Max Planck Institute For Mathematik

Bonn, Germany

Max Planck Institute For Mathematik

Bonn, Germany

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Dimofte T.,Institute for Advanced Study | Dimofte T.,Cambridge College | Dimofte T.,Max Planck Institute For Mathematik
Advances in Theoretical and Mathematical Physics | Year: 2013

We construct from first principles the operators ÂM that annihilate the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on knot complements M. The operator ÂM is a quantization of a knot complement's classical A-polynomial AM(l,m). The construction proceeds by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in topological quantum field theory to put them back together. We advocate in particular that, properly interpreted, "gluing = symplectic reduction." We also arrive at a new finite-dimensional state integral model for computing the analytically continued "holomorphic blocks" that compose any physical Chern-Simons partition function. © 2013 International Press.

Dimofte T.,Institute for Advanced Study | Gaiotto D.,Institute for Advanced Study | Gukov S.,California Institute of Technology | Gukov S.,Max Planck Institute For Mathematik
Communications in Mathematical Physics | Year: 2014

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N = 2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that Sb 3 partition functions of two mirror 3d N = 2 gauge theories are equal. Three-dimensional N = 2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional N = 2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing. © 2013 Springer-Verlag Berlin Heidelberg.

Gukov S.,California Institute of Technology | Gukov S.,Max Planck Institute For Mathematik
Journal of High Energy Physics | Year: 2016

Abstract: Interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different “topological sectors” for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) dowain walls? Analyzing these questions in a wide variety of contexts — from counting RG walls to AdS/CFT correspondence — will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott’s version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and “topological obstructions” for the strong form of the C-theorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong C-theorem happens and predicts “phase transitions” along the RG flow when the topological obstruction is non-trivial. Along the way, we also find new conformal manifolds in well-known 4d CFT’s and point out connections with the superconformal index and classifying spaces of global symmetry groups. © 2016, The Author(s).

D'Adderio M.,Max Planck Institute For Mathematik | Moci L.,TU Berlin
European Journal of Combinatorics | Year: 2012

We prove that the Ehrhart polynomial of a zonotope is a specialization of the arithmetic Tutte polynomial introduced by Moci (2012) [16]. We derive some formulae for the volume and the number of integer points of the zonotope. © 2012 Elsevier Ltd.

Ohta S.,Kyoto University | Ohta S.,Max Planck Institute For Mathematik | Sturm K.-T.,University of Bonn
Archive for Rational Mechanics and Analysis | Year: 2012

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance. © 2012 Springer-Verlag.

Manin Y.I.,Max Planck Institute For Mathematik
Selecta Mathematica, New Series | Year: 2016

Encoding, transmission and decoding of information are ubiquitous in biology and human history: from DNA transcription to spoken/written languages and languages of sciences. During the last decades, the study of neural networks in brain performing their multiple tasks was providing more and more detailed pictures of (fragments of) this activity. Mathematical models of this multifaceted process led to some fascinating problems about “good codes” in mathematics, engineering, and now biology as well. The notion of “good” or “optimal” codes depends on the technological progress and criteria defining optimality of codes of various types: error-correcting ones, cryptographic ones, noise-resistant ones etc. In this note, I discuss recent suggestions that activity of some neural networks in brain, in particular those responsible for space navigation, can be well approximated by the assumption that these networks produce and use good error-correcting codes. I give mathematical arguments supporting the conjecture that search for optimal codes is built into neural activity and is observable. © 2016 The Author(s)

Manin Y.I.,Max Planck Institute For Mathematik
Journal of Physics: Conference Series | Year: 2014

This paper is a survey based upon the talk at the satellite QQQ conference to ECM6, 3Quantum: Algebra Geometry Information, Tallinn, July 2012. It is dedicated to the analogy between the notions of complexity in theoretical computer science and energy in physics. This analogy is not metaphorical: I describe three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.

Mautner C.,Max Planck Institute For Mathematik
Selecta Mathematica, New Series | Year: 2014

We give geometric descriptions of the category {Mathematical expression} of rational polynomial representations of {Mathematical expression} over a field {Mathematical expression} of degree {Mathematical expression} for {Mathematical expression}, the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category {Mathematical expression} and the Schur functor. © 2014 Springer Basel.

Manin Y.I.,Max Planck Institute For Mathematik
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

Consider the set of all error-correcting block codes over a fixed alphabet with q letters. It determines a recursively enumerable set of points in the unit square with coordinates (R,δ):= (relative transmission rate, relative minimal distance). Limit points of this set form a closed subset, defined by R ≤ α q (δ), where α q (δ) is a continuous decreasing function called asymptotic bound. Its existence was proved by the author in 1981, but all attempts to find an explicit formula for it so far failed. In this note I consider the question whether this function is computable in the sense of constructive mathematics, and discuss some arguments suggesting that the answer might be negative. © 2012 Springer-Verlag.

Weisse A.,Max Planck Institute For Mathematik
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics | Year: 2013

Iterative methods that operate with the full Hamiltonian matrix in the untrimmed Hilbert space of a finite system continue to be important tools for the study of one- and two-dimensional quantum spin models, in particular in the presence of frustration. To reach sensible system sizes such numerical calculations heavily depend on the use of symmetries. We describe a divide-and-conquer strategy for implementing translation symmetries of finite spin clusters, which efficiently uses and extends the "sublattice coding" of H. Q. Lin. With our method, the Hamiltonian matrix can be generated on-the-fly in each matrix vector multiplication, and problem dimensions beyond 1011 become accessible. © 2013 American Physical Society.

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