Arithmetic of Algebraic Groups; Commutative Algebra; Non-archimedian Fields

1. PhD Thesis (Completed 1, In progress 2):

Student Name: Claudio Bravo.
Institution: Universidad de Chile.
Thesis Title: "Fracciones continuadas y regiones fundamentales para subgrupos de PGL2" (Continued fractions and fundamental regions for subgroups of PGL2)
Defense date: November, 2021 (expected).

Student Name: Ignacio Saavedra.
Institution: Universidad de Chile.
Thesis Title: "Cálculo de complejos cocientes del edificio de Bruhat-Tits para PGLn" (Computations of quotient complexes of the Bruhat-Tits building for PGL_n).
Defense date: March, 2020 (expected).

Student Name: Patricio Quiroz.
Institution: Universidad de Chile.
Thesis Title: "Cálculo de la norma espinorial para formas anti-hermitianas sobre un álgebra de cuaterniones de división sobre un cuerpo local diádico" (Computation of the spinor norm for skew-hermitian forms on a quaternion division algebra over a local dyadic field)
Defense date: January, 2013.
Current position: Dr Quiroz is currently teaching at Universidad Federico Santa María..

2. Master Thesis (Completed 4)

Student Name: Marco Godoy.
Institution: Pontificia Universidad Católica de Chile (PUC).
Thesis Title: "Acciones del grupo de unidades de órdenes de Eichler, sobre el Arbol de Bruhat-Tits, asociados a divisores de grado 1 y 2" (actions of the unit group of an Eichler order on the Bruhat-Tits tree associated to divisors of degree 1 and 2).
Defense date: January, 2019.

Student Name: Claudio Bravo.
Institution: Universidad de Chile.
Thesis Title: "Cálculo de la posición relativa de ramas de cuaterniones puros con producto simetrizado no nulo sobre álgebras de cuaterniones" (Computations of the relative position for branches of pure quaternions with non-vanishing simetrized product in quaternion algebras).
Defense date: March, 2018.
Current position: C. Bravo is currently a PhD student at Universidad de Chile. He also teaches at Universidad Federico Santa María.

Student Name: Jaime Contreras.
Institution: Universidad de Chile.
Thesis Title: "Cuerpo de representación para incrustaciones optímales" (Representation fields for optimal embeddings).
Defense date: July,,2015.
Current position: J. Contreras currently teaches at Universidad Diego Portales y is a Mathematics Tutor at Universidad San Sebastián.

Student Name: Ignacio Saavedra.
Institution: Universidad de Chile.
Thesis Title: "Cálculos del cuerpo de representación para órdenes explícitos" (Computation of the representation field for explicit orders).
Defense date: January,,2014.
Current position: I. Saavedra is currently a PhD. student at Universidad de Chile. He also has a part-time teaching position at Universidad Diego Portales.

3. Research units at Universidad de Chile.. (Completed: 4):

i) Student Name: Claudio Bravo. Title: "Aplicaciones de la teoría de grafos a algebras locales y globales" (Applications of graph theory to local and global algebras). Year: 2015.
ii) Student Name: Maximiliano Sandoval. Title: "Epsilonización" (Epsilonization). Year: 2014.
iii) Student Name: Camilo Vera. Title: "Funciones aritméticas, multiplicativas, y fórmula de inversión de Moebius" (Multiplicative arithmetic functions and Moebius inversion formula). Year: 2012
iv) Student Name: Gonzalo Jiménez. Title: "Reticulados y su comportamiento bajo completaciones" (Lattices and its behavior under completions). Year: 2012

1. General audience talks:

i) Workshop: "De ranas saltarinas y piezas de ajedrez: El algoritmo de Euclides" (On jumping frogs and chess pieces: Euclid's Algorithm). 2 sessions of 1 hr. each. MateA Math Camp. December, 2018. An activity for female high-school students.
ii) Short Course: "Arboles y órdenes" (trees and orders). 3 sessions of 1 hr. each, IV Encuentro Chileno de Teoría de Números (IV Chilean Number Theory Meeting), May, 2018.
iii) Talk: "Reticulados locales y globales" (Local and global lattices) 50 minutes talk. PUC Mathematics Colloquium (A student seminar). July, 2016.
Iv) Talk: "De la cinta de Moebius a los puentes de Koenigsberg" (From Moebius strip to Koenisberg bridges) 50 minutes talk,Science week 2009, (October, 2009. An activity of the Facultad de Ciencia at Universidad de Chile.

2. Mathematics Olympiads activities:

i) I have been part of the organizing committee for the Chilean National Mathematics Olympiads since 2009. This activity includes designing tests, reviewing student answers and selecting winners. I have helped to design tests since 2005.
ii) I have been involved twice in international competitions at the Problem Tribunal, once at the XIX South Cone Olympiad in 2008 and once at the XXXI Iberoamerican Mathematics Olympiad in 2016.
iii) I trained the Chilean team between 2003 and 2006.

3. Other activities for general public:

i) I was invited to the radio program "Quiero Ser Científico" (I want to be a scientist), at our Faculty's radio at Universidad de Chile. The emission date was July 24, 2017.
ii) In 2016 I wrote an article with the title "Ajedrez Matemático, Una aritmética bidimensional" (Mathematical chess: a bi-dimensional arithmetic) for the magazine "revista del profesor de Matemáticas" RPMAT (Magazine for the Mathematics teacher). The article was accepted for publication, but not published due to a lack of continuity for the magazine.
iii) In 2015 I made a "scientific capsule". A short video where I explain generalities about my research. It has been on CNN-Chile and it is available in YouTube.

4. Main invited talks (Main invited conferences and plenary talks, ).

i) Invited Talk: Geometry of the Bruhat-Tits tree: Classifying orders and lattices, Conference on the arithmetic theory of quadratic forms, January, 2019, Seoul, South Korea. Conference organized by the Seoul National University (SNU).
ii) Invited Talk: Geometry of the Bruhat-Tits tree: Branches of orders and field extensions, Conference:Quadratic forms in Chile 2018, January 2018, Talca, Chile.
iii) Invited Talk: Invariant sets and quotient graphs, CM-2014 Satellite Conference on Integral Quadratic Forms and Related Topics, August, 2014, Gjeoungyu, South Korea
iv) Invited Talk: Some applications of the theory of representations of orders, International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms, December, 2013, Puerto Natales, Chile.
v) Invited Talk: Algebraic dynamical systems and p-adic fields, First Joint Meeting AMS- SOMACHI, December, 2010. Pucón, Chile.
vi) Invited Talk: Skew-hermitian forms and representation fields, First Joint Meeting AMS- SOMACHI, December, 2010. Pucón, Chile.
vii) Invited Talk: Spinor class fields over function fields, Workshop on the Arithmetic of Quadratic Forms and Integral Lattices, December, 2010. Futrono, Chile, a satellite congress of the First Joint Meeting AMS-SOMACHI.
viii) Short course: Algebraic ergodic systems. Summer school at "Instituto de Matemática Pura e Aplicada" IMPA (Pure and Applied Mathematics Institute), January, 2004. Rio de Janeiro, Brasil.
ix) Invited Talk: Integral dyadic skew-Hermitian forms. International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms, December, 2002. Talca-Pucón, Chile.
x) Plenary Conference: Lattices and cohomology, LXX Encuentro de la Sociedad de Matemáticas de Chile (LXX Meeting of the Chilean Math Society). December, 2000. Punta de Tralca, Chile.

5. Other international talks:

i) Talk: Spinor image and the geometry of trees, XXII Coloquio Latinoamericano de Álgebra (XXII Latinamerican Algebra Colloquuium) [08/2017]. Quito, Ecuador.
ii) Talk: Representation Field Computations via Bruhat-Tits trees, XIX Coloquio Latinoamericano de Álgebra (XIX Latinamerican Algebra Colloquuium) [12/2012]. Pucón, Chile.
iii) Talk: Cuerpos de Representación y Aplicaciones (Representation fields and applications), IV Congreso Latinoamericano de Matemáticos (IV Latin-american Mathematicians' Congress) [08/2012], Córdova, Argentina.
iv) Talk: Spinor class fields for lattices in function fields, XVIII Coloquio Latinoamericano de Álgebra (XVIII Latin-american Algebra Colloquuium) [6/2009]. São Pedro, Brasil.
v) Talk: Spinor class fields and representations. XVI Congreso Latinoamericano de Álgebra (XVI Latin-american Algebra Colloquuium) [8/2001]. Córdoba, Argentina.

6. Other national talks:

i) Talk: Fibrados vectoriales y grafos cocientes cuaterniónicos (Fiber bundles and quaternionic quotient graphs), Seminario de aritmética y geometría de Valparaíso (Valparaiso arithmetic and geometry seminar), [04/2017]. Valparaíso, Chile.
ii) Talk: Ramas del árbol de Bruhat-Tits (Branches of the Bruhat-Tits tree), Seminario de aritmética y geometría de Valparaíso (Valparaiso arithmetic and geometry seminar), [10/2016]. Valparaíso, Chile.
iii) Talk: Ramas y extensiones de cuerpos (Branches and field extensions), LXXXIV Encuentro de la Sociedad de Matemáticas de Chile (LXXXIV Meeting of the Chilean Math Society), [11/2015]. Pucón, Chile.
iv) Talk: Ordenes, árboles y transformaciones de Moebius (Orders, trees and Moebius transformations), II Encuentro Chileno de Teoría de Números (II Chilean Number Theory Meeting) [05/2015], Valparaíso, Chile.
v) Talk: Sobre ciertas ramas del árbol de Bruhat-Tits (On certain branches of the Bruhat-Tits tree), XXVII Jornada de Matemática de la Zona Sur (XXVII South Zone Mathematics Days), [04/2015], Chillán, Chile.
vi) Talk: Raices cúbicas de la unidad en órdenes positivos definidos (cubic roots of unity in positive definite orders), LXXXIII Encuentro de la Sociedad de Matemáticas de Chile (LXXXIII Meeting of the Chilean Math Society), [12/2014]. Quinamávida, Chile.
vii) Talk: Spinor class fields for generalized Eichler orders, LXXXII Encuentro de la Sociedad de Matemáticas de Chile (LXXXII Meeting of the Chilean Math Society), [11/2013]. Olmué, Chile.
viii) Talk: Calculando grafos cocientes cuaterniónicos via representaciones de órdenes (Computing quaternion quotient graph via representations of orders). XXVI Jornada de Matemática de la Zona Sur (XXVI South Zone Mathematics Days), [04/2013], Quinamávida, Chile.
ix) Talk: Ramas de órdenes cíclicos (Branches of cyclic orders), LXXXI Encuentro de la Sociedad de Matemáticas de Chile (LXXXI Meeting of the Chilean Math Society), [11/2012]. Olmué, Chile
x) Talk: Cuerpos de representación maximales (Maximal representation fields), XXV Jornada Matemática de la Zona Sur (XXV South Zone Mathematics Days), (04/2012). Concepción, Chile.
xi) Talk: Reticulados y órdenes (Lattices and orders), Monthly Number Theory Seminar PUC-UV [11/2011]. Pontificia Universidad Católica de Chile (PUC), Santiago, Chile.
xii) Talk: Árboles y cuerpos de representación (trees and representation fields), IX Algebra and Number Theory Workshop [12/2011]. Santa Cruz, Chile.
xiii) Talk: Cuerpo de representación para órdenes conmutativos (Representation field for commutative orders), XXIV Jornada Matemática de la Zona Sur (XXIV South Zone Mathematic Days), [04/2011]. Pucón, Chile.
xiv) Talk: Representation fields for skew-hermitian forms, LXXIX Encuentro de la Sociedad de Matemáticas de Chile (LXXIX Meeting of the Chilean Math Society), [11/2009]. Olmué, Chile.
xv) Talk: Spinor class fields over function fields, VII Algebra and Number Theory Workshop, [12/2008]. Talca, Chile.
xvi) Talk: Spinor class fields and sheaves, LXXVIII Encuentro de la Sociedad de Matemáticas de Chile (LXXVIII Meeting of the Chilean Math Society), [11/2008). Valparaiso, Chile.
xvii) Talk: On existence of representation fields, XVII Congreso Matemático Capricornio (XVII Capricorn Math Congress), [7/2008]. Iquique, Chile.
xviii) Talk: On permutations on a finite field, LXXVII Encuentro de la Sociedad de Matemáticas de Chile (LXXVII Meeting of the Chilean Math Society), [11/2007]. Punta de Tralca, Chile.
xix) Talk: Maximal orders and representations, VI Algebra and Number Theory Workshop [12/2006]. Talca, Chile.
xx) Talk: Spinor class fields: A counterexample, LXXVI Encuentro de la Sociedad de Matemáticas de Chile (LXXVI Meeting of the Chilean Math Society), [10/2006]. Olmué, Chile.
xxi) Talk: Género espinorial cuaterniónico y extensiones de cuerpos (Quaternionic spinor genus and field extensions), LXXV Encuentro
de la Sociedad de Matemáticas de Chile (LXXV Meeting of the Chilean Math Society), [11/2005]. Olmué, Chile.
xxii) Talk: Hilbert symbols and cohomology, XIX Jornadas de Matemáticas de la Zona Sur (XIX South Zone Mathematic Days), [4/2005]. Talca, Chile.
xxiii) Talk: Anillos, ideales, y sistemas dinámicos (Rings, ideals and dynamical systems), IV Algebra and Number Theory Workshop [12/2003]. Talca, Chile.
xxiv) Talk: La ecuación {Sigma}_i x^{n(i)}y^{m(i)}=0 en característica 2 (The equation {Sigma}_i x^{n(i)}y^{m(i)}=0 in characteristic 2), LXXIII Encuentro de la Sociedad de Matemáticas de Chile (LXXIII Meeting of the Chilean Math Society), [10/2003]. Vicuña, Chile.
xxv) Talk: Sistemas algebraicos, r-mixing y geometría de sucesiones excepcionales (r-mixing algebraic systems and geometry of exceptional sequences). LII Coloquio Nacional de Sistemas Dinámicos (LII National Dynamical Systems Seminar), [8/2003]. Santiago, Chile.
xxvi) Talk: Sistemas ergódicos algebraicos (Algebraic ergodic systems). LI Coloquio Nacional de Sistemas Dinámicos (LII National Dynamical Systems Seminar), [10/2002]. Santiago, Chile.
xxvii) Talk: Cuerpos de clases espinoriales y representaciones (Spinor class fields and representations). III Algebra and Number Theory Workshop [12/2001]. Talca, Chile.
xxviii) Talk: Reticulados, cohomología y clasificación (Lattices, cohomology and classification). II Algebra and Number Theory Workshop (12/2000). Talca, Chile.