Search results for "math.MP"
showing 10 items of 115 documents
Instanton Counting, Quantum Geometry and Algebra
2020
The aim of this memoir for "Habilitation \`a Diriger des Recherches" is to present quantum geometric and algebraic aspects of supersymmetric gauge theory, which emerge from non-perturbative nature of the vacuum structure induced by instantons. We start with a brief summary of the equivariant localization of the instanton moduli space, and show how to obtain the instanton partition function and its generalization to quiver gauge theory and supergroup gauge theory in three ways: the equivariant index formula, the contour integral formula, and the combinatorial formula. We then explore the geometric description of $\mathcal{N} = 2$ gauge theory based on Seiberg-Witten geometry together with it…
Integrating over quiver variety and BPS/CFT correspondence
2019
We show the vertex operator formalism for the quiver gauge theory partition function and the $qq$-character of highest-weight module on quiver, both associated with the integral over the quiver variety.
Hysteresis Model of Unconscious-Conscious Interconnection: Exploring Dynamics on m-Adic Trees
2015
The theoretical model outlined in this paper, has been experimentally validated by: H. Kim ,J-Y. Moon ,G.A. Mashour & U. Lee, ''Mechanisms of hysteresis in human brain networks during transitions of consciousness and unconsciousness: Theoretical principles and empirical evidence'', PLOS-Computational Biology, August 30, 2018, https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1006424; International audience; In this brief note, we focus attention on a possible implementation of a basic hysteretic pattern (the Preisach one), suitably generalized, into a formal model of unconscious-conscious interconnection and based on representation of mental entities by m-adic numbers. …
On the semiclassical limit of the defocusing Davey-Stewartson II equation
2018
Inverse scattering is the most powerful tool in theory of integrable systems. Starting in the late sixties resounding great progress was made in (1+1) dimensional problems with many break-through results as on soliton interactions. Naturally the attention in recent years turns towards higher dimensional problems as the Davey-Stewartson equations, an integrable generalisation of the (1+1)-dimensionalcubic nonlinear Schrödinger equation. The defocusing Davey-Stewartson II equation, in its semi-classical limit has been shown in numerical experiments to exhibit behavior that qualitatively resembles that of its one-dimensional reduction, namely the generation of a dispersive shock wave: smooth i…
Families of rational solutions to the KPI equation of order 7 depending on 12 parameters
2017
International audience; We construct in this paper, rational solutions as a quotient of two determinants of order 2N = 14 and we obtain what we call solutions of order N = 7 to the Kadomtsev-Petviashvili equation (KPI) as a quotient of 2 polynomials of degree 112 in x, y and t depending on 12 parameters. The maximum of modulus of these solutions at order 7 is equal to 2(2N + 1)2= 450. We make the study of the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6. When all these parameters grow, triangle and ring structures are obtained.
Rational solutions to the KPI equation of order 7 depending on 12 parameters
2018
We construct in this paper, rational solutions as a quotient of two determinants of order 2N = 14 and we obtain what we call solutions of order N = 7 to the Kadomtsev-Petviashvili equation (KPI) as a quotient of 2 polynomials of degree 112 in x, y and t depending on 12 parameters. The maximum of modulus of these solutions at order 7 is equal to 2(2N + 1) 2 = 450. We make the study of the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6. When all these parameters grow, triangle and ring structures are obtained.
Rational solutions to the KdV equation depending on multi-parameters
2021
We construct multi-parametric rational solutions to the KdV equation. For this, we use solutions in terms of exponentials depending on several parameters and take a limit when one of these parameters goes to 0. Here we present degenerate rational solutions and give a result without the presence of a limit as a quotient of polynomials depending on 3N parameters. We give the explicit expressions of some of these rational solutions.
Degenerate Riemann theta functions, Fredholm and wronskian representations of the solutions to the KdV equation and the degenerate rational case
2021
International audience; We degenerate the finite gap solutions of the KdV equation from the general formulation given in terms of abelian functions when the gaps tend to points, to get solutions to the KdV equation given in terms of Fredholm determinants and wronskians. For this we establish a link between Riemann theta functions, Fredholm determinants and wronskians. This gives the bridge between the algebro-geometric approach and the Darboux dressing method.We construct also multi-parametric degenerate rational solutions of this equation.
Empirical measures and Vlasov hierarchies
2013
The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent 1 on the i…
The geodesic X-ray transform with matrix weights
2019
Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary, and a detailed analysis of layer stripping arguments ba…