp:
A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001), 297-346 http://arxiv.org/abs/math/0008220Richard Kenyon, Andrei Okounkov, Scott Sheffield Dimers and Amoebae, Ann. Math. 163 (2006), no. 3, 1019--1056 http://arxiv.org/abs/math-ph/0311005
A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001), 297-346 http://arxiv.org/abs/math/0008220Richard Kenyon, Andrei Okounkov, Scott Sheffield Dimers and Amoebae, Ann. Math. 163 (2006), no. 3, 1019--1056 http://arxiv.org/abs/math-ph/0311005
Association in the course directory
MALS
Last modified: Fr 31.08.2018 08:54
(dimers) or rhombi. If we fix a bounded region in the plane and then choose
a tiling of this region by dominos (or by rhombi) at random: how does
it look like? What is the mathematics of these random tilings?While, at a first glance, these question do not even seem to make
sense, one notices quickly what could be meant if one looks at random
tilings of large regions (see http://www.mat.univie.ac.at/~kratt/limshap1.pdf.'>http://www.mat.univie.ac.at/~kratt/limshap1.pdf">http://www.mat.univie.ac.at/~kratt/limshap1.pdf.
Indeed, in the recent past, Richard Kenyon, Andrei Okounkov, Scott
Sheffield and co-authors have proved fascinating theorems, which show
that random tilings are not so "random" at all (as the pictures do
indeed suggest), but obey laws which can be described mathematically
in very precise ways.Starting from the "Lectures on dimers" by Richard Kenyon, which
provide an overview of the underlying mathematical theory of tilings,
we shall introduce ourselves to this area. This area draws upon
combinatorics, probability theory, and complex analysis.
It is therefore advantageous to be aquainted with the basics
in these areas.