In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

/pdf/cluster-algebras-i-foundations-3bc4xdvh4r.pdf

Cluster algebras I: Foundations

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. 
The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.

Cluster algebras II: Finite type classification

Determinantal point processes (DPPs) are elegant probabilistic models of repulsion that arise in quantum physics and random matrix theory. In contrast to traditional structured models like Markov random fields, which become intractable and hard to approximate in the presence of negative correlations, DPPs offer efficient and exact algorithms for sampling, marginalization, conditioning, and other inference tasks. We provide a gentle introduction to DPPs, focusing on the intuitions, algorithms, and extensions that are most relevant to the machine learning community, and show how DPPs can be applied to real-world applications like finding diverse sets of high-quality search results, building informative summaries by selecting diverse sentences from documents, modeling non-overlapping human poses in images or video, and automatically building timelines of important news stories.

/pdf/determinantal-point-processes-for-machine-learning-4jsyx0d5k9.pdf

Determinantal point processes for machine learning

We develop a new approach to cluster algebras, based on the notion of an upper cluster algebra defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [7], we show that under an assumption of ``acyclicity,'' a cluster algebra coincides with its upper counterpart and is finitely generated; in this case, we also describe its defining ideal and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to an upper cluster algebra explicitly defined in terms of relevant combinatorial data.

Cluster algebras III: Upper bounds and double Bruhat cells

Quantum theory has found a new field of application in the realm of information and computation during recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some notable aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as examples of the impact of quanta on the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement made possible in information processing by a quantum computer. Finally the authors provide some examples of current experimental realizations for quantum computers and future prospects.

https://tu-freiberg.de/sites/default/files/media/institut-fuer-theoretische-physik-10451/Lehre/Quantentheorie2/qm-comp.pdf

Information and computation: Classical and quantum aspects

From the Publisher:
Among the most exciting developments in science today is the design and construction of the quantum computer. Its realization will be the result of multidisciplinary efforts, but ultimately, it is mathematics that lies at the heart of theoretical quantum computer science.Mathematics of Quantum Computation brings together leading computer scientists, mathematicians, and physicists to provide the first interdisciplinary but mathematically focused exploration of the field's foundations and state of the art. Each section of the book addresses an area of major research, and does so with introductory material that brings newcomers quickly up to speed. Chapters that are more advanced include recent developments not yet published in the open literature.Information technology will inevitably enter into the realm of quantum mechanics, and, more than all the atomic, molecular, optical, and nanotechnology advances, it is the device-independent mathematics that is the foundation of quantum computer and information science. Mathematics of Quantum Computation offers the first up-to-date coverage that has the technical depth and breadth needed by those interested in the challenges being confronted at the frontiers of research.

Mathematics of Quantum Computation

In this paper we study universality for quantum gates acting on qudits. Qudits are states in a Hilbert space of dimension d where
d can be any integer ≥ 2. We determine which 2-qudit gates V have the
properties: (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection
of all 1-qudit gates together with V produces all n-qudit gates exactly.
We show that (i) and (ii) are equivalent conditions on V , and they hold
if and only if V is not a primitive gate. Here we say V is primitive if
it transforms any decomposable tensor into a decomposable tensor. We
discuss some applications and also relations with work of other authors.

Universal quantum gates

This volume, dedicated to Kostant on his 65th birthday, is a collection of 22 papers by international leaders in the fields of Lie theory, geometry, algebra and math physics. The papers broadly reflect the range of Kostant's work.

Lie theory and geometry : in honor of Bertram Kostant

Lie Theory and Geometry

Immanants are polynomial functions of n by n matrices attached to irreducible characters of the symmetric group S_n, or equivalently to Young diagrams of size n. Immanants include determinants and permanents as extreme cases. Valiant proved that computation of permanents is a complete problem in his algebraic model of NP theory, i.e., it is VNP-complete. We prove that computation of immanants is VNP-complete if the immanants are attached to a family of diagrams whose separation is $\Omega(n^\delta)$ for some $\delta>0$. We define the separation of a diagram to be the largest number of overhanging boxes contained in a single row. Our theorem proves a conjecture of Buergisser for a large variety of families, and in particular we recover with new proofs his VNP-completeness results for hooks and rectangles.

Complexity and Completeness of Immanants

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