- 28.11.2018, Yuriy Nemish (Vienna, Austria)
Local laws for polynomials of Wigner matrices
We consider general self-adjoint polynomials in several independent random matrices whose entries are
centered and have constant variance. Under some numerically checkable conditions, we establish the optimal
local law, i.e., we show that the empirical spectral distribution on scales just above the eigenvalue
spacing follows the global density of states which is determined by free probability theory. First, we give
a brief introduction to the linearization technique that allows to transform the polynomial model into a linear
one, which has simpler correlation structure but higher dimension. After that we show that the local law holds
up to the optimal scale for the generalized resolvent of the linearized model, which also yields the local law
for polynomials. Finally, we show how the above results can be applied to prove the optimal bulk local law for
two concrete families of polynomials: general quadratic forms in Wigner matrices and symmetrized products of
independent matrices with i.i.d. entries. This is a joint work with Laszlo Erdös and Torben Krüger.
- 24.10.2018, Konrad Schrempf (Vienna University)
From Linear to Semidefinite Programming, or: Using Noncommutativity for Optimization
Linear Programming (LP) is a well known theory
for optimization and easy to apply, many implementations of
the classical Simplex tableau or interior point methods are
available. Semidefinite Programming (SDP) is its natural
generalization by using matrices instead of vectors and
linear matrix-inequalities (LMIs) instead of classical
inequalities. Quadratic Programming (QP) and optimization
with zero-one-variables can be treated with SDP.
I will give a brief introduction to the
basic concepts and illustrate the formulation of a small
problem to be able to use one of the existing SDP solvers.
- 17.10.2018, Xumin Wuang (Besancon, France)
Invariant Markov semigroups on the quantum homogeneous spaces
We find that there is a one to one correspondence between invariant
Markov semigroups on compact quantum groups and invariant Markov
semigroups on coidalgebras. For strongly continuous Markov semigroups,
we can always find a generator. Specially, for invariant Markov
semigroups on these spheres O^×_N (×=classical, half-liberated or
free case) which is considered as a coidalgebra of C(O^×_N), we can
show these eigenvector spaces and associated eigenvalues of generators.
These eigenvalues can be described by some orthogonal polynomials.
- 04.10.2018, 11 am, Johannes Hoffmann (Aachen, Germany)
Theoretical and Computational Aspects of Ore Localization
Any multiplicative subset S of a commutative ring R induces a localization ring S^{-1}R in
which every element of S becomes invertible, abstracting the process of constructing the rationals
from the integers. Ore localization is a generalization of this classical concept to non-commutative
rings. The main idea is the same and most of the results can be extended, but the price we have to
pay comes down to restrictions on S. In this talk, we give an introduction to Ore localization both
from theoretical and constructive points of view. We start from an axiomatic definition and move on
to Ore's construction utilizing Ore sets instead of multiplicative sets. Different Ore sets can
induce the same localization and our search for a canonical Ore set leads to the notion of saturation
closure: applied to an Ore set S we get a normal form that induces the same localization, but has
better structural properties, for example it gives us a complete description of all elements of R that
become units when passing to S^{-1}R. As a further instance of saturation closure in this context we
consider the local closure problem and its applications in differential and difference equations.
Finally we discuss our efforts on making Ore localization computationally effective in the framework of
so-called G-algebras and their implementation in the computer algebra system Singular.
- 28.06.2018, Giusi Alfano (TU Berlin)
Random channel matrix models in the transition from the 4th to the 5th generation of mobile telephony
Performance analysis and design of wireless and optical communications systems
resort to random matrix theory since mid ‘90’s, with the advent of the so-called MIMO
(Multiple-Input-Multiple-Output) technology paradigm.
It (mostly) accounts to equip both the transmit as well as the receive side of a wireless
link with multiple antennas; this way, a matrix (with randomly distributed entries)
becomes the natural model for the coupling between transmitted and received signal,
up to the presence of thermal noise.
System working assumptions for the 3rd generation of mobile telephony led to the
adoption of complex zero-mean Gaussian matrices with iid entries as suitable models
for the radio channel effects on the transmitted signal. The possible presence of
correlation among the antennas and/or a non-zero mean for some of the entries were
eventually accommodated within a generalized Wishart setting in the finitedimensional
case and, correspondingly, into a free probability framework in the largesystem
regime.
Upon introduction of small-cell paradigms with 4th generation, channel matrices were
modeled as products of independent Ginibre matrices, with properly adjusted
parameters. The overall model being still unitarily invariant, both a finite-size analysis
of the spectral properties of such channel matrices, as well as asymptotic spectrum
theorems have been derived.
Forthcoming wireless generation will also adopt millimeter waves (mmWave) for data
transmission, and the resulting channel matrices will lack unitarily invariance.
The talk is divided in two parts: first, finite size analysis of a MIMO system in presence
of progressive scattering, modeled through a mixed product of random and
deterministic matrices of arbitrary size is presented, together with evaluation of
ergodic and outage capacity. A log-gas representation of the matrix model is also
discussed.
In the last part, a channel model for mmWave MIMO will be detailed, focusing on the
practical issues impacting on the marginals of the entries, and on the symmetries of
the law of the resulting matrix product.
- 06.06.2018, Adrian Celestino (CIMAT, Mexico)
Eigenvalues of random matrices with discrete spectrum
In this talk we will present the ideas of Collins, Hasebe and Sakuma (2015)
for studying the spectrum of random matrices which are obtained as self-adjoint
polynomials in random matrices of two types: random matrices with discrete
spectrum in the limit, and random matrices which have joint limiting distribution
and are globally rotationally invariant. They proved that the random eigenvalues
of this random matrix model converge almost surely to the eigenvalues of a
deterministic compact operator by showing that the mixed moments of the two types
of random matrices (with respect the non-normalized trace) satisfy a rule called
cyclic monotone independence. Finally, some numerical experiments are presented
for some polynomials whose eigenvalues can be computed explicitly by cyclic
monotone independence.
- 23.05.2018, Tobias Mai, Sheng Yin (Saarbrücken)
The free field: zero divisors, Atiyah property and realizations via unbounded operators
We report on some results obtained recently in joint work with R. Speicher.
We consider noncommutative rational functions as well as matrices in polynomials in noncommuting
variables in two settings: in an algebraic context the variables are formal variables, and their
rational functions generate the ''free field''; in an analytic context the variables are given by
operators from a finite von Neumann algebra and the question of rational functions is treated within
the affiliated unbounded operators. Our main result shows that for a ''good'' class of operators --
namely those for which the free entropy dimension is maximal -- the analytic and the algebraic theory
are isomorphic. This means in particular that any non-trivial rational function can be evaluated as an
unbounded operator for any such good tuple and that those operators don't have zero divisors. On the
matrix side, this means that matrices of polynomials which are invertible in the free field are also
invertible as matrices over unbounded operators when we plug in our good operator tuples. We also address
the question how this is related to the strong Atiyah property. The above yields a quite complete picture
for the question of zero divisors (or atoms in the corresponding distributions) for operator tuples with
maximal free entropy dimension.
- 25.04.2018, Pierre Tarrago (CIMAT, Mexico)
Subordination for the free deconvolution
The classical deconvolution of measures is an important problem which
consists in recovering the distribution of a random variable from the
knowledge of the random variable modified by an independent noise with
known distribution. In this talk, I will discuss the free version of
this problem: how can we recover the distribution of a non-commutative
random variable from the knowledge of the distribution of the random
variable modified by the addition (or multiplication) of a free
independent noise? Since large independent random matrices in general
positions are approximately free, an answer to the former question is
a first step in the extraction of the spectral distribution of a large
matrix from the knowledge of the matrix with an additive or
multiplicative noise.
Contrary to the classical case, the free convolution is not described
by an integral kernel like the Fourier transform. This problem has
been circumvented by Biane, Voiculescu, Belinschi and Bercovici which
developed a fixed point method called subordination. I will explain
how this method can be used to reduce the free deconvolution problem
to a classical one. This is a joint work with Octavio Arizmendi
(CIMAT) and Carlos Vargas (CIMAT).
- 18.04.2018, Simon Lentner (Hamburg)
Finite Tensor Categories and Correlation Functions
I want to demonstrate, how correlation functions (from the view of quantum field theory) are linked
with and controlled by the representation theory of finite algebras. Main examples are the Ising model,
spin chains or percolation theory. My own current research in this matter focusses on the case where
this representation theory is non-semisimple, and in particular the representation of a quantum group.
- 11.04.2018, Torben Krüger (Bonn)
Spectral Universality for Random Matrices: From the Global to the Local Scale
The spectral statistics of large dimensional self-adjoint random matrices often exhibits universal
behavior. On the global spectral scale the density of states depends only on the first two moments
of the matrix entries and follows a universal shape at all its singular points, i.e. whenever it
vanishes. On the local scale the joint distribution of a finite number of eigenvalues depends only
on the symmetry type of the random matrix (Wigner-Dyson-Mehta spectral universality). We
present recent results and methods that establish such spectral universality properties from the
global down to the smallest spectral scale for a wide range of random matrix models, including
matrices with general expectation and correlated entries.
[Joint work with the Erdös group at IST Austria]
- 21.03.2018, Elba Garcia-Failde (Max Planck Institute, Bonn)
Enumerative interpretation of higher order free cumulants
In this talk, we call ordinary maps a certain type of graphs embedded n surfaces, in contrast to fully simple maps, which we introduce as maps in which the boundaries do not touch each other nor themselves. It is well-known that the generating series of ordinary maps satisfy a universal recursive procedure, called topological recursion (TR). We prove that after applying the symplectic transformation of exchanging $x$ and $y$ in the initial data of the TR (the spectral curve), the TR correlators enumerate fully simple maps. We give explicit combinatorial proofs for disks and cylinders, recovering R-transform formulas already known in the context of free probability for first and second order free cumulants. For the rest of topologies, our proof relies on a matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. The TR restricted to genus zero gives recursive formulas for the higher order free cumulants and suggests the possibility of a universal theory of approximate higher order free cumulants taking into account the higher genus amplitudes. We also give a universal relation between fully simple and ordinary maps involving double monotone Hurwitz numbers. In particular, we obtain an ELSV-like formula for strictly monotone double Hurwitz numbers with arbitrary ramification profile over $0$ and $(2,\ldots,2)$ over $\infty$.
-
February 13, 2018, 4:15 pm, Marek Bozejko (Wroclaw, Poland)
Positive definite functions on Coxeter groups with applications to non-commutative probability
In my talk we will consider the following subjects:
1. Classical Riesz product construction as a special construction of a
probability measure on a torus (compact Abelian group).
2. Riesz product on Rademacher-Cantor groups, dihedral groups,
permutation groups and free groups and connection with Boolean and
(conditionally) free probability.
3. The length functions on permutation groups and more general on
Weyl-Coxeter groups (W,S).
4. The set of Coxeter generators S is a weak Sidon set in arbitrary
Weyl-Coxeter group (W,S).
5. Kchintchine inequalities with some applications to operator spaces.
References:
1. M.Bozejko, S.Gal, W.Mlotkowski, Positive definite functions on Coxeter
groups with applications to operator spaces and noncommutative
probability, Comm. Math. Phys. 2018.
2. M.Bozejko, W.Ejsmont and T.Hasebe, Fock space associated to Coxeter
groups of type B, J.Funct. Anal. 269,1769-1795,2015.
3. M.Bozejko, W.Ejsmont and T.Hasebe, Noncommutative probability of type
D, Inter.J.Math,28(2),2017.
4. M. Bozejko and R. Speicher, Completely positive maps on Coxeter
groups, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994)
97-120.
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