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Seminar information archive
Seminar information archive ～04/19｜Today's seminar 04/20 | Future seminars 04/21～
GCOE Seminars
14:30-16:00 Room #370 (Graduate School of Math. Sci. Bldg.)
Takuma Akimoto (Keio university, Global environmental leaders program)
Distributional behaviors of time-averaged observables in anomalous diffusions (subdiffusion and superdiffusion) (ENGLISH)
Takuma Akimoto (Keio university, Global environmental leaders program)
Distributional behaviors of time-averaged observables in anomalous diffusions (subdiffusion and superdiffusion) (ENGLISH)
[ Abstract ]
In anomalous diffusions attributed to a power-law distribution,
time-averaged observables such as diffusion coefficient and velocity of drift are intrinsically random. Anomalous diffusion is ubiquitous phenomenon not only in material science but also in biological transports, which is characterized by a non-linear growth of the mean square displacement (MSD).
(subdiffusion: sublinear growth, super diffusion: superlinear growth).
It has been known that there are three different mechanisms generating subdiffusion. One of them is a power-law distribution in the trapping-time distribution. Such anomalous diffusion is modeled by the continuous time random walk (CTRW). In CTRW, the time-averaged MSD grows linearly with time whereas the ensemble-averaged MSD does not. Using renewal theory, I show that diffusion coefficients obtained by single trajectories converge in distribution. The distribution is the Mittag-Leffler (or inverse Levy) distribution [1,2].
In superdiffusion, there are three different mechanisms. One stems from positive correlations in random walks; the second from persistent motions in random walks, called Levy walk; the third from very long jumps in random walks, called Levy flight.
If the persistent time distribution obeys a power law with divergent mean in Levy walks, the MSD grows as t^2 whereas the mean of positions is zero. When an external bias is added in Levy walks, the response to bias (velocity of drift) appears in the distribution, which is what we term a distributional response [3]. The distribution is the generalized arcsine distribution.
These distributional behaviors open a new window to dealing with the average (ensemble or time average) in single particle tracking experiments.
[1] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. 101, 058101 (2008).
[2] T. Miyaguchi and T. Akimoto, Phys. Rev. E 83, 031926 (2011).
[3] T. Akimoto, Phys. Rev. Lett. 108, 164101 (2012)
In anomalous diffusions attributed to a power-law distribution,
time-averaged observables such as diffusion coefficient and velocity of drift are intrinsically random. Anomalous diffusion is ubiquitous phenomenon not only in material science but also in biological transports, which is characterized by a non-linear growth of the mean square displacement (MSD).
(subdiffusion: sublinear growth, super diffusion: superlinear growth).
It has been known that there are three different mechanisms generating subdiffusion. One of them is a power-law distribution in the trapping-time distribution. Such anomalous diffusion is modeled by the continuous time random walk (CTRW). In CTRW, the time-averaged MSD grows linearly with time whereas the ensemble-averaged MSD does not. Using renewal theory, I show that diffusion coefficients obtained by single trajectories converge in distribution. The distribution is the Mittag-Leffler (or inverse Levy) distribution [1,2].
In superdiffusion, there are three different mechanisms. One stems from positive correlations in random walks; the second from persistent motions in random walks, called Levy walk; the third from very long jumps in random walks, called Levy flight.
If the persistent time distribution obeys a power law with divergent mean in Levy walks, the MSD grows as t^2 whereas the mean of positions is zero. When an external bias is added in Levy walks, the response to bias (velocity of drift) appears in the distribution, which is what we term a distributional response [3]. The distribution is the generalized arcsine distribution.
These distributional behaviors open a new window to dealing with the average (ensemble or time average) in single particle tracking experiments.
[1] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. 101, 058101 (2008).
[2] T. Miyaguchi and T. Akimoto, Phys. Rev. E 83, 031926 (2011).
[3] T. Akimoto, Phys. Rev. Lett. 108, 164101 (2012)
2012/05/02
Operator Algebra Seminars
16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Yuhei Suzuki (Univ. Tokyo)
A measurable group theoretic solution to von Neumann's Problem (after Gaboriau and Lyons) (JAPANESE)
Yuhei Suzuki (Univ. Tokyo)
A measurable group theoretic solution to von Neumann's Problem (after Gaboriau and Lyons) (JAPANESE)
2012/05/01
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Hisashi Kasuya (The University of Tokyo)
Minimal models, formality and hard Lefschetz property of
solvmanifolds with local systems (JAPANESE)
Hisashi Kasuya (The University of Tokyo)
Minimal models, formality and hard Lefschetz property of
solvmanifolds with local systems (JAPANESE)
2012/04/27
Seminar on Probability and Statistics
15:00-16:10 Room #006 (Graduate School of Math. Sci. Bldg.)
NOMURA, Ryosuke (Graduate school of Mathematical Sciences, Univ. of Tokyo)
Convergence conditions on step sizes in temporal difference learning (JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/02.html
NOMURA, Ryosuke (Graduate school of Mathematical Sciences, Univ. of Tokyo)
Convergence conditions on step sizes in temporal difference learning (JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/02.html
2012/04/25
Operator Algebra Seminars
16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Takuya Takeishi (Univ. Tokyo)
Bost-Connes system and class field theory (JAPANESE)
Takuya Takeishi (Univ. Tokyo)
Bost-Connes system and class field theory (JAPANESE)
2012/04/24
Numerical Analysis Seminar
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Hideaki Ishikawa (Semiconductor Leading Edge Technologies, Inc.)
Quantum mechanics and numerical analysis (JAPANESE)
[ Reference URL ]
http://www.infsup.jp/utnas/
Hideaki Ishikawa (Semiconductor Leading Edge Technologies, Inc.)
Quantum mechanics and numerical analysis (JAPANESE)
[ Reference URL ]
http://www.infsup.jp/utnas/
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Dylan Thurston (Columbia University)
Combinatorial Heegaard Floer homology (ENGLISH)
Dylan Thurston (Columbia University)
Combinatorial Heegaard Floer homology (ENGLISH)
[ Abstract ]
Heegaard Floer homology is a powerful invariant of 3- and 4-manifolds.
In 4 dimensions, Heegaard Floer homology (together with the
Seiberg-Witten and Donaldson equations, which are conjecturally
equivalent), provides essentially the only technique for
distinguishing smooth 4-manifolds. In 3 dimensions, it provides much
geometric information, like the simplest representatives of a given
homology class.
In this talk we will focus on recent progress in making Heegaard Floer
homology more computable, including a complete algorithm for computing
it for knots.
Heegaard Floer homology is a powerful invariant of 3- and 4-manifolds.
In 4 dimensions, Heegaard Floer homology (together with the
Seiberg-Witten and Donaldson equations, which are conjecturally
equivalent), provides essentially the only technique for
distinguishing smooth 4-manifolds. In 3 dimensions, it provides much
geometric information, like the simplest representatives of a given
homology class.
In this talk we will focus on recent progress in making Heegaard Floer
homology more computable, including a complete algorithm for computing
it for knots.
2012/04/23
Algebraic Geometry Seminar
17:10-18:40 Room #122 (Graduate School of Math. Sci. Bldg.)
Takehiko Yasuda (Osaka University)
Motivic integration and wild group actions (JAPANESE)
Takehiko Yasuda (Osaka University)
Motivic integration and wild group actions (JAPANESE)
[ Abstract ]
The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant
and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is
an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.
Then I pushed forward with their study by generalizing the motivic integration to
Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of
the birational geometry of stacks.
However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),
and the wild (= not tame) case has remained unexplored.
In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group
of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized
arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of
Artin-Schreier extensions of $k((t))$, the field of Laurent series.
The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant
and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is
an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.
Then I pushed forward with their study by generalizing the motivic integration to
Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of
the birational geometry of stacks.
However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),
and the wild (= not tame) case has remained unexplored.
In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group
of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized
arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of
Artin-Schreier extensions of $k((t))$, the field of Laurent series.
2012/04/21
Harmonic Analysis Komaba Seminar
13:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Yutaka, Terasawa (Graduate School of Mathematical Sciences, University of Tokyo) 13:30-15:00
Dyadic, classical and martingale harmonic analysis (JAPANESE)
A_\\\\infty constants between BMO and weighted BMO (JAPANESE)
Yutaka, Terasawa (Graduate School of Mathematical Sciences, University of Tokyo) 13:30-15:00
Dyadic, classical and martingale harmonic analysis (JAPANESE)
[ Abstract ]
Yohei Tsutsui (Waseda University) 15:30-17:00A_\\\\infty constants between BMO and weighted BMO (JAPANESE)
[ Abstract ]
2012/04/20
Seminar on Probability and Statistics
14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)
KOIKE, Yuta (Graduate school of Mathematical Sciences, Univ. of Tokyo)
On the asymptotic mixed normality of the pre-averaged Hayashi-Yoshida
estimator with random and nonsynchronous sampling (JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/01.html
KOIKE, Yuta (Graduate school of Mathematical Sciences, Univ. of Tokyo)
On the asymptotic mixed normality of the pre-averaged Hayashi-Yoshida
estimator with random and nonsynchronous sampling (JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/01.html
2012/04/18
Number Theory Seminar
16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)
Alan Lauder (University of Oxford)
Explicit constructions of rational points on elliptic curves (ENGLISH)
Alan Lauder (University of Oxford)
Explicit constructions of rational points on elliptic curves (ENGLISH)
[ Abstract ]
I will present an algorithm for computing certain special
values of p-adic L-functions, and discuss an application to
the efficient construction of rational points on elliptic curves.
I will present an algorithm for computing certain special
values of p-adic L-functions, and discuss an application to
the efficient construction of rational points on elliptic curves.
Operator Algebra Seminars
16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Koichi Shimada (Univ. Tokyo)
Classification of Group Actions on Factors (after Masuda) (JAPANESE)
Koichi Shimada (Univ. Tokyo)
Classification of Group Actions on Factors (after Masuda) (JAPANESE)
2012/04/17
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Eriko Hironaka (Florida State University)
Pseudo-Anosov mapping classes with small dilatation (ENGLISH)
Eriko Hironaka (Florida State University)
Pseudo-Anosov mapping classes with small dilatation (ENGLISH)
[ Abstract ]
A mapping class is a homeomorphism of an oriented surface
to itself modulo isotopy. It is pseudo-Anosov if the lengths of essential
simple closed curves under iterations of the map have exponential growth
rate. The growth rate, an algebraic integer of degree bounded with
respect to the topology of the surface, is called the dilatation of the
mapping class. In this talk we will discuss the minimization problem
for dilatations of pseudo-Anosov mapping classes, and give two general
constructions of pseudo-Anosov mapping classes with small dilatation.
A mapping class is a homeomorphism of an oriented surface
to itself modulo isotopy. It is pseudo-Anosov if the lengths of essential
simple closed curves under iterations of the map have exponential growth
rate. The growth rate, an algebraic integer of degree bounded with
respect to the topology of the surface, is called the dilatation of the
mapping class. In this talk we will discuss the minimization problem
for dilatations of pseudo-Anosov mapping classes, and give two general
constructions of pseudo-Anosov mapping classes with small dilatation.
2012/04/16
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Makoto Miura (University of Tokyo)
Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)
Makoto Miura (University of Tokyo)
Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)
[ Abstract ]
Minuscule Schubert varieties admit the flat degenerations to projective
Hibi toric varieties, whose combinatorial structure is explicitly
described by finite posets. In this talk, I will explain these toric
degenerations and discuss the mirror symmetry for complete intersection
Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.
Minuscule Schubert varieties admit the flat degenerations to projective
Hibi toric varieties, whose combinatorial structure is explicitly
described by finite posets. In this talk, I will explain these toric
degenerations and discuss the mirror symmetry for complete intersection
Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yusuke Okuyama (Kyoto Institute of Technology)
Fekete configuration, quantitative equidistribution and wanderting critical orbits in non-archimedean dynamics
(JAPANESE)
Yusuke Okuyama (Kyoto Institute of Technology)
Fekete configuration, quantitative equidistribution and wanderting critical orbits in non-archimedean dynamics
(JAPANESE)
2012/04/14
Monthly Seminar on Arithmetic of Automorphic Forms
13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Kazutoshi Kariyama (Onomichi city university) 13:30-14:30
Explicit formula for the formal degree of the discrete series representations of GL_m(D). (JAPANESE)
Keijyu Souno (Math.-Sci., Tokyo Univ.) 15:00-16:00
Moments of the derivatives of the Riemann zeta function (JAPANESE)
Kazutoshi Kariyama (Onomichi city university) 13:30-14:30
Explicit formula for the formal degree of the discrete series representations of GL_m(D). (JAPANESE)
Keijyu Souno (Math.-Sci., Tokyo Univ.) 15:00-16:00
Moments of the derivatives of the Riemann zeta function (JAPANESE)
[ Abstract ]
In my talk, we consider the integral moments of the derivatives of the Riemann zeta function on the critical line. We give certain lower bounds for these moments under the assumption of the Riemann hypothesis.
In my talk, we consider the integral moments of the derivatives of the Riemann zeta function on the critical line. We give certain lower bounds for these moments under the assumption of the Riemann hypothesis.
2012/04/13
Seminar on Probability and Statistics
14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)
KAMATANI, Kengo (Graduate School of Engineering Science, Osaka University)
Asymptotic properties of MCMC for cumulative link model (JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/00.html
KAMATANI, Kengo (Graduate School of Engineering Science, Osaka University)
Asymptotic properties of MCMC for cumulative link model (JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/00.html
2012/04/11
Operator Algebra Seminars
16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Shweta Sharma (Univ. Paris Sud)
Mathematical Aspects of Fractional Quantum Hall Effect (ENGLISH)
Shweta Sharma (Univ. Paris Sud)
Mathematical Aspects of Fractional Quantum Hall Effect (ENGLISH)
Number Theory Seminar
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Damian Rossler (CNRS, Universite de Toulouse)
Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)
Damian Rossler (CNRS, Universite de Toulouse)
Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)
[ Abstract ]
Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).
Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).
2012/04/10
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Takuya Sakasai (The University of Tokyo)
On homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra (JAPANESE)
Takuya Sakasai (The University of Tokyo)
On homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra (JAPANESE)
[ Abstract ]
We discuss homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra
with particular stress on their abelianizations (degree 1 part).
Then, by using a theorem of Kontsevich,
we give some applications to rational cohomology of the moduli spaces of
Riemann surfaces and metric graphs.
This is a joint work with Shigeyuki Morita and Masaaki Suzuki.
We discuss homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra
with particular stress on their abelianizations (degree 1 part).
Then, by using a theorem of Kontsevich,
we give some applications to rational cohomology of the moduli spaces of
Riemann surfaces and metric graphs.
This is a joint work with Shigeyuki Morita and Masaaki Suzuki.
2012/04/09
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Kazushi Ueda (Osaka University)
On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)
Kazushi Ueda (Osaka University)
On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)
[ Abstract ]
In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.
If the time permits, I will also discuss an application to monodromy of hypergeometric functions.
In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.
If the time permits, I will also discuss an application to monodromy of hypergeometric functions.
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Shigeharu TAKAYAMA (University of Tokyo)
Effective estimate on the number of deformation types of families of canonically polarized manifolds over curves
(JAPANESE)
Shigeharu TAKAYAMA (University of Tokyo)
Effective estimate on the number of deformation types of families of canonically polarized manifolds over curves
(JAPANESE)
2012/04/04
PDE Real Analysis Seminar
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Jens Hoppe (Sogang University / KTH Royal Institute of Technology)
Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)
Jens Hoppe (Sogang University / KTH Royal Institute of Technology)
Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)
[ Abstract ]
We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.
For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided.
We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.
For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided.
2012/04/03
Monthly Seminar on Arithmetic of Automorphic Forms
13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Kazutoshi Kariyama (Onomichi city university) 13:30-14:30
Explicit formula for the formal degree of the discrete series representations of GL_m(D). (JAPANESE)
Keijyu Souno (Math.-Sci., Tokyo Univ.) 15:00-16:00
Moments of the derivatives of the Riemann zeta function (JAPANESE)
Kazutoshi Kariyama (Onomichi city university) 13:30-14:30
Explicit formula for the formal degree of the discrete series representations of GL_m(D). (JAPANESE)
Keijyu Souno (Math.-Sci., Tokyo Univ.) 15:00-16:00
Moments of the derivatives of the Riemann zeta function (JAPANESE)
[ Abstract ]
In my talk, we consider the integral moments of the derivatives of the Riemann zeta function on the critical line. We give certain lower bounds for these moments under the assumption of the Riemann hypothesis.
In my talk, we consider the integral moments of the derivatives of the Riemann zeta function on the critical line. We give certain lower bounds for these moments under the assumption of the Riemann hypothesis.
2012/03/23
Operator Algebra Seminars
16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)
Alex Kumjian (University of Nevada, Reno)
Higher Rank Graph $C^*$-algebras (ENGLISH)
Alex Kumjian (University of Nevada, Reno)
Higher Rank Graph $C^*$-algebras (ENGLISH)
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