Past seminars

Abstract: Appeals to randomness in number-theoretic constructions are common in modern scientific literature. Well-known names such as V.I. Arnold, M. Katz, Yu.G. Sinai, and T. Tao have all contributed to this field. Unfortunately, these approaches rely on various heuristics, often non-trivial and elegant. A new analytical approach has been proposed to address the question of randomness and complexity in individual deterministic sequences. The approach demonstrates the expected high complexity of quadratic residues, as well as the unexpectedly low complexity of primes. Technically, this approach is based on a new construction of the entropy of a single trajectory of a dynamical system. This entropy occupies an intermediate position between classical metric Kolmogorov-Sinai entropy and topological entropy. All necessary definitions will be provided during the presentation.

Abstract: The topic of the presentation is the connection between the Large Deviation Principle (LDP) for the invariant measure of a random process and the LDP for the same process in the space of trajectories. It is shown that if the trajectory action functional has a certain structure and the family of invariant measures is exponentially tight, then the LDP for invariant measures follows from the trajectory LDP, regardless of other properties of the random process. The action functional for the invariant measure is characterized in terms of the solution to the max-balance equations, which arise as a limit in the sense of large deviations of the equilibrium equations for the invariant measure. The non-uniqueness of the equilibrium position of the corresponding dynamic system is allowed. As an application, the LDP for the invariant measure of a diffusion process with jumps is considered.

Abstract: We consider diffusion processes associated with forward and backward Cauchy problem for various types of nonlinear parabolic equations. First, we consider the backward Cauchy problem and construct the required processes as solutions of certain forward-backward stochastic differential equations. Given FBSDEs we state conditions that allow to prove the existence and uniqueness of their solutions and state connections with the original problem. To construct approximate numerical solution of the FBSDE we use the possibility to reduce the FBSDE to a correspondent stochastic control problem. Finally, to solve this stochastic control problem we apply the technique of neural networks. This approach allows to deal with PDEs in spaces of large dimensions.

Next we consider the forward Cauchy problem for a nonlinear PDE and discuss some possibilities to reduce it to the backward Cauchy problem in order to apply the previous results to it. As an example we apply the above approach to construction of an optimal portfolio in the Heston model. 

Abstract: Polymeric microgels are typically understood as "soft" colloidal particles with a mesh structure, swollen in a solvent. Their size ranges from several tens of nanometers to several microns. Polymeric microgels can serve as a soft, permeable, and responsive to external stimuli alternative to solid colloidal particles for stabilizing water-oil emulsions. The physical reason for the adsorption of microgels at the interface of two immiscible liquids is the screening of unfavorable "oil-water" contacts by adsorbed subchains. In other words, adsorption leads to a reduction in the surface tension between the liquids. The ability of adsorbed microgels to mix two immiscible liquids is demonstrated. It is also shown that in compositionally asymmetric mixtures (oil as the minor component), microgels can absorb oil, with the concentration of oil inside the microgels being orders of magnitude higher than outside. Therefore, microgels can serve as absorbers and concentrators of liquids dissolved in water. The report also discusses the peculiarities of microgel adsorption on solid and porous surfaces. 

Abstract: The presentation discusses linear stochastic control systems with the assumption that their coefficients depend on time. These systems correspond to models of processes from various application areas (physical, financial-economic, engineering, etc.). In this context, the objective functionals have an integral quadratic form. An analysis of control tasks over an infinite time interval is conducted based on the use of criteria that generalize known ergodic criteria for long-term averages. Specific classes of control systems are examined as examples: systems with a variable diffusion matrix, various types of discounting in the objective functional, and stochastic time scales.

Abstract: Leibniz defined the integral as the "sum of an infinitely large number of infinitely small quantities." In this presentation, we will explain how to correctly define the Leibniz integral sum when we consider "quantity" to refer to a class of asymptotically equivalent sequences, following the principle of comparison: if all the terms of the first integral sum do not exceed their corresponding terms in the second, then the first integral sum does not exceed the second. The integral of a function with respect to the Lebesgue measure, defined through Leibniz integral sums generated by increasingly finer sequences of partitioning intervals, coincides with the Kurzweil-Henstock and Denjoy-Perron integrals. The Leibniz integral for an interval in the form of Stieltjes-type f(x)dg(x) is defined for any functions of finite variation, even in the presence of common points of discontinuity. This is the case when Stieltjes integral sums do not have a limit.

Abstract: The talk will provide a general overview and key concepts of polymer physics, its connection with classical problems of statistical physics and mathematical statistics. A brief excursion into the short history of polymer science development will be given, with the first mention of long chain molecular structures dating back to 1922. I will also present the view of polymer physics on the most complex polymer systems – biomacromolecules (proteins, DNA/RNA). Among a number of current topics in polymer physics, the following will be briefly covered:

• Reconstruction of chromatin conformations based on a known contact map, "folding" problems, and designing protein sequences;

• The concept of entanglements in polymer systems and the problem of creating oriented fibers;

• "Active matter" and the peculiarities of polymers made of active units.