University of Zagreb

Random temporal graphs are a version of the classical Erdős-Rényi random graph $G(n,p)$ where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time stamps. We are interested in the asymptotics for the distances in such graphs, mostly in the regime of interest where the average degree $np$ is of order $\log n$ ('near' the phase transition).

More specifically, we will discuss the asymptotic lengths of increasing paths: the lengths of the shortest and longest paths between typical vertices, as well as the maxima between any two vertices; this also covers the (temporal) diameter. In the regime $np \gg \log n$, longest increasing paths were studied by Angel, Ferber, Sudakov and Tassion.

The talk contains joint work with Nicolas Broutin and Gábor Lugosi.

I will present some results from the newly developed theory of wavelets; based on the joint work with the following authors:

P.M. Luthy, H.Šikić, F.Soria, G.L.Weiss, E.N.Wilson.One-DimensionalDyadic Wavelets.Mem. Amer. Math. Soc. 280 (2022), no 1378, ix+152 pp.

About two and a half decades ago and based on the influential book by Fernandez and Weiss, an approach was developed to study wavelets from the point of view of their connections with Fourier analysis. The idea was to study wavelets and other reproducing function systems via the four basic equations that characterized various properties of wavelet systems, like frame and basis properties, completeness, orthogonality, etc. Despite hundreds of research papers and the impressive development of the theory of such systems, some questions remain open even in the basic case of dyadic wavelets on the real line. Among the most thorough treatments that we provide in this lengthy paper is the issue of the understanding of the low-pass filters associated with the MRA structure. In this talk, I will focus on some of these results. As it turned out, a more general and abstract approach to the problem, using the study of dyadic orbits and the newly introduced Tauberian function, reveals several interesting properties and opens an interesting context for some older results

Unitary highest weight modules were classified in the 1980s by Enright-Howe-Wallach and independently by Jakobsen. The classification is based on a version of the Dirac inequality, but the proofs also require a number of other techniques and are quite involved. We present a much simpler proof based on a different version of the Dirac inequality. This is joint work with Vladimir Soucek and Vit Tucek.
 

It is well-known that only a limited number of the fluid flow problems can be solved (or approximated) by the solutions in the explicit form. To derive such solutions, we usually need to start with (over)simplified mathematical models and consider ideal geometries on the flow domains with no distortions introduced. However, in practice, the boundary of the fluid domain can contain various small irregularities (rugosities, dents, etc.) being far from the ideal one. Such problems are challenging from the mathematical point of view and, in most cases, can be treated only numerically. The analytical treatments are rare because introducing the small parameter as the perturbation quantity in the domain boundary forces us to perform tedious change of variables. Having this in mind, our goal is to present recent analytical results on the effects of a slightly perturbed boundary on the fluid flow through a channel filled with a porous medium. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter $\varepsilon$ and arbitrary smooth function. The porous medium flow is described by the Darcy-Brinkman model which can handle the presence of a boundary on which the no-slip condition for the velocity is imposed. Using asymptotic analysis with respect to $\varepsilon$, we formally derive the effective model in the form of the explicit formulae for the velocity and pressure. The obtained asymptotic approximation clearly shows the nonlocal effects of the small boundary perturbation. The error analysis is also conducted providing the order of accuracy of the asymptotic solution. We will also address the problem of the solute transport through a semi-infinite channel filled with a fluid saturated sparsely packed porous medium. A small perturbation of magnitude $\varepsilon$ is applied on the channel's walls on which the solute particles undergo a first-order chemical reaction. The effective model for solute concentration in the small-Péclet-number-regime is derived using asymptotic analysis with respect to $\varepsilon$. The obtained mathematical model clearly indicates the influence of the porous medium, chemical reaction and boundary distortion on the effective flow.

This is a joint work with Eduard Marušić-Paloka (University of Zagreb).

We will give the results on the models of thin plates and rods in nonlinear elasticity by doing simultaneous homogenization and dimensional reduction. In the case of bending plate we are able to obtain the models only under periodicity assumption and assuming some special relation between the periodicity of the material and thickness of the body. In the von K\'arm\'an regime of rods and plates and in the bending regime of rods we are able to obtain the models in the general non-periodic setting. In this talk we will focus on the derivation of the rod model in the bending regime without any assumption on periodicity.

We will briefly review the notions of Dirac cohomology and of $A_{\mathfrak{q}}(\lambda)$ modules of real reductive groups, and recall a formula for the Dirac cohomology of an $A_{\mathfrak{q}}(\lambda)$ module. Then we will discuss to what extent an $A_{\mathfrak{q}}(\lambda)$ module is determined by its Dirac cohomology. This is joint work with Jing-Song Huang and David Vogan.