p-adic mathematics has found broad applications in theoretical physics and biology, attracting considerable interest in areas such as quantum mechanics, string theory, quantum gravity, spin-glass theory, and systems biology. One of its key applications is describing hierarchical energy landscapes, which play a central role in understanding relaxation phenomena in complex systems, including glasses, clusters, and proteins. 

After introducing the notions of symmetry and differential equations, we review the possibilities of symmetry methods and highlight their advantages in the theory of differential equations and mathematical physics. As an illustrative example, we discuss the history of the system of Nyzhnyk equations and its variety of models and its applications and provide an overview of its extended symmetry analysis developed in our work. In particular, we focus on the dispersionless Nyzhnyk equation. Namely, we construct essential megaideals of the maximal Lie invariance algebra of this equation. Using the original version of the algebraic megaideals-based method, we compute the point- and contact-symmetry pseudogroups of this equation as well as the point-symmetry pseudogroups of its Lax representation and the original real symmetric dispersionless Nizhnik system. This is the first example in the literature, where there is no need to use the direct method for completing the computation. We also discuss geometric properties of the dispersionless Nyzhnyk equation that completely define it. Lie reductions of this equation are classified, which results in wide families of its new closed-form invariant solutions. In addition, we present results on hidden generalized symmetries, hidden cosymmetries and hidden conservation laws of this equation.

It is well-known that the classical resolution of singularities, even for curves, has some disagreeable features when one considers its effects on sheaves and especially on derived categories. Kuznetsov has proposed the notion of categorical resolution which should repair the situation. Analogous question is of importance in the theory of rings, algebras and modules, as M. Auslander noted.
We consider an approach to categorical resolution of singularities of rings and schemes (including non-commutative) based on the notion of minors and recollements (or bilocalizations). It provides a categorical resolution together with its semi-orthogonal decomposition. The latter gives estimates for the Rouquier dimension of the resolution and, hence, to that of the original derived category.
This approach gives an immediate result in 1-dimensional case, as well as for several special rings (Bäckström rings, gentle algebras, nodal orders and curves). Moreover, it often allows to describe the structure of the corresponding derived categories and/or establish its tilting to finite dimensional algebras.

In this talk, I will present a modern perspective on the analysis of Painlevé equations. The Painlevé I equation is the simplest of the six nonlinear ordinary differential equations discovered by Paul Painlevé in 1902. For many decades, the Painlevé equations were regarded as objects of purely mathematical interest.
This situation changed in 1976, when physicists discovered that Painlevé equations describe the critical behavior of the Ising model, revealing deep connections between nonlinear differential equations and statistical mechanics. Since then, Painlevé equations have appeared in numerous areas of mathematical physics.
In the 21st century, new and unexpected links between Painlevé equations and quantum field theory have emerged. These developments have provided powerful new methods that benefit both mathematics and physics.
As an example, I will explain how the Holomorphic Anomaly Equation, arising in topological string theory, can be used to solve the Painlevé I equation in terms of modular forms. Remarkably, modular forms, classical objects originating in 19th-century mathematics and now central to modern number theory, naturally appear in the solutions.

During this talk, I will explicitly solve some important polynomial equations over integers modulo a prime. They are important because their solutions parametrize all geometric objects called 1-foliations. Interestingly, this quickly becomes solving explicit differential equations in algebra, not analysis. This computation is a part of my current research about geometry modulo p. Nevertheless, we won't use anything crazy; only polynomials, derivatives, and integers modulo p.

Miguel Alcubierre proposed a model of faster-than-light space travel in 1994 [1]. He imagined a scenario in which a traveler is locally at rest in a compact region of flat space-time, and around this region there is a distortion that moves the traveler at a velocity - relative to a distant observer - that exceeds the speed of light. While this is a very exciting prospect, since then many researchers have noted difficulties with such models - including construction, stability, and the exotic stress-energy required. The latter is often cited as "negative mass/energy", or more formally as "violation of the weak energy condition" - and have led to arguments [2] that models such as Alcubierre's are physically impossible. However, the no-go theorem presented in [2] assumes that the space-time is free of singularities. In general relativity, different space-times can be joined at co-dimension 1 boundaries. The result is generally a singularity that supports a thin membrane at that boundary. Such a scenario is already familiar to us as Brane-World Cosmology. In recent work [3], I constructed a model of faster-than-light space travel by joining regions of different space-time along thin membranes and proved that there is no violation of the weak energy condition. In technical terms, arguments such as [2] that models of superluminal warp drive require a violation of the weak energy condition are based upon the Landau-Raychaudhuri equation, which describes the change of the divergence of a congruence of null geodesics. The need for a positive contribution to the derivative of this divergence becomes a requirement that a particular contraction of the stress-energy tensor be negative. However, in my model the difference in extrinsic curvature across the boundary-membrane can yield a positive contribution to the Landau-Raychaudhuri equation, eliminating that requirement.

Mathematicians like to associate a "fundamental group" to a geometric object X (for instance an algebraic variety). Some of you know the (absolute) Galois group of a field, or the topological fundamental group of a well-behaved topological space. The idea is always that the group classifies all the "coverings" of X. In algebraic geometry, it is also possible to define fundamental groups, and a deep and difficult question in this context is the so-called "anabelian geometry": What geometric information about the object X does its fundamental group carry? Some results are known only in dimension zero (for a point) and in dimension 1 (for a curve).
After giving a very brief introduction to this subject, I will try to explain how a detour into some non-archimedean geometry can bring an interesting light to this topic, especially when the base field is algebraically closed. The talk will stay elementary, I will not assume the audience to be familiar with algebraic geometry, fundamental groups, anabelian theories nor with non-archimedean geometry. My goal will be to explain some general picture, and for this reason I will not enter into a lot of technical details.

Automatic sequences give rise to one of the most basic models of computation and have remarkable links to various areas of mathematics, including dynamics, algebra and logic. Many properties of these sequences have been extensively studied. In my talk, I will focus on the perspective of combinatorial number theory, or more specifically - higher order Fourier analysis. Together with Jakub Byszewski and Clemens Müllner we obtained a decomposition result which allows us to express any (complex-valued) automatic sequence as the sum of a structured part, which is easy to work with, and a part which is pseudorandom or Gowers uniform. This has immediate applications to the asymptotic count of additive patterns, as well as less direct application to arithmetical subword complexity (joint with Müllner) and quantitative variants of Cobham's theorem (upcoming paper with Adamczewski and Müllner).

In my talk, I present a construction of the wreath product of inverse semigroups. Before exploring some of its basic properties, we will look at the origin of the term and its connections to Kyiv. The talk is intended as a gentle introduction to the subject, so no special background knowledge is required.

A contraction of a Lie algebra is a limiting procedure that transforms one Lie algebraic structure into another. Such transformations play an important role in mathematical physics, as they provide systematic links between different physical theories, describe symmetry reductions, and arise naturally in the analysis of invariant differential equations.
In this talk, I will present several definitions of Lie algebra contractions and discuss criteria for their existence, including new results. We will analyze the Hasse diagrams describing contraction relations among low-dimensional and nilpotent Lie algebras over the real and complex fields, obtained in our recent investigations. In addition, I will present an algorithm for constructing generalized Inönü–Wigner contractions and illustrate the approach with a number of explicit examples.

Differential equations arising in algebraic geometry have surprising arithmetic properties. We will show examples, discuss the nature of these phenomena and demonstrate their appearance in string theory in physics.

Differential equations arising in algebraic geometry have surprising arithmetic properties. We will show examples, discuss the nature of these phenomena and demonstrate their appearance in string theory in physics.