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Vol 75, No 5 (2020)
- Year: 2020
- Articles: 11
- URL: https://ogarev-online.ru/0042-1316/issue/view/7518
Yang–Baxter algebras, convolution algebras, and Grassmannians
Abstract
This paper surveys a new actively developing direction in contemporary mathematics which connects quantum integrable models with the Schubert calculus for quiver varieties: there is a purely geometric construction of solutions to the Yang–Baxter equation and their associated Yang–Baxter algebras which play a central role in quantum integrable systems and exactly solvable (integrable) lattice models in statistical physics. A simple but explicit example is given using the classical geometry of Grassmannians in order to explain some of the main ideas. The degenerate five-vertex limit of the asymmetric six-vertex model is considered, and its associated Yang–Baxter algebra is identified with a convolution algebra arising from the equivariant Schubert calculus of Grassmannians. It is also shown how our methods can be used to construct quotients of the universal enveloping algebra of the current algebra $\mathfrak{gl}_2[t]$ (so-called Schur-type algebras) acting on the tensor product of copies of its evaluation representation $\mathbb{C}^2[t]$. Finally, our construction is connected with the cohomological Hall algebra for the $A_1$-quiver.Bibliography: 125 titles.
Uspekhi Matematicheskikh Nauk. 2020;75(5):3-58
3-58
Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes
Abstract
This survey presents results on the stability of elevation solitary waves in axisymmetric elastic membrane tubes filled with a fluid. The elastic tube material is characterized by an elastic potential (elastic energy) that depends non-linearly on the principal deformations and describes the compliant elastic media. Our survey uses a simple model of an inviscid incompressible fluid, which nevertheless makes it possible to trace the main regularities of the dynamics of solitary waves. One of these regularities is the spectral stability (linear stability in form) of these waves. The basic equations of the ‘axisymmetric tube – ideal fluid’ system are formulated, and the equations for the fluid are averaged over the cross-section of the tube, that is, a quasi-one-dimensional flow with waves whose length significantly exceeds the radius of the tube is considered. The spectral stability with respect to axisymmetric perturbations is studied by constructing the Evans function for the system of basic equations linearized around a solitary wave type solution. The Evans function depends only on the spectral parameter $\eta$, is analytic in the right-hand complex half-plane $\Omega^+$, and its zeros in $\Omega^+$ coincide with unstable eigenvalues. The problems treated include stability of steady solitary waves in the absence of a fluid inside the tube (the case of constant internal pressure), together with the case of local inhomogeneity (thinning) of the tube wall, the presence of a steady fluid filling the tube (the case of zero mean flow) or a moving fluid (the case of non-zero mean flow), and also the problem of stability of travelling solitary waves propagating along the tube with non-zero speed.Bibliography: 83 titles.
Uspekhi Matematicheskikh Nauk. 2020;75(5):59-100
59-100
Adjunction in 2-categories
Abstract
The aim of the paper is to introduce an approach to the theory of 2-categories which is based on systematic use of the Grothendieck construction and the Segal Machine and to show how adjunction questions can be investigated by means of this approach and what its connections are with more traditional approaches. As an application, the derived Morita 2-category and the Fourier–Mukai 2-category over a Noetherian ring are constructed and the embedding of the latter in the former is demonstrated.Bibliography: 15 titles.
Uspekhi Matematicheskikh Nauk. 2020;75(5):101-152
101-152
Fenchel–Nielsen coordinates and Goldman brackets
Abstract
It is explicitly shown that the Poisson bracket on the set of shear coordinates defined by V. V. Fock in 1997 induces the Fenchel–Nielsen bracket on the set of gluing parameters (length and twist parameters) for pair-of-pants decompositions of Riemann surfaces $\Sigma_{g,s}$ with holes. These structures are generalized to the case of Riemann surfaces $\Sigma_{g,s,n}$ with holes and bordered cusps.Bibliography: 49 titles.
Uspekhi Matematicheskikh Nauk. 2020;75(5):153-190
153-190
Ramsey theory in the $n$-space with Chebyshev metric
Uspekhi Matematicheskikh Nauk. 2020;75(5):191-192
191-192
Extreme value theory for triangular arrays of dependent random variables
Uspekhi Matematicheskikh Nauk. 2020;75(5):193-194
193-194
The spectral radius of a certain parametric family of functional operators
Uspekhi Matematicheskikh Nauk. 2020;75(5):195-196
195-196
Online algorithm for aggregating experts' predictions with unbounded quadratic loss
Uspekhi Matematicheskikh Nauk. 2020;75(5):197-198
197-198
Quantisation ideals of nonabelian integrable systems
Uspekhi Matematicheskikh Nauk. 2020;75(5):199-200
199-200
Anatolii Iserovish Neishtadt (on his 70th birthday)
Uspekhi Matematicheskikh Nauk. 2020;75(5):201-208
201-208
Caucasus Mathematical Olympiad
Uspekhi Matematicheskikh Nauk. 2020;75(5):209-211
209-211