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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.

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

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. Source

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

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. Source

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

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. Source

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

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. Source

Sakovich S.,Max Planck Institute for Mathematics
Journal of Mathematical Physics

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. Source

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