Seminários e cursos curtos

Categorical algebra is a fundamental branch of mathematics that lies at the intersection of category theory and algebra. On the one hand, it captures the fruitful properties and structures studied in algebra via category theory. On the other hand, it investigates the global categorical properties that algebraic objects enjoy when collected together. Both these endeavors are essential to extend and transport the fundamental concepts and theorems of algebra to different and broader settings. In this talk, we present an innovative theory that generalizes categorical algebra to the framework of 2-dimensional category theory. This has the notable advantage that the second dimension can be used both to weaken conditions that are too strict in nature and to refine algebraic invariants, obtaining a richer theory which encompasses a broader range of examples. Furthermore, 2-dimensional categorical algebra is essential to effectively compare different algebraic categories with each other.

This talk is based on a joint work with Elena Caviglia and Zurab Janelidze.

Abelian categories and triangulated categories provide fundamental frameworks to study homological and cohomological problems across algebraic geometry, topology and representation theory.

In this talk we will explain how we can study the 2-category AbCat of abelian categories and the 2-category Triang of triangulated categories through the lenses of 2-dimensional categorical algebra. Surprisingly, through these lenses, AbCat and Triang look extremely similar.

We will show that the important notions of Serre subcategories and Serre quotients of abelian categories precisely correspond respectively with the 2-dimensional kernels and cokernels in AbCat. In a similar way, thick triangulated subcategories and Verdier localizations of triangulated categories are exactly the 2-kernels and the 2-cokernels in Triang.

Furthermore, even more striking similarities between the two contexts arise when characterizing these 2-kernels and 2-cokernels in terms of categorical properties satisfied by their underlying functors.

These results will allow us to show that both the 2-categories AbCat and Triang are exact, in appropriate 2-dimensional senses. In particular, AbCat is 2-Puppe exact in a 2-dimensional sense, while Triang satisfies the weaker exactness property of a 2-homological category.

This talk is based on a joint work in progress with Zurab Janelidze, Luca Mesiti and Ulo Reimaa.

Free boundary problems naturally arise in a variety of models from material science, particularly in connection with phase transitions and interfacial phenomena. In this talk, we discuss a class of variational problems that lead to singular PDEs in heterogeneous and irregular media. We focus on models whose intrinsic singularities are shaped by the medium itself, so that variations in the underlying structure directly influence the geometric behavior of solutions in a highly nonlinear fashion. On the analytical side, this leads to the development of a layered regularity theory, designed to remain robust across different regimes. In more structured settings, we establish $C^{1,\alpha}$-regularity of the free boundary, showing how increased organization of the medium is reflected in the qualitative behavior of solutions and their interfaces.

The development of the theory of martingale Hardy spaces has been strongly influenced by the classical theory. Because of this it is inevitable to compare results of this theory to those on classical Hardy spaces and during the decades this new direction was developing in this way and many similarities between these theories have already been found. But nowadays a lot of new results were obtained in the theory of martingale Hardy spaces which are new in classical case.

This lecture is devoted to review theory of martingale Hardy spaces. In particular, we give atomic decomposition of these spaces and show how this theorem simplify the proofs of boundedness of any sub-linear operators on these spaces. For the illustration of bounded operators on the martingale Hardy spaces we define Vilenkin groups and their characters which are called Vilenkin functions. We also define Fourier series with respect to Vilenkin system and maximal operators related to these operators. Moreover, we define modulus of continuity in martingale Hardy spaces and derive necessary and sufficient conditions in the terms of modulus of continuity such that partial sums with respect to one Vilenkin-Fourier series converge in $H^p$ norm. We also investigate classical Hardy spaces, modulus of continuity in these spaces and present one to one analogues of these results for partial sums in this classical case.

At the end of this presentation we state some open problem how to generate these two similar results for Hardy spaces on some locally compact Abelian groups and partial sums with respect to character functions of these groups. Moreover, we also state similar open problems for variable martingale Hardy spaces. Finally, we will choose opposite approach and will try to state and investigate results for partial sums of the two-dimensional Fourier series in martingale Hardy spaces for classical case of Hardy spaces.