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Instituto de Matemáticas Aplicadas UCV
Lomelí, Luis

Lomelí, Luis

Doctor en Matemáticas, Purdue University, Indiana, Estados Unidos

luis.lomeli@pucv.cl

(+56) 32 227 4007

Pagina del academico

Langlands Program and L-functions

Langlands functoriality conjectures predict generalized reciprocity laws between two areas of current research, namely, Number Theory and Representation Theory. Classically, Artin reciprocity and Tate’s thesis are covered by the abelian case of Class Field Theory. The Langlands Program addresses, among other aspects, the open problem posed by the non-abelian case. Artin L-functions on the Galois side and automorphic L-functions on the representation theoretic side carry important arithmetic information that is to be preserved by Langlands correspondences. In particular, we study automorphic L-functions via the Langlands-Shahidi method over function fields and have obtained new cases of Langlands functoriality and important applications.

Programa de Langlands y funciones L

Las conjeturas de Langlands consisten en leyes de reciprocidad generalizadas entre dos áreas de investigación actual, la teoría de números moderna y la teoría de representaciones. En la teoría de cuerpos de clases se estudia el caso abeliano, tesis de Tate y reciprocidad de Artin. El programa de Langlands aborda, entre otros aspectos, el problema abierto que nos plantea el caso no abeliano. Las funciones L asociadas a representaciones de Galois, por parte de la teoría de números, y las funciones L automorfas, por parte de la teoría de representaciones, contienen información aritmética importante la cual debe ser preservada por las conjeturas de Langlands. En particular, hacemos un estudio de las funciones L automorfas por medio del método de Langlands-Shahidi sobre un cuerpo de funciones y obtenemos casos y aplicaciones importantes en la funtorialidad de Langlands.

CV

Work experience:

March 2016 – Present: Profesor Asociado, Instituto de Matemáticas PUCV.

Prof. Lomelí has held visiting and postdoctoral positions at the following institutions:

IHÉS, MPIM, MSRI, University of Oklahoma, Purdue University and the University of Iowa.

Ph.D.:

2007 Purdue University.

Coordinator:

Ph.D. Research Line in Number Theory, Instituto de Matemáticas PUCV.

SNI level 1

Sistema Nacional de Investigadores, CONACYT, México.

Grants:

2020 — 2021: “Number Theory: Interconnections with Algebra, Combinatorics and Representation Theory”, MATH-AmSud 20-MATH-06. CNRS, CONICYT, COLCIENCIAS. Coordinator (Chile).

2018: “Local Langlands correspondence for GL(n) and the Langlands-Shahidi method”. CONICYT MEC Grant No. 80170039, Chile. Sponsoring Scientist.

2017 — 2020: “Langlands Program and L-functions”. FONDECYT Standard Grant No. 1171583, Chile. Principal Investigator.

Academic year 2016: Project VRIEA/PUCV 039.367, Chile. Principal Investigator.

Publications

L. Lomelí, “The LS method over funcion fields”, preprint.

L. Lomelí, “Ramanujan Conjecture and Riemann Hypothesis for the unitary groups over function fields”, preprint.

L. Lomelí, “Langlands Program and Ramanujan Conjecture: a survey”, to appear in Contemporary Mathematics A.M.S.

G. Henniart and L. Lomelí, “Asai cube L-functions and the local Langlands conjecture”, to appear in Journal of Number Theory. Published online (2020).

L. Lomelí, “Rationality and holomorphy of Langlands-Shahidi L-functions over function fields”, Mathematische Zeitschrift 291 (2019), 711-739.

W. T. Gan and L. Lomelí, Globalization of supercuspidal representations over function fields and applications, Journal of the European Mathematical Society, 20 (2018), 2813-2858.

Lomelí, L. A., On automorphic L-functions in positive characteristic, Annales de l’Institut Fourier 66 (2016), 1733-1771.

R. Ganapathy and L. Lomelí, On twisted exterior and symmetric square gamma factors, Annales de l’Institut Fourier 65 (2015), 1105-1132.

L. Lomelí, The LS method for the classical groups in positive characteristic and the Riemann Hypothesis, American Journal of Mathematics 137 (2015), 473-496.

G. Henniart and L. Lomelí, Uniqueness of Rankin-Selberg factors, Journal of Number Theory 133 (2013), 4024-4035.

G. Henniart and L. Lomelí, Characterization of gamma factors: the Asai case, International Mathematics Research Notices Vol. 2013, No. 17, pp. 4085-4099.

G. Henniart and L. Lomelí, Local-to-global extensions for GL(n) in non-zero characteristic: a characterization of symmetric and exterior square gamma factors, American Journal of Mathematics, 133, (2011), pp. 187-196.

L. Lomelí, Functoriality for the classical groups over function fields, International Mathematics Research Notices, Vol. 2009, No. 22, pp. 4271-4335.

Courses

Current course

Algebraic Number Theory, First Semester 2020.

Courses taught at IMA PUCV

Algebra II, Second Semester 2019.

Topics in Number Theory, First Semester 2019.

Number Theory, Second Semester 2018.

Trees and Buildings, First Semester 2018.

Introduction to Linear Algebra and Differencial Equations, First Semester 2017.

Analytic Number Theory, Second Semester 2016.

Measure Theory, First Semester 2016.

Students

Ph.D. Students:

Héctor del Castillo.

Emilio Améstica.

Master’s Students:

Javier Navarro.

Cristóbal Torres (2020).

Cristian Pérez (2019).

Nicolás Álamos (2018).

Workshop / Conference

Workshop:

Nov 29 – Dec 4, 2020: “Langlands Program: Number Theory and Representation Theory”
Banff, Casa México Oaxaca.

Number Theory and Representation Theory Day:

Jan 15, 2020: “Jornada de Teoría de Números y Representaciones”, Universidad de Santiago de Chile.

 

Study Group / Mini Courses

Study Group (En Español)

Modular Forms, First Semester 2020.

Mon, Fri, 12:20 hours (online).

Mini Course (In English):

Jan 13 and 17, 2020: Anne-Marie Aubert (CNRS, Sorbonne Université), “Local Langlands correspondence for depth zero representations of the classical groups”, Instituto de Matemáticas PUCV, Valparaíso.

Study Group

Chevalley Groups, Second Semester 2019.

p-adic Fourier Analysis and Representation Theory of p-adic Groups, First Semester 2019.

Representation Theory, Second Semester 2018.

Mini Course:

July 3, 5, 10 and 12, 2018: Guy Henniart (Université d’Orsay), “Carayol representations and ramification”, Instituto de Matemáticas PUCV, Valparaíso.

Study Group

The Local Langlands Conjecture for GL(2), First Semester 2018.

The Local Langlands Conjecture for GL(2), Second Semester 2017.

Pi Day

“Día de Pi 2020”, Instituto de Matemáticas PUCV, Valparaíso.