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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.

Cooridnator:

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

SNI level 1

Sistema Nacional de Investigadores, CONACYT, México.

Grants:

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

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

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

Publications

L. Lomelí, “Langlands Program and Ramanujan Conjecture: a survey”, submitted.

G. Henniart and L. Lomelí, “Asai cube L-functions and the local Langlands conjecture”, submitted.

L. Lomelí, “The Langlands-Shahidi method over function fields: the Ramanujan conjecture and the Riemann Hypothesis for the unitary groups”, submitted.

L. Lomelí, “Rationality and holomorphy of Langlands-Shahidi L-functions over function fields”, Mathematische Zeitschrift, published online (2018), https://doi.org/10.1007/s00209-018-2100-7.

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, 5, 1733-1771, 2016

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.

Students

Graduate Students:

Héctor del Castillo. Ph.D. Student in the Langlands Program.

Undergraduate Students:

Alejandra Pérez Manríquez (co-director). Undergraduate dissertation: “Cuerpos locales no arquimedeanos F y representaciones irreducibles de F y F*”, 2017.

Cristóbal Torres Rojas (co-director). Undergraduate dissertation: “Medidas de Haar y Caracter Modular”,  2017.

Courses

Current course

Number Theory, Second Semester 2018.

Courses taught at IMA PUCV

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.

Schedule

Arithmetic and Geometry in Valparaíso

AGV Seminar.

Traditionally meets at 14:30 hrs on Wednesdays.

Study Group in Representation Theory

Meets regularly at 14:00 hrs on Tuesdays.