Mathematical and Theoretical Physics, dedicated to Ludwig Faddeev

28 May 2024, 09:45 → 31 May 2024, 18:30 Europe/Moscow

PDMI/--- - 311 (PDMI)

       

Ludwig Dmitrievich Faddeev. 23 March 1934 -26 February 2017 Ludwig Dmitrievich Faddeev for RAN.pdf Scientific heritage of Ludwig Faddeev

Reports

• Andrei K. Pogrebkov__Quantum aspect of the classical dynamics.pdf
• Nikolay M. Bogoliubov, Cyril Malyshev__ Correlation Functions of Heisenberg XX Chain and.pdf
• Sergey Derkachov__Calculation of multiloop Feynman diagrams and diagonalization.pdf
• Tatiana Suslina__Homogenization of the Periodic Schrödinger-type Equations.pdf
• Victor M. Buchstaber__Lie algebras of multidimensional Schrödinger operators.pdf
• Vladimir Korepin and Kun Hao__Transcendental Numbers in Quantum Spin Chains.pdf

Tuesday, 28 May

09:45 → 10:15 Registration/coffee

10:15 → 10:30 Opening

10:30 → 11:30 Об одной проблеме, возникающей при вычислении корреляционных функций для восьмивершинной модели

Работа о восьмивершинной модели, написанная Л.Д.Фаддеевым совместно с Л.А.Тахтаджяном, служит важнейшим источником квантового метода обратной задачи. В частности, результаты этой работы использовались Е.К.Скляниным для определения его знаменитой алгебры. Разработанный в результате математический аппарат мы использовали , совместно с Боосом, Джимбо и Мивой, для вычисления корреляционных функций. В нашем подходе их вычисление использует следы над алгеброй Склянина, алгоритмическое вычисление которых до сих пор не до конца понятно.
Цель настоящего доклада состоит в привлечении внимания к этой интересной , на мой взгляд, проблеме.

Speaker: Fedor Smirnov (LPTHE)

11:30 → 12:00 Coffee break

12:00 → 13:00 New linearization formula for q-ultraspherical polynomials

For the product Cm(x; β|q)Cn(x; β′|q) of two continuous q-ultraspherical polynomials with different parameters β and β′, explicit formulae are presented for the coefficients of its expansion in the basis Ck(x; β|q) in the case β′ = qβ−1. We also discuss a generalization to multivariate Jack and Macdonald polynomials and its relation to Pieri formulas and Q-operators.

Speaker: Evgeny Sklyanin (UNiversity of York, UK)

13:00 → 15:00 Lunch

15:00 → 16:00 Applications of the quantum dilogarithm

The quantum dilogarithm is a special function of two variables that finds various applications, including quantum topology and lattice integrable models of quantum field theory and statistical mechanics. Although a special case of that function was introduced in 1886 by Hölder, its deep connections to quantum world were revealed only in early 1990's after the discovery of the quantum five term identity by Ludwig Faddeev. I will review some of the properties of the quantum dilogarithm, its generalizations to the context of locally compact Abelian groups, and applications in spectral theory, quantum integrable systems and quantum topology.

Speaker: Rinat Kashaev (Geneva U., Switzerland)

16:00 → 16:30 Coffee break

16:30 → 17:30 Zamolodchikov-Faddeev algebra in Ruijsenaars hyperbolic model

Ruijsenaars hyperbolic model was discovered in the mid-1980s as a many-body interacting system describing solitons trajectories for the famous sine-Gordon equation. The Hamiltonians of the quantum problem are multi-dimensional difference operators with shifts of coordinates in imaginary direction, and in 2012 Hallnäs and Ruijsenaars managed to explicitly construct their eigenfunctions. Recently in the joint work with Derkachov, Kharchev and Khoroshkin we proved several properties of these eigenfunctions, including orthogonality, completeness and some symmetries. In the talk I will discuss the main tools we used and how they are related to the famous Zamolodchikov-Faddeev algebra.

Speaker: Nikita Belousov (Steklov Mathematical Institute, St. Petersburg)

17:30 → 18:30 Homogenization of the periodic Schrödinger-type equations

In L_2 (R^d;C^n ), we consider a selfadjoint strongly elliptic second-order differential operator A_ε. It is assumed that the coefficients of A_ε are periodic and depend on x/ε, where ε > 0 is a small parameter. We study the behavior of the operator exponential e^(-iA_ε τ) for small ε and τ∈ R. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation i∂_τ u_ε (x,τ) = (A_ε u_ε )(x,τ) with the initial data from a special class.

Speaker: Tatiana Syslina (St. Petersburg State University)

18:30 → 20:25 Welcome Party

Wednesday, 29 May

10:30 → 11:30 On stratified structure of integrable dynamics

Speaker: Nicolai Reshetikhin (Tsinghua University)

11:30 → 12:00 Coffee break

12:00 → 13:00 Calculation of multiloop Feynman diagrams and diagonalization of commuting operators

We consider calculation of two families of multiloop Feynman diagrams: zig-zag diagrams and Basso-Dixon diagrams. The first family gives contribution to the beta-function of standard ϕ4 theory, and the second family appear as leading contribution in fishnet field theory introduced by V.Kazakov and O.Gurdogan. The problem of calculation of such type of diagrams can be reformulated as problem of diagonalization of some family of commuting operators. Diagonalization can be performed using methods of quantum integrable systems.

Speaker: Sergey Derkachev (PDMI , St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences)

13:00 → 15:00 Lunch

15:00 → 16:00 Can one hear the shape of the Archimedean prime?

Speaker: Alain Connes (College de France)

16:00 → 16:30 Coffee break

16:30 → 17:30 Scale Invariance, Conformal Invariance, and Ricci Flow

Speaker: Edward Witten (IAS, Princeton)

17:30 → 18:30 Transcendental Numbers in Spin Chains

Speaker: Vladimir Korepin (Stony Brook University)

Thursday, 30 May

10:30 → 11:30 Обобщения гамильтоновой динамики Дирака

Speaker: Valery Kozlov (Steklov Mathemarical Institute, Moscow)

11:30 → 12:00 Coffee break

12:00 → 13:00 Скрученное тензорное произведение, гладкие алгебры и некоммутативное разрешение особых кривых

Speaker: Dmitri Orlov (Steklov Mathemarical Institute, Moscow)

13:00 → 15:00 Lunch

15:00 → 16:00 Teichmuller spaces, cocycles, and second class constraints

Speaker: Anton Alekseev (University of Geneva)

16:00 → 16:30 Coffee break

16:30 → 17:30 Wronski map and positive Grassmannians

The B. and M.Shapiro conjecture states that if the Wronskian of polynomials with complex coefficients have only real roots, then the span of those polynomials has a basis given by polynomials with real coefficients. The conjecture has several important reformulations in real algebraic geometry. The conjecture was proved (also for quasi-exponentials, products of polynomials and exponentials of linear functions) by E.Mukhin, A.Varchenko and myself using the completeness of the Bethe ansatz for the gl(N) Gaudin model and the symmetry of the Gaudin Hamiltonians with respect to the tensor Shapovalov form.
Recently, S.Karp and K.Purbhoo showed that the span of polynomials in question belongs to the totally positive part of the Grassmannian define by the Taylor expansion at a real point greater than all roots of the Wronskian. In a joint work with S.Karp and E.Mukhin we extended this statement to quasi-exponentials and showed that this positivity statement corresponds to the positivity of higher transfer-matrices of Gaudin model corresponding to polynomial irreducible representations of gl(N) introduced by A.Alexandrov, S.Leurent, Z.Tsuboi, and A.Zabrodin.

Speaker: Vitaliy Tarasov (Steklov Mathematical Institute, St. Petersburg)

17:30 → 18:30 Combinatorial 2d topological conformal field theory from a local cyclic A-infinity algebra

I will explain a construction of a combinatorial 2d TCFT, assigning partition functions to triangulated cobordisms (as chain maps between spaces of states), in such a way that a Pachner flip induces a Q-exact change. More generally, the partition function becomes a nonhomogeneous closed cochain on the “flip complex.” One has a combinatorial counterpart of the BV operator G_{0,-} arising from evaluating the theory on a special 1-cycle on the flip complex of the cylinder. The local input for the model is a cyclic A-infinity algebra, with the operation m_3 playing the role of the BRST-primitive G of the stress-energy tensor T=Q(G).

I will also describe a way to incorporate invariance-up-to-homotopy with respect to the second 2d Pachner move (stellar subdivision/aggregation). This version of the model is based on secondary polytopes of Gelfand-Kapranov-Zelevinsky and uses a certain enhancement (by extra homotopies) version of a cyclic A-infinity algebra as input

The talk is based on a joint work with Andrey Losev and Justin Beck, https://arxiv.org/pdf/2402.04468.pdf.

Speaker: Pavel Mnev (University of Notre Dame)

19:00 → 22:00 Banquet

Friday, 31 May

10:30 → 11:30 Lie algebras of multidimensional Schrodinger operators

Speaker: Viktor Buchstaber (Steklov Mathemarical Institute, Moscow)

11:30 → 12:00 Coffee break

12:00 → 13:00 Correlation Functions of Heisenberg XX Chain and Enumeration of Constrained Plane Partitions

Speakers: Nikolai Bogoliubov (Steklov Mathematical Institute, St. Petersburg), Сyril Malyshev (Steklov Mathematical Institute, St. Petersburg)

13:00 → 15:00 Lunch

15:00 → 16:00 Quantum effects in classical dynamics

Speaker: Andrei Pogrebkov (Steklov Mathematical Insitute; Skoltech)

16:00 → 16:30 Coffee break

16:30 → 17:30 On Determinants of Integrable Operators with Shifts

In the talk, some recent results of the asymptotic analysis of the Fredholm determinants of integral operators appearing in the study of the correlation functions in non-free fermion exactly solvable quantum field models will be presented. Specially, we shall consider the shifted sine-kernel as a case study. The talk is based on the earlier works of the speaker with V. Korepin, A. Izergin, N. Slavnov and K. Koslowski and on the ongoing project with T. Bothner, A. Simon and K. Kozlowski.

Speaker: Alexander Its (Indiana Universitry Indianapolis)

17:30 → 18:30 On a cutoff regularization in the coordinate representation

Speaker: Aleksandr Ivanov (Steklov Mathematical Institute, St. Petersburg)