Duality Theory and The Fractional Calculus of Variations (Delfim F. M. Torres)

Utilizing the duality theory of fractional calculus, originally introduced by Caputo and Torres in 2015, we are able to transform left fractional operators into right fractional operators and vice versa through duality [01].

In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or right operators.

The emergence of these novel fractional integration by parts formulas inspires the introduction of a new calculus of variations, where only one type of fractional derivative (left or right) is present. This applies to both the problem formulation and the corresponding necessary optimality conditions.

As a practical application, we present a new Lagrangian that relies solely on left-hand side fractional derivatives. The fractional variational principle derived from this Lagrangian leads us to the equation of motion for a dissipative/damped system [02].

[01] M. C. Caputo and D. F. M. Torres, Duality for the left and right fractional derivatives, Signal Process. 107 (2015), 265--271. https://doi.org/10.1016/j.sigpro.2014.09.026

[02] D. F. M. Torres, The duality theory of fractional calculus and a new fractional calculus of variations involving left operators only, Mediterr. J. Math. 21 (2024), no. 3, Paper No. 106, 16 pp. https://doi.org/10.1007/s00009-024-02652-x

Optimal domains for the Cheeger inequality (Giuseppe Buttazzo)

We study a generalized form of the Cheeger inequality by considering the shape functional 

where the original Cheeger case corresponds to p=2 and q=1. Here λp(Ω) denotes the principal eigenvalue of the Dirichlet p-Laplacian. The infimum and the supremum of Fp,q are discussed, together with the existence of optimal domains. Some open problems will be illustrated as well.

Transport problems related to the Hamilton-Jacobi equation (Konstantin Khanin)

Solutions to the Hamilton-Jacobi equation are obtained by using the Lax-Oleinik variational principle. 

The velocity field is generated by characteristics which are minimizers of the Lax-Oleinik action functional until they merge with shocks. The problem of constructing transport dynamics after merging with shocks can be positively solved in the case of inviscid Burgers equation corresponding to quadratic Hamiltonians. However, in the case of non-quadratic Hamiltonians the problem remains widely open. We shall discuss several approaches related to various regularization strategies.

One of them is based on viscosity regularization. The other one is related to adding a small noise. We shall discuss a different approach which involves adding both a small viscosity term and a properly calibrated white noise.  

Calculus of Variations on the Class of Convex Functions and Contact Geometry (Lev Lokutsievskiy)

Suppose we have a classical functional from the calculus of variations, with the goal of minimizing it among the set of convex functions. A well-known example is Newton's unsolved minimal resistance problem. Another set of one-dimensional examples arises from microeconomics.

Any such problem can be expressed in three equivalent forms: the initial form, the conjugate form, and the contact form. The talk is devoted to the contact form, which is expressed in the space of 1-jets and proves to be very helpful. For instance, any one-dimensional problem can be easily solved in the contact form using standard methods.

Finite Time Dynamics and Predictions for Strongly Chaotic Systems (Leonid Bunimovich)

In statistical theory of dynamical systems we are used to deal with asymptotic in time laws (various limit theorems where time lends to infinity). It seems to be the only natural approach because questions of a type how dynamics at time 17 differs from dynamics  at time 25 indeed sound (not a little) crazy.  For instance, in equilibrium statistical mechanics main questions essentially deal with phase transitions (a number of equilibrium states) rather than with evolution. It turned out that there are reasonable questions about a finite time transport in phase spaces of strongly chaotic systems, which could be answered. 

Extremal problems and monotone rearrangement on averaging classes (Pavel Zatitskii)

We will discuss integral extremal problems on the so-called averaging classes of functions, meaning classes defined in terms of averages of their elements, such as BMO, VMO, and Muckehoupt weights. A typical extremal problem we consider involves an integral inequality, such as the John--Nirenberg inequality for BMO. One common way to formulate such questions is using Bellman functions. It turns out that such Bellman functions are solutions to specific boundary value problems, formulated in terms of convex geometry. We will also discuss the monotone rearrangement operator acting on the averaging classes, which arises naturally in this context and is useful when solving extremal problems.

Integrable convex billiards (Michael Bialy)

I shall discuss the Birkhoff-Poritsky conjecture for classical billiards and possible generalizations to other billiard models. I shall also emphasize the open questions in this area.

Bicycling geodesics, elastic curves, and the filament equation (Sergei Tabachnikov)

Bicycle is modeled as a directed segment of a fixed length that can move so that the velocity of the rear end is always aligned with the segment. A bicycle path is a motion of the segment, subject to this nonholonomic constraint, and the length of the path, by definition, is the length of the front track. This defines a problem of sub-Riemannian geometry, and one wants to describe the respective geodesics. I shall discus three variations on this theme: the planar bicycle motion, the bicycle motion in multidimensional Euclidean space, and the planar motion of a 2-linkage (a tricycle?) Somewhat unexpectedly, these problems are closely related with the filament equation, a completely integrable system of soliton type, whose soliton curves appear as the sub-Riemannian geodesics.

How close are we to factorise an arbitrary nonsingular matrix function? (Gennady Mishuris)

The factorisation of arbitrary nonsingular matrix functions on the unit circle remains an open problem in the field of factorisation theory. Current theoretical results are limited to specific classes of matrix functions. Moreover, numerical methods face significant constraints due to the well-known instability of small perturbations in matrix functions unless the Gohberg-Krein-Bojarski criterion is satisfied. This criterion requires that the difference between the largest and smallest partial indices does not exceed one. Since no general method exists to construct partial indices, this has posed a significant barrier to numerical approaches, severely limiting the practical applications of this otherwise powerful technique. We discuss how recent advancements in the area address these challenges. We argue that, paradoxically, numerical approaches offer a path to resolving the longstanding factorization problem in its entirety.

L. Ephremidze, I. Spitkovsky. (2020) On explicit Wiener-Hopf factorization of 2 × 2 matrices in the vicinity of the given matrix. Proc. R. Soc. A 476 (2238): 20200027.

N.V. Adukova, V.M. Adukov, G. Mishuris. (2024) An effective criterion for a stable factorisation of strictly nonsingular 2 × 2 matrix functions. Utilisation of the ExactMPF package. Proc. R. Soc. A 480(2299): https://doi.org/10.1098/rspa.2024.0116

Geodesics on vibrating manifolds (Mark Levi)

I will give a geometrical explanation of the counterintuitive “Indian rope trick” which amounts to the stabilization of the inverted multiple pendulum caused by imposed vibration of the pivot point. The geometrical reason for this phenomenon becomes clear by treating the problem as a “geodesic” on a vibrating manifold.

Affine control problems with a chain recurrent drift (Sergey Kryzhevich)

Consider a family of smooth vector fields on a compact manifold satisfying the Hörmander condition: the Lie algebra generated by those vector fields spans the whole tangent space of the manifold. Take the flow, generated by one of those vector fields (suppose it is chain recurrent) and consider its small perturbation in the direction of linear combinations. The main question is the following: given initial conditions, can we attain any other point of the manifold, going along the trajectories of perturbed systems?

 We discuss this problem starting from the classical Chow - Rashevskii Theorem to the modern results, counterexamples, and open problems.

The joint spectral radius problem: convex geometric tools (Vladimir Protasov)

The concept of the joint spectral radius of matrices has been used since late 1980s  as a measure of stability of linear switching dynamical systems.  Later it has found important applications in the theory of  functional equations with a contraction of the argument, in combinatorics, number theory, fractals,   random power series, etc. However, the computation or even estimation of the joint spectral radius is  hard. It was shown by Blondel and Tsitsiklis in 2000 that this problem is in general algorithmically undecidable. Nevertheless, recent geometrical methods make it possible to efficiently estimate  this value or even to find it precisely for the vast majority of matrix families.  These methods are  based on the study of invariant convex functionals of dynamical systems.

Existence of maximum of time averaged harvesting in the KPP-model  with permanent and impulse harvesting (Alexey Davydov)

We consider a distributed renewable resource on a smooth closed manifold whose dynamics are provided by the Kolmogorov-Petrovskii-Piskunov and Fisher equation, including non-local one. Under natural constraints on the parameters of the equation and on periodic or impulsed harvesting of this resource, we show the existence of a  strategy that provides the maximum time averaged  benefit.

How to make convex sets (Vera Roshchina)

Convex sets can appear deceptively simple, however constructing convex sets with desired structure can be a challenging task. This challenge is exemplified by major open questions about the existence of convex sets with specific properties. Amongst the most famous are perhaps the polynomial Hirsch and generalised Lax conjectures; even in three-dimensions the Dürer conjecture about the existence of nonoverlapping unfoldings of polytopes, and the Carathéodory conjecture about the minimal number of umbilics on a sufficiently smooth convex set are still unresolved.

I will introduce some recent ideas that can help build convex sets with prescribed structure and demonstrate how these ideas are used to construct examples in conic optimisation.

The talk is based on recent collaborative work with Bruno Lourenço (Institute of Statistical Mathematics, Japan), James Saunderson (Monash University, Australia) and Levent Tunçel (University of Waterloo, Canada).

If a Minkowski Finsler billiard is projective, then it is a standard billiard (Alexey Glutsyuk)

We discuss two generalizations of standard Euclidean billiards:

1) Projective billiards, introduced by S.Tabachnikov in 1997, which generalize billiards on all the space forms of constant curvature and planar outer billiards.

2) Minkowski Finsler billiards, which were introduced in a joint paper by S.Tabachnikov and E.Gutkin in 2002.

We discuss two well-known conjectures on classical billiards:

- Ivrii’s Conjecture stating that the set of periodic orbits has measure zero;

- Birkhoff Conjecture stating that the only integrable planar billiards are ellipses.

We present their states of art for the projective billiards and state of Ivrii’s Conjecture for triangular orbits in Minkowski Finsler billiards.

We show that if in a Minkowski Finsler billiard the reflection law is projective, then it is a standard billiard up to affine transformation. This is a joint result with V.Matveev.

Straightedge without compass. A solution to Hilbert's problem (Arseniy Akopyan)

The theory of compass-and-straightedge construction goes back to ancient Greeks, it played a crucial role in the life of Carl Gauss, and was a core of mathematics up to the 20-the century. The accepted view was that the theory is complete and answers for all questions were found. But this is not the case. I will tell you about my joint work with Roman Fedorov on the solution of a problem posed by David Hilbert more than a century ago: given two non-intersecting circles, is it possible to construct their centers using only a straightedge? The answer is unexpected and the solution consists mostly from pictures and cartoons.