This book introduces key topics on Geometric Invariant Theory, a technique to obtaining quotients in algebraic geometry with a good set of properties, through various examples. It starts from the classical Hilbert classification of binary forms, advancing to the construction of the moduli space of semistable holomorphic vector bundles, and to Hitchin's theory on Higgs bundles. The relationship between the notion of stability between algebraic, differential and symplectic geometry settings is also covered.
Unstable objects in moduli problems -- a result of the construction of moduli spaces -- get specific attention in this work. The notion of the Harder-Narasimhan filtration as a tool to handle them, and its relationship with GIT quotients, provide instigating new calculations in several problems. Applications include a survey of research results on correspondences between Harder-Narasimhan filtrations with the GIT picture and stratifications of the moduli space of Higgs bundles.
Graduate students and researchers who want to approach Geometric Invariant Theory in moduli constructions can greatly benefit from this reading, whose key prerequisites are general courses on algebraic geometry and differential geometry.

Introduces key topics on Geometric Invariant Theory through examples and applications Covers Hilbert classification of binary forms and Hitchin's theory on Higgs bundles Takes particular note of unstable objects in module problems

Autorentext

Alfonso Zamora Saiz is a professor at the School of Computer Science Engineering, Technical University of Madrid, Spain. Holding a PhD in algebraic geometry from the Complutense University of Madrid (2013), he has been a visiting PhD student at Cambridge University and Columbia University, a postdoc at the IST in Lisbon, lecturer at the California State University Channel Islands and a professor at the CEU San Pablo University in Madrid. He has also worked as a quantitative analyst in the industry. His research interests include algebra, geometry and topology in pure math, as well as data analytical applications and mathematics education. Carlos Quesada González holds a PhD in applied mathematics from the Complutense University of Madrid, Spain. He is a professor and Vice-dean of the School of Business at the CEU San Pablo University in Madrid, where he has helped to establish the Business Intelligence degree. He also teaches master courses on big data for finance and collaborates as a statistical analyst with Grant-Thornton. Lluís Hurtado Gil is a professional data scientist at eDreams ODIGEO and holds a PhD in astrophysics from Valencia University (2016). He was a professor of statistics and econometrics for three years at CEU San Pablo University, where he also served as secretary of the Statistics and Applied Mathematics Department. He has published works on econometrics for undergraduate students and research papers on statistical applications in modern astrophysics with R code. Currently, he continues to collaborate with the International J-PAS Survey, investigating the physics of the accelerating universe. Professionally, he has specialized in the application of stochastic processes to digital marketing. Diego Mondéjar Ruiz is a professor at the Department of Applied Mathematics and Statistics, CEU San Pablo University in Madrid, Spain. He obtained his PhD in mathematics from the Complutense University of Madrid with a thesis on topological data analysis and computational topology in 2015. In addition to having been a visiting PhD student at Stanford University and the University of Pennsylvania, he has taught mathematics, statistics and programming courses at several universities.

Inhalt
Preface.- Introduction.- Preliminaries.- Geometric Invariant Theory.- Moduli Space of Vector Bundles.- Unstability Correspondence.- Stratifications on the Moduli Space of Higgs Bundles.- References.- Index.