Weingarten surfaces arising from soliton theory

Ankara : Department of Mathematics and Institute of Engineering and Sciences, Bilkent University, 1999.Thesis (Master's) -- Bilkent University, 1999.Includes bibliographical references leaves 43-45.In this work we presented a method for constructing surfaces in associated with the symmetries of Gauss-Mainardi-Codazzi equations. We show that among these surfaces the sphere has a unique role. Under constant gauge transformations all integrable equations are mapped to a sphere. Furthermore we prove that all compact surfaces generated by symmetries of the sine-Gordon equation are homeomorphic to sphere. We also construct some Weingarten surfaces arising from the deformations of sine-Gordon, sinh-Gordon, nonlinear Schrödinger and modified Korteweg-de Vries equations.Ceyhan, ÖzgürM.S

Feynman Integrals and Motives of Configuration Spaces

We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications Conf Γ(X) of arrangements of subvarieties associated to the subgraphs of a Feynman graph Γ, with X a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of X and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful compactification of the form defined by the unrenormalized Feynman amplitude has singularities along a hypersurface, whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification. A regularization of the Feynman integrals can be obtained by modifying the cycle of integration, by replacing the divergent locus with a Leray coboundary. The ambiguities are then defined by Poincaré residues. While these residues give periods associated to the cohomology of the exceptional divisors and their intersections, the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification

Tropical Backpropagation

This work introduces tropicalization, a novel technique that delivers tropical neural networks as tropical limits of deep ReLU networks. Tropicalization transfers the initial weights from real numbers to those in the tropical semiring while maintain- ing the underlying graph of the network. After verifying that tropicalization will not affect the classification capacity of deep neural networks, this study introduces a tropical reformulation of backpropagation via tropical linear algebra. Tropical arithmetic replaces multiplication operations in the network with additions and addition operations with max, and therefore, theoretically, reduces the algorithmic complexity during the training and inference phase. We demonstrate the latter by simulating the tensor multiplication underlying the feed-forward process of state- of-the-art trained neural network architectures and compare the standard forward pass of the models with the tropical ones. Our benchmark results show that tropi- calization speeds up inference by 50 %. Hence, we conclude that tropicalization bears the potential to reduce the training times of large neural networks drastically

Open string theory and planar algebras

In this note we show that abstract planar algebras are algebras over the topological operad of moduli spaces of stable maps with Lagrangian boundary conditions, which in the case of the projective line are described in terms of real rational functions. These moduli spaces appear naturally in the formulation of open string theory on the projective line. We also show two geometric ways to obtain planar algebras from real algebraic geometry, one based on string topology and one on Gromov-Witten theory. In particular, through the well known relation between planar algebras and subfactors, these results establish a connection between open string theory, real algebraic geometry, and subfactors of von Neumann algebras.Comment: 13 pages, LaTeX, 7 eps figure

ALGEBRAIC RENORMALIZATION AND FEYNMAN INTEGRALS IN CONFIGURATION SPACES

Abstract. This paper continues our previous study of Feynman integrals in configuration spaces and their algebro-geometric and motivic aspects. We consider here both massless and massive Feynman amplitudes, from the point of view of potential theory. We consider a variant of the wonderful compactification of configuration spaces that works simultaneously for all graphs with a given number of vertices and that also accounts for the external structure of Feynman graph. As in our previous work, we consider two version of the Feynman amplitude in configuration space, which we refer to as the real and complex versions. In the real version, we show that we can extend to the massive case a method of evaluating Feynman integrals, based on expansion in Gegenbauer polynomials, that we investigated previously in the massless case. In the complex setting, we show that we can use algebro-geometric methods to renormalize the Feynman amplitudes, so that the renormalized values of the Feynman integrals are given by periods of a mixed Tate motive. The regularization and renormalization procedure is based on pulling back the form to the wonderful compactification and replace it with a cohomologous one with logarithmic poles. A complex of forms with logarithmic poles, endowed with an operator of pole subtraction, determine a Rota–Baxter algebra on the wonderful compactifications. We can then apply the renormalization procedure via Birkhoff factorization, after interpreting the regularization as an algebra homomorphism from the Connes–Kreimer Hopf algebra of Feynman graphs to the Rota–Baxter algebra. We obtain in this setting a description of the renormalization group. We also extend the period interpretation to the case of Dirac fermions and gauge bosons. Content