Seminar SAAAJ, Ljubljana 25.11.: Povzetki
Noncommutative noetherian prime rings with bounded factorizations Daniel Smertnig
Abstract. A prime ring $R$ is a bounded factorization (BF) domain if for every non-zero-divisor a of R, there exists a nonnegative integer l(a) such every factorization of a into atoms (irreducible elements) has length at most l(a). Commutative noetherian domains are easily seen to be BF-domains, but the proofs involve prime ideals or localizations. In a noncommutative noetherian prime ring every non-zero-divisor still factors into a product of atoms. However, it is open whether such a prime ring has BF. We present some recent sufficient conditions. This is joint work with J. Bell, K. Brown, and Z. Nazemian.
On theta correspondence Marcela Hanzer
Abstract. We give an overview of theta correspondence, from the classical theta functions to the modern approach as a tool for relating representations on different groups. We comment on the interplay of theta correspondence with other ingredients of the Langlands programme.
Jordan product preserving maps on simple Jordan algebras Nik Stopar
Abstract. Every simple finite-dimensional formally real Jordan algebra is isomorphic to one of the following: algebra of n x n hermitian matrices over the real numbers R, complex numbers C or quaternions H, algebra of 3 × 3 hermitian matrices over the octonions O, or algebra of spin factors Spin_n with n > 1. In this talk we describe maps on this type of algebras with the property ϕ(A ◦ B) = ϕ(A) ◦ ϕ(B) for all elements A and B, i.e., maps that preserve the Jordan product. Although we do not assume additivity or bijectivity, it turns out that any such map is either a real linear automorphism or a constant map. On the way to our main result we also discuss several auxiliary results, including a description of zero Jordan product preserving maps, a generalization of Uhlhorn’s theorem to octonions, and some results on ortographs of simple Jordan algebras. In the octonionic case we briefly mention the fundamental theorem of octonionic projective geometry and its connections to our results. This is joint work with Gregor Dolinar (University of Ljubljana) and Bojan Kuzma (University of Primorska).
Weak solutions in fluid-structure interactions: Cauchy and periodic problems Boris Muha
Abstract Fluid-structure interactions (FSI) play a pivotal role in various natural and technological phenomena. Mathematically, FSI problems are described by a coupled nonlinear system of partial differential equations, where the Navier-Stokes equations are coupled to elasto-dynamics across the moving interface. In this presentation, we will provide an overview of the most recent advancements in the theory of weak solutions for FSI problems. Specifically, we will explore the challenges and nuances encountered when transitioning from solving the Cauchy problem to obtaining periodic solutions. To illustrate these concepts, we will examine a simple heat-wave system, which serves as a prototypical example of fluid-structure interactions. For this system, we will present some new existence results for periodic solutions. The presented results are joint work with S. Schwarzacher and J. Webster.
