Maria Rita Casali
Research fields:
- Geometric Topology of low dimensional manifolds
- Representations of PL-manifolds via combinatorial tools (knots and branched coverings, framed links, Kirby diagrams, edge-colored graphs....) and procedure for passing from one representation method to another
- Classification of PL-manifolds by means of "regular genus" and comparison between different PL-invariants, with particular regard to dimension 4 and dimension 5
- Search for combinatorial moves which are able to realize PL-homeomorphism between the represented manifolds (also in the case of non-empty boundary)
- Computational approach to the study of PL-manifolds, involving elaboration and implementation of generation and classification algorithms, in order to obtain essential catalogues of colored triangualtions of PL-manifolds, in dimension 3 and 4
- Comparison between dfferent notions of "complexity", with particular reference to the estimation of Matveev complexity by means of crystallization theory
- Applications of the GEM theory (making use of edge-colored graphs representing singular manifolds, or PL manifolds with non-empty boundary) to the theory of random tensor models (which is of interest for theoretical physics as an approach to quantum gravity in arbitrary dimension)
- Study of trisections of PL 4-manifolds induced by gems, and extension to the boundary case; estimation of trisection genus for compact 4-manifolds associated to Kirby diagrams; relationship between trisection genus and regular genus
List of scientific publications
Triennial reserch report 2018-2021
Triennial research report 2015-2018
Triennial research report 2012-2015
Teaching activity verification document
--------------------
SOME RESEARCH PRESENTATIONS/SURVEYS:
-
Seminar on “ GEM-INDUCED TRISECTIONS OF COMPACT 4-MANIFOLDS " held on 30 June 2022 at the Congress "Conformal Geometry and Low Dimensional Manifolds", Avila (Spain): see the slides of the Seminar.
-
Conference on “ KIRBY DIAGRAMS AND 5-COLORED GRAPHS REPRESENTING COMPACT 4-MANIFOLDS " held by invitation on 06 September 2021 within the KTRT online Seminars organized by V.O. Manturov (Moscow Institute of Physics and Technology, RU): see the first part and the second part of the slides of the Conference.
-
Presentation at the Joint Meeting UMI-SIMAI-PTM, held in Wroclaw (Polonia), from 17 to 20 September 2018, within the Special Session dedicated to ``Geometric Topology, Manifolds, and Group Actions": see the slides of the seminar “ REPRESENTING AND CLASSIFYING COMPACT PL 4-MANIFOLDS VIA REGULAR 5-COLORED GRAPHS ”, held by invitation of the organizers.
-
Lecture held - by invitation - at the Workshop “ TRIANGULATION ”, at the Mathematical Research Institute of Oberwolfach (Germany) on May 4, 2012: see the extended abstract (published on Oberwolfach Reports: DOI: 10.4171/OWR/2012/24) or the slides of the lecture: abstract - slides.
-
Survey on crystallization theory, foundations and current results in dimension four, presented during the Workshop “ COLORED GRAPHS AND RANDOM TENSORS ”, jointly organized with the theoretical physicist Vincent Rivasseau, at the LPT of Orsay - France (“Laboratoire de Physique Théorique d’Orsay”) on 14 and 15 January 2016.
See:
- the Workshop scheduling: program
- the slides of the lecture “Coloured graphs representing PL n-manifolds: moves and code”: lecture_1
- the slides of the lecture “Crystallizations of PL 4-manifolds”: lecture_2 .
-
A further survey on results achieved in dimension 4 appears in the paper “ CLASSIFYING PL 4-MANIFOLDS VIA CRYSTALLIZATIONS: RESULTS AND OPEN PROBLEMS ” (by M.R. Casali – P. Cristofori – C. Gagliardi), published in the book: "A Mathematical Tribute to Professor José María Montesinos Amilibia”, Universidad Complutense Madrid (2016), ISBN: 978-84-608-1684-3: survey_4dim
-----------------------