In this talk, I will begin by reviewing Fraïssé's construction method, a model-theoretic technique for constructing infinite homogeneous structures, and its foundational role in the field. I will then discuss key correspondences between the combinatorial properties of these structures and the group-theoretic properties of their automorphism groups. The focus will then shift to the Urysohn universal metric space (and Urysohn sphere), introduced by Pavel Urysohn in the 1920s. This remarkable space is a homogeneous metric space into which any separable metric space can be isometrically embedded. Finally, I will present selected results and observations concerning its isometry group, emphasizing their properties and broader implications.

In this talk, we explore the saturation number of monomial ideals in the polynomial ring \( S = K[x_1, \ldots, x_n] \) over a field \( K \). We focus on various classes of ideals, including \(\mathbf{c}\)-bounded strongly stable, Veronese-type, irreducible monomial ideals, and monomial ideals in two variables. Additionally, we examine the saturation numbers of ordinary and symbolic powers of specific families of monomial ideals, relating them to the saturation numbers of their irreducible components.

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Mathematical modeling is a powerful tool for understanding complex phenomena across different fields. This talk explores two areas: bubble dynamics and battery modeling. In fluid dynamics, we study cavitation bubble collapse near boundaries using boundary integral and finite volume methods, with applications in medicine and cleaning. In electrochemistry, we examine battery behavior using physics-based models like the Doyle-Fuller-Newman framework, essential for improving energy storage. By presenting these different fields, we show how modeling helps solve diverse problems. We'll discuss our findings and highlight areas needing further research due to time, knowledge, or resource limitations. This talk aims to encourage collaboration and inspire new research in both bubble dynamics and battery modeling.

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In this talk we will discuss some recent interesting connections between properties of sets of nonnegative integers with different kinds of geometries. Starting with a primitive integer vector, we examine certain associated functions and relate the setting to statements in convex and algebraic geometries.

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We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, n ≥ 3. The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it follows that if {fn}∞ n=1 is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion H(fn)), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence {fn}∞ n=1 with fn → f locally uniformly and with lim supn→∞ H(fn) < H(f). Our main result shows this is true for affine mappings. Addressing conjectures of F.W. Gehring and Iwaniec, we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each α < √ 2 there is fn → f locally uniformly with f affine and α lim sup n→∞ H(fn) < H(f) We conjecture √ 2 to be best possible.

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In this seminar, we first recap the α-energy functional, Euler-Lagrange operator and α stress-energy tensor. Second, it is shown that the critical points of the α-energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an α-harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal α-harmonic maps are minimal submanifolds. Then, the stability of any α-harmonic map from a Riemannian manifold to a Riemannian manifold with non-positive Riemannian curvature is demonstrated. Finally, the instability of α-harmonic maps from a compact manifold to a standard unit sphere is investigated.

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We look at a class of nonassociative algebras that can be defined using skew polynomials, and discuss their structure and applications. These algebras can be seen as canonical generalizations of central simple algebras, and some behave surprisingly similar to their classical "cousins". They are employed to build codes used for wireless digital data transmission, e.g. in mobile phones, laptops or portable TVs. To make the talk accessible to a broad audience, we will explain the main ideas using the example of Hamilton's quaternion algebra. We will explain how to generalize the construction of Hamilton's quaternion algebra to nonassociative four-dimensional algebras, and how to use them in space-time block coding.

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Arithmetic Kleinian group is an important branch of low dimensional geometry and topology. One of the significant side of it, is two generating arithmetic groups. In our research, we will complete Maclaclan and Gaven martin work for all possible Arithmetic groups which generated by two elements $(f; g)$ of order $(p; q)$. They calculated $(p; q)$- cases for $p; q < 6$. But for $p; q < 6$, cases still have remainded. Here we try to introduce some Theorems and Methods to complete our work.

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For two Banach algebras A and B and a nonzero multiplicative linear functional ϴ on B, Monfared introduced the ϴ-Lau product B. In this paper, we investigate and study the notions of ϕ-biprojectivity, ϕ-biflatness and ϕ-Johnson amenability of B and their relations with A and B. As an application, we characterize ϕ-biflatness and ϕ-biprojectivity for ϴ-Lau product of Banach algebras related to locally compact groups and discrete semigroups. 

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Imagine a future where advanced quantum computers are a reality. What will they be capable of? Quantum computers have the potential to perform a variety of valuable tasks, but they also pose significant privacy risks. One of the most frequently discussed concerns is their ability to break current cryptographic methods, meaning they could potentially decrypt confidential messages sent over the internet. If we continue to rely on today's technology, secure and private online communication may become impossible. In this talk, we will explore this issue and discuss proactive steps we can take to protect ourselves in a world with quantum computing.

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For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we will demonstrate formulas for the densities of primes subject to GRH for which the index of the reduction group has a given value. Moreover, we will completely classify the cases of rank one, torsion groups for which the density vanishes and the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we will mention some partial results.

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Transferring geometric problems into the languages of permutations groups. Our goal is to describe the geometric problem about spaces and then transfer it into the theory of permutation groups by using the fundamental groups. The Riemann existence theorem provides a one to one correspondence between them. As a result, we can deal with finite primitive groups instead of indecomposable meromorphic functions.

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 This research is primarily concerned with the convexity properties of distortion functionals -particularly the linear distortion - defined on quasiconformal homeomorphisms of domains in Euclidean n-spaces, though we will mainly stick to three-dimensions. The principal application is in identifying the lower semi-continuity of distortion on uniformly convergent limits of sequence of quasiconformal mappings. For example, given the curve family or analytic definitions of quasiconformality – discussed in this research – it is known that if {𝒇𝑛} is a sequence of K-quasiconformal mappings (and here K depends on the particular distortion but is the same for every element of the sequence) which converges to a function 𝒇, then the limit function is also K-quasiconformal. 

Despite a widespread belief that this was also true the geometric definition of quasiconformal-ity (via the linear distortion 𝐻(𝒇) defined below) Tadeusz Iwaniec gave a specific surprising example to show that the linear distortion function is not lower semicontinuous. The main aim of this research is to show that this failure of lower semicontinuity is actually far more common. Perhaps even generic in the sense that it might be true under mild restrictions on a quasiconformal 𝒇, there may be a sequence if {𝒇𝑛} with 𝒇𝑛→𝒇 uniformly and with the below property 

lim𝑛→∞sup𝐻(𝒇𝑛)→𝐻(𝒇). 

The main result in my thesis is to show this is true for wide class of linear mapping. 

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This work presented some novel systems for chaotic systems‎, ‎the so-called parallel systems‎. ‌‎Indeed ‎we will define a parallel system for a chaotic system where the‎ ‎attractor obtained from the new system exhibits complex chaotic dynamics as well‎. The application of these systems is discussed.  For instance, It is shown with the help of these systems, the synchronisation problem between two systems can be transformed into an optimal control problem‎.

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