Bonn, Germany

The Max Planck Institute for Mathematics is a research institute specializing in mathematics located in Bonn, Germany. It is named in honor of the German physicist Max Planck. The MPIM is one of the 80 institutes in the Max Planck Society .The Institute, having emerged from the collaborative research center called Theoretical Mathematics , was founded by Friedrich Hirzebruch in 1980 and he acted as the director of the institute until his retirement in 1995. Currently, the institute is managed by a board of four directors consisting of Werner Ballmann , Gerd Faltings, Peter Teichner and Don Zagier. Friedrich Hirzebruch was, and Yuri Manin is, acting as emeriti. Wikipedia.

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News Article | March 16, 2016

British number theorist Andrew Wiles has received the 2016 Abel Prize for his solution to Fermat’s last theorem — a problem that stumped some of the world’s greatest minds for three and a half centuries. The Norwegian Academy of Science and Letters announced the award — considered by some to be the 'Nobel of mathematics' — on 15 March. Wiles, who is 62 and now at the University of Oxford, UK, will receive 6 million kroner (US$700,000) for his 1994 proof of the theorem, which states that there cannot be any positive whole numbers x, y and z such that xn + yn = zn, if n is greater than 2. Soon after receiving the news on the morning of 15 March, Wiles told Nature that the award came to him as a “total surprise”. That he solved a problem considered too hard by so many — and yet a problem relatively simple to state — has made Wiles arguably “the most celebrated mathematician of the twentieth century”, says Martin Bridson, director of Oxford's Mathematical Institute — which is housed in a building named after Wiles. Although his achievement is now two decades old, he continues to inspire young minds, something that is apparent when school children show up at his public lectures.  “They treat him like a rock star,” Bridson says. “They line up to have their photos taken with him.” Wiles's story has become a classic tale of tenacity and resilience. While a faculty member at Princeton University in New Jersey in the 1980s, he embarked on a solitary, seven-year quest to solve the problem, working in his attic without telling anyone except for his wife. He went on to make a historic announcement at a conference in his hometown of Cambridge, UK, in June 1993, only to hear from a colleague two months later that his proof contained a serious mistake. But after another frantic year of work — and with the help of one of his former students, Richard Taylor, who is now at the Institute for Advanced Study in Princeton — he was able to patch up the proof. When the resulting two papers were published in 1995, they made up an entire issue of the Annals of Mathematics1, 2. But after Wiles's original claim had already made front-page news around the world, the pressure on the shy mathematician to save his work almost crippled him. “Doing mathematics in that kind of overexposed way is certainly not my style, and I have no wish to repeat it,” he said in a BBC documentary in 1996, still visibly shaken by the experience. “It’s almost unbelievable that he was able to get something done” at that point, says John Rognes, a mathematician at the University of Oslo and chair of the Abel Committee. “It was very, very intense,” says Wiles. “Unfortunately as human beings we succeed by trial and error. It’s the people who overcome the setbacks who succeed.” Wiles first learnt about French mathematician Pierre de Fermat as a child growing up in Cambridge. As he was told, Fermat formulated his eponymous theorem in a handwritten note in the margins of a book in 1637: “I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain,” he wrote (in Latin). “I think it has a very romantic story,” Wiles says of Fermat's idea. “The kind of story that catches people’s imagination when they’re young and thinking of entering mathematics.” But although he may have thought he had a proof at the time, only a proof for one special case has survived him, for exponent n = 4. A century later, Leonhard Euler proved it for n = 3, and Sophie Germain's work led to a proof for infinitely many exponents, but still not for all. Experts now tend to concur that the most general form of the statement would have been impossible to crack without mathematical tools that became available only in the twentieth century. In 1983, German mathematician Gerd Faltings, now at the Max Planck Institute for Mathematics in Bonn, took a huge leap forward by proving that Fermat's statement had, at most, a finite number of solutions, although he could not show that the number should be zero. (In fact, he proved a result viewed by specialists as deeper and more interesting than Fermat's last theorem itself; it demonstrated that a broader class of equations has, at most, a finite number of solutions.) To narrow it to zero, Wiles took a different approach: he proved the Shimura-Taniyama conjecture, a 1950s proposal that describes how two very different branches of mathematics, called elliptic curves and modular forms, are conceptually equivalent. Others had shown that proof of this equivalence would imply proof of Fermat — and, like Faltings' result, most mathematicians regard this as much more profound than Fermat’s last theorem itself. (The full citation for the Abel Prize states that it was awarded to Wiles “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.”) The link between the Shimura–Taniyama conjecture and Fermat's theorum was first proposed in 1984 by number theorist Gerhard Frey, now at the University of Duisburg-Essen in Germany. He claimed that any counterexample to Fermat's last theorem would also lead to a counterexample to the Shimura–Taniyama conjecture. Kenneth Ribet, a mathematician at the University of California, Berkeley, soon proved that Frey was right, and therefore that anyone who proved the more recent conjecture would also bag Fermat's. Still, that did not seem to make the task any easier. “Andrew Wiles is probably one of the few people on Earth who had the audacity to dream that he can actually go and prove this conjecture,” Ribet told the BBC in the 1996 documentary. Fermat's last theorem is also connected to another deep question in number theory called the abc conjecture, Rognes points out. Mathematician Shinichi Mochizuki of Kyoto University's Research Institute for Mathematical Sciences in Japan claimed to have proved that conjecture in 2012, although his roughly 500-page proof is still being vetted by his peers. Some mathematicians say that Mochizuki's work could provide, as an extra perk, an alternative way of proving Fermat, although Wiles says that sees those hopes with scepticism. Wiles helped to arrange an Oxford workshop on Mochizuki's work last December, although his research interests are somewhat different. Lately, he has focused his efforts on another major, unsolved conjecture in number theory, which has been listed as one of seven Millennium Prize problems posed by the Clay Mathematics Institute in Oxford, UK. He still works very hard and thinks about mathematics for most of his waking hours, including as he walks to the office in the morning. “He doesn’t want to cycle,” Bridson says. “He thinks it would be a bit dangerous for him to do it while thinking about mathematics.”

Fritz T.,Max Planck Institute for Mathematics
New Journal of Physics | Year: 2010

We consider a temporal version of the Clauser-Horne-ShimonyHolt (CHSH) scenario using projective measurements on a single quantum system. It is known that quantum correlations in this scenario are fundamentally more general than correlations obtainable with the assumptions of macroscopic realism and non-invasive measurements. In this paper, we also educe some fundamental limitations of these quantum correlations. One result is that a set of correlators can appear in the temporal CHSH scenario if and only if it can appear in the usual spatial CHSH scenario. In particular, we derive the validity of the Tsirelson bound and the impossibility of Popescu-Rohrlich (PR)-box behavior. The strength of possible signaling also turns out to be surprisingly limited, giving a maximal communication capacity of approximately 0.32 bit. We also find a temporal version of Hardy's non-locality paradox with a maximal quantum value of 1/4. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

Jurco B.,Max Planck Institute for Mathematics
International Journal of Geometric Methods in Modern Physics | Year: 2011

We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. If (L → M → N) is a Lie 2-crossed module and Y → X is a surjective submersion then an (L → M → N)-bundle 2-gerbe over X is defined in terms of a so-called (L → M → N)-bundle gerbe over the fiber product Y[2] = Y × XY, which is an (L → M)-bundle gerbe over Y[2] equipped with a trivialization under the change of its structure crossed module from L → M to 1 → N, and which is subjected to further conditions on higher fiber products Y[3], Y[4] and Y[5]. String structures can be described and classified using 2-crossed module bundle 2-gerbes. © 2011 World Scientific Publishing Company.

Michalek M.,Polish Academy of Sciences | Michalek M.,Max Planck Institute for Mathematics
Journal of Combinatorial Theory. Series A | Year: 2013

Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals defining the varieties. Conjectural bounds on the degree in which these ideals are generated were given by Sturmfels and Sullivant (2005) [25, Conjectures 29, 30]. We prove that for the 3-Kimura model, corresponding to the group G=Z2×Z2, the projective scheme can be defined by an ideal generated in degree 4. In particular, it is enough to consider degree 4 phylogenetic invariants to test if a given point belongs to the variety. We also investigate G-models, a generalization of abelian group-based models. For any G-model, we prove that there exists a constant d, such that for any tree, the associated projective scheme can be defined by an ideal generated in degree at most d. © 2013.

Marcolli M.,California Institute of Technology | Paolucci A.M.,Max Planck Institute for Mathematics
Complex Analysis and Operator Theory | Year: 2011

We consider representations of Cuntz-Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron-Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets. © 2009 The Author(s).

Manschot J.,Max Planck Institute for Mathematics | Manschot J.,University of Bonn
Communications in Number Theory and Physics | Year: 2012

Generating functions of BPS invariants for N = 4 U(r) gauge theory on a Hirzebruch surface with r ≤ 3 are computed. The BPS invariants provide the Betti numbers of moduli spaces of semistable sheaves. The generating functions for r = 2 are expressed in terms of higher level Appell functions for a certain polarization of the surface. The level corresponds to the self-intersection of the base curve of the Hirzebruch surface. The non-holomorphic functions are determined, which added to the holomorphic generating functions provide functions, which transform as a modular form.

Manschot J.,Max Planck Institute for Mathematics | Manschot J.,University of Bonn
Letters in Mathematical Physics | Year: 2013

Bogomolnyi-Prasad-Sommerfield (BPS) invariants are computed, capturing topological invariants of moduli spaces of semi-stable sheaves on rational surfaces. For a suitable stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch surface Σℓ takes the form of a product formula. BPS invariants for other stability conditions and other rational surfaces are obtained using Harder-Narasimhan filtrations and the blow-up formula. Explicit expressions are given for rank ≤ 3 sheaves on Σℓ and the projective plane ≤2. The applied techniques can be applied iteratively to compute invariants for higher rank. © 2013 Springer Science+Business Media Dordrecht.

Martius G.,Max Planck Institute for Mathematics | Der R.,Max Planck Institute for Mathematics | Ay N.,Max Planck Institute for Mathematics | Ay N.,Santa Fe Institute
PLoS ONE | Year: 2013

Information theory is a powerful tool to express principles to drive autonomous systems because it is domain invariant and allows for an intuitive interpretation. This paper studies the use of the predictive information (PI), also called excess entropy or effective measure complexity, of the sensorimotor process as a driving force to generate behavior. We study nonlinear and nonstationary systems and introduce the time-local predicting information (TiPI) which allows us to derive exact results together with explicit update rules for the parameters of the controller in the dynamical systems framework. In this way the information principle, formulated at the level of behavior, is translated to the dynamics of the synapses. We underpin our results with a number of case studies with high-dimensional robotic systems. We show the spontaneous cooperativity in a complex physical system with decentralized control. Moreover, a jointly controlled humanoid robot develops a high behavioral variety depending on its physics and the environment it is dynamically embedded into. The behavior can be decomposed into a succession of low-dimensional modes that increasingly explore the behavior space. This is a promising way to avoid the curse of dimensionality which hinders learning systems to scale well. © 2013 Martius et al.

Sakovich S.,Max Planck Institute for Mathematics
Journal of Mathematical Physics | Year: 2011

We find a transformation which relates a new third-order integrable nonlinear evolution equation, introduced recently by Qiao, with the well-known modified Korteweg-de Vries equation. Then we use this transformation to derive smooth soliton solutions of the new equation from the known rational and soliton solutions of the old one. © 2011 American Institute of Physics.

Levin A.,Max Planck Institute for Mathematics | Olshanetsky M.,Max Planck Institute for Mathematics | Smirnov A.,Institute of Theoretical and Experimental Physics | Zotov A.,Institute of Theoretical and Experimental Physics
Communications in Mathematical Physics | Year: 2012

We consider topologically non-trivial Higgs G-bundles over Riemann surfaces Σ g with marked points and the corresponding Hitchin systems. We show that if G is not simply-connected, then there exists a finite number of different sectors of the Higgs bundles endowed with the Hitchin Hamiltonians. They correspond to different characteristic classes of the underlying bundles defined as elements of H 2(Σ g, Z{stroke}(G)), (Z{stroke}(G)) is a center of G). We define the conformal version CG of G - an analog of GL(N) for SL(N), and relate the characteristic classes with degrees of CG-bundles. We describe explicitly bundles in the genus one (g=1) case. If Σ 1 has one marked point and the bundles are trivial then the Hitchin systems coincide with Calogero-Moser (CM) systems. For the nontrivial bundles we call the corresponding systems the modified Calogero-Moser (MCM) systems. Their phase space has the same dimension as the phase space of the CM systems with spin variables, but less number of particles and greater number of spin variables. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, and define the phase spaces and the Poisson structure using dynamical r-matrices. The latter are completion of the classification list of Etingof-Varchenko corresponding to the trivial bundles. To describe the systems we use a special basis in the Lie algebras that generalizes the basis of 't Hooft matrices for sl(N). We find that the MCM systems contain the standard CM subsystems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the stable holomorphic bundles with non-trivial characteristic classes. © 2012 Springer-Verlag Berlin Heidelberg.

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