Univerza na Primorskem
Fakulteta za matematiko, naravoslovje in informacijske tehnologije
Več informacij o projektu / More info about the project
Opis / Description
SLO:
Ta projekt raziskuje orbite podprostorov pod delovanjem klasičnih grup v Galoisovi geometriji, pri čemer se osredotoča na njihovo interakcijo zalgebrskimi varietetami nad končnimi polji. Medtem ko so dejanja klasičnih grup v nizkodimenzionalnih prostorih dobro razumljena, ostaja njihovovedenje v višjih dimenzijah večinoma neraziskano. Študija nestandardnih delovanj grup na Segrejevih in Veronesejevih varietetah zagotavlja novevpoglede v strukturo tenzorskih dekompozicij, sekantne varietete in geometrijske invariante, pomembne za kombinatoriko in algebraično geometrijo.
Ključni cilj je preučiti orbitalno strukturo podprostorov v ambientalnem prostoru teh sort, s posebnim poudarkom na klasifikaciji ravnin vambientalnem prostoru Veronesejeve ploskve in njihovi povezavi z mrežami kvadrikov. Razumevanje teh orbit ima posledice za težave s tenzorskimrangom in sekantne varietete, kar prispeva k širšemu preučevanju algebraične geometrije nad končnimi polji.
Projekt vključuje tudi računalniško komponento, ki izboljšuje obstoječe metode za klasifikacijo orbit z uporabo programskega paketa FinInG. Zrazvojem računalniških tehnik za primere majhnega obsega želimo vzpostaviti rezultate dokazov o konceptu, ki podpirajo globlje teoretičnorazumevanje visokodimenzionalnih grupnih delovanj. Rezultati bodo okrepili znanje o končni geometriji, algebraičnih varietetah in njihovih strukturnihlastnostih.
EN:
This project investigates orbits of subspaces under classical group actions in Galois geometry, focusing on their interaction with algebraic varieties over finite fields. While the actions of classical groups on low-dimensional spaces are well understood, their behaviour in higher dimensions remains largely unexplored. The study of non-standard group actions on Segre and Veronese varieties provides new insights into the structure of tensor decompositions, secant varieties, and geometric invariants relevant to combinatorics and algebraic geometry.
A key objective is to examine the orbit structure of subspaces in the ambient space of these varieties, with particular emphasis on the classification of planes in the ambient space of the Veronese surface and their connection to nets of quadrics. Understanding these orbits has implications for tensor rank problems and secant varieties, contributing to the broader study of algebraic geometry over finite fields.
A key objective is to examine the orbit structure of subspaces in the ambient space of these varieties, with particular emphasis on the classification of planes in the ambient space of the Veronese surface and their connection to nets of quadrics. Understanding these orbits has implications for tensor rank problems and secant varieties, contributing to the broader study of algebraic geometry over finite fields.
The project also includes a computational component, improving existing methods for classifying orbits using the FinInG software package. By developing computational techniques for small-scale cases, we aim to establish proof-of-concept results that support a deeper theoretical understanding of high-dimensional group actions. The outcomes will enhance knowledge in finite geometry, algebraic varieties, and their structural properties.