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SUMMARY:School "Randomness online"
DTSTART;VALUE=DATE-TIME:20201104T132000Z
DTEND;VALUE=DATE-TIME:20201108T162000Z
DTSTAMP;VALUE=DATE-TIME:20240226T132959Z
UID:indico-event-40@indico.eimi.ru
DESCRIPTION:Online school "Randomness online"\n\n\n \n The video records
of the lectures are here\n \n \n The lecture notes are below on this page
\n \n\n\nNovember 4 – 8\, 2020\n\nThis is an introductory online school
for the EIMI thematic program “New Trends in Mathematical Stochastics
” to be held in St. Petersburg in 2021 (rescheduled from 2020 due to COV
ID-19 pandemic). The aim of the school is to introduce young researchers t
o several important topics in stochastic geometry and probability theory i
ncluding random polytopes\, random matrices\, and floating bodies. Senior
researchers interested in learning these topics are also welcome.\n\nThe s
chool consists of 3 courses 4 lectures each. The lecture courses are:\n\n
\n Random Polytopes\, Zakhar Kabluchko (University of Münster)\n Tridiago
nal Random Matrices\, Manjunath Krishnapur (Indian Institute of Science)\n
Floating body\, Approximation by polytopes and Data depth\, Elisabeth Wer
ner (Case Western Reserve University)\n\n\n\nLecture Courses:\n\n\nRandom
Polytopes (Zakhar Kabluchko)\n\nA polytope is a convex hull of finitely ma
ny points in Euclidean space. By taking these points to be random\, we obt
ain random polytopes. Examples include convex hulls of independent identic
ally distributed random points (including the so-called Gaussian polytope
which arises if the points have standard Gaussian distribution)\, convex h
ulls of multidimensional random walks\, random projections of regular poly
topes\, and many others.\n\nWe shall be interested in computing expectatio
ns of various functionals of such polytopes\, for example the volume\, the
number of faces\, internal and external angles\, and some others. It turn
s out that there are many unexpected interrelations between these function
als. For example\, Baryshnikov and Vitale observed that the number of face
s has the same distribution for Gaussian polytopes as for projections of r
egular polytopes.\n\nThe main tool used in our computations is the integra
l geometry of convex cones. We shall introduce the participants to this su
bject. In particular\, we shall give various definitions of intrinsic volu
mes for convex cones. Also\, we shall address some problems of classical
geometry. For example\, we shall compute the number of parts in which $
n$ hyperplanes in general position divide the $d$-dimensional space. As sh
own by Wendel\, this problem is equivalent to the following one: compute t
he probability that the Gaussian polytope contains the origin. \n\nRandom
polytopes\, random cones\, and their integral geometric characteristics a
ppear naturally in the Grassmannian approach to linear programming suggest
ed by A. M. Vershik. We shall provide some examples following the paper by
Amelunxen\, Lotz\, McCoy and Tropp https://arxiv.org/abs/1303.6672\n\n\nT
ridiagonal Random Matrices (Manjunath Krishnapur)\n\nThis lecture series w
ill be about the use of tridiagonal random matrices in the study of the lo
g gas on the real line (particles on the line in a quadratic potential wel
l\, repelling by logarithmic interaction energy).\n\nStarting with the inn
ovative ideas of Trotter and then Dumitriu-Edelman who found these models\
, we shall cover some of the basic results such as Selberg integral\, Wign
er's semicircle law\, Furedi-Komlos and then go on to study the behaviour
of particles at the edge\, moderate deviations for the largest eigenvalue\
, the beta Tracy-Widom distributions of Ramirez-Rider-Virag via operator l
imits.\n\nOther related topics that may appear are Szego's theorem on asy
mptotics of zeros of orthogonal polynomials and Kerov's theorem on separat
ion of interlaced roots of orthogonal polynomials. In the end\, we hope to
talk about some recent works on law of iterated logarithm in exponential
last passage percolation (due to Ledoux and Basu–Ganguly–Hegde–K.)\,
using the results from earlier lectures and a well-known mapping of LPP w
ith Wishart random matrices.\n\nOverall\, it is expository in nature and a
imed at graduate students and does not assume prior knowledge of random ma
trix theory.\n\nReferences:\n\n1) Holcomb\, D. and Virag\, B.\, Operator l
imits of random matrices\, https://people.kth.se/~holcomb/ShortCourseNotes
.pdf\n\n2) Krishnapur\, M.\, Random matrix theory (notes from a course)\,
http://math.iisc.ac.in/~manju/RMT17/RMT_2017.pdf\n\n\nFloating body\, Appr
oximation by polytopes and Data depth (Elisabeth Werner)\n\nTwo important
closely related notions in affine convex geometry are the floating body an
d the affine surface area of a convex body.\n\nThe floating body of a conv
ex body is obtained by cutting off caps of volume less or equal to a fixed
positive constant. Taking the right-derivative of the volume of the float
ing body gives rise to an affine invariant\, the affine surface area. Th
is was established for all convex bodies in all dimensions by Schuett and
Werner. There is a natural inequality associated with affine surface area\
, the affine isoperimetric inequality\, which states that among all convex
bodies\, with fixed volume\, affine surface area is maximized for ellipso
ids.\n\nDue to its important properties\, which make them effective and po
werful tools\, affine surface area and floating body are omnipresent in
geometry and have applications in many other areas of mathematics\, e.g.\,
in problems of approximation of convex bodies by polytopes and for the
notion of halfspace depth for multivariate data from statistics.\n\nht
tps://indico.eimi.ru/event/40/
LOCATION:Zoom
URL:https://indico.eimi.ru/event/40/
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