Homepage of Oliver Lorscheid

NameOliver Lorscheid
FunctionProfessor (pesquisador associado)
InstituteInstituto Nacional de Matematica Pura e Aplicada
AdressRoom 350
Estrada Dona Castorina 110
CEP: 22460-320, Rio de Janeiro
RJ, Brazil
Phone+55 21 2529 5150
Emailoliver at impa dot br

On sabbatical in Bonn

Until February 2020, I am on a sabbatical at the Max Planck Institute for Mathematics in Bonn.


My general area of research is algebraic and arithmetic geometry, with focus on F1-geometry and its applications to tropical geometry and matroids, quiver Grassmannians and automorphic forms. A more comprehensive list of my research interests can be found in my CV.

The title of my PhD thesis is Toroidal automorphic forms over function fields. My advisor was Gunther Cornelissen at the University of Utrecht.

The title of my Diploma thesis is Comnpleteness and compactness for varieties over local fields. My advisor was Florian Pop at the University of Bonn.


Since 2012, I am organizing the Colóquio de Geometria e Aritmética do Rio de Janeiro together with Carolina Araujo, Cecília Salgado, Eduardo Esteves, Nivaldo Medeiros.

Together with Ethan Cotterill, Alicia Dickenstein and Nicolas Puignau, I have organized the workshop Tropical Geometry in the Tropics at IMPA during May 11-15, 2015.

Together with Asher Auel, Cecília Salgado and Antony Várilly-Alvarado, I have organized the workshop Rational Points at IMPA during May 25-29, 2015.

Together with Markus Reineke and Thorsten Weist, I have organized the workshop Workshop on Quiver Grassmannians and their Applications at the University of Wuppertal during March 21-24, 2017.

Together with Margarida Melo, Johannes Nicaise and Sam Payne, I have organized the ICM stellite conference Tropical Geometry and Moduli Spaces in the Hotel La Plage in Cabo Frio, close to Rio de Janeiro, during August 13-17, 2018.

Link to regular seminars at IMPA.

Teaching at IMPA

2018.3: Algebra 2 (terca- e quinta-feira, 15h30--17h00, na sala 349)

2018.2: Blueprints and Tropical Scheme Theory

2017.2: Commutative Algebra

2017.1: Algebraic Number Theory

2016.3: Geometria Analítica Não-arquimediana e Geometria Tropical

2016.3: Tópicos de Álgebra: Introdução ao Programa de Langlands.

2016.2: Algebra 1

2016.2: Curvas elipticas (curso de leitura)

2015.3: Algebra 2 (junto com Carolina Araujo)

2015.3: Teoria dos corpos dos classe (curso de leitura)

2015.3: Variedades toricos (curso de leitura, junto com Carolina Araujo)

2015.3: Topos theory (study seminar, joint with Reimundo Heluani)

2015.3: Tropical moduli of curves (study seminar, joint with Ethan Cotterill and Marco Pacini)

2015.2: Palestras sobre geometria tropical (parte do curso "Topicos de algebra")

2015.1: Introduction to Algebraic Number Theory

2014.3: Algebra 2

2014.3: Espaços de moduli e fibrados vetoriais (curso de leitura)

2014.2: Topicos classicos de geometria algébrica (curso de leitura, junto com Carolina Araujo)

2014.1: Minicourse "F1-geometry and applications"

2013.3: Geometria Algébrica I

For more information, see IMPA's calender.


Tropical geometry over the tropical hyperfield.

Higher balancing for locally tropically convex tropical varieties (with Emilio Peixoto).

Descartes' rule of signs, Newton polygons, and polynomials over hyperfields (with Matt Baker).

The moduli space of matroids (with Matt Baker).

Scheme theoretic tropicalization.

Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials (with Thorsten Weist).


Representation type via Euler characteritics and singularieties of quiver Grassmannians (with Thorsten Weist). To appear in the Bulletin of the London Mathematical Society.

The geometry of blueprints. Part II. Forum Math. Sigma 6, e20, 90pp., 2018.

F1 for everyone. Jahresber. Dtsch. Math.-Ver. 120(2), 83-116, 2018.

Plücker relations for quiver Grassmannians (with Thorsten Weist). Algebr. Represent. Theory, 22(1):211--218, 2019.

Cech cohomology over F1^2 (with Jaret Flores and Matt Szczesny). J. Algebra 485, 269-287, 2017.

Quiver Grassmannians of extended Dynkin type D - Part 1: Schubert systems and decompositions into affine spaces (with Thorsten Weist). To appear in the Memoirs of the American Mathematical Society.

On the relation between hyperrings and fuzzy rings (with Jeffrey Giansiracusa and Jaiung Jun). Beitr. Algebr. Geom. 58(4), 735-764, 2017.

Quasicoherent sheaves on projective schemes over F1 (with Matt Szczesny). J. Pure Appl. Algebra 222(6):1337-1354, 2018.

Blue schemes as relative schemes after Toen and Vaquie. J. Algebra 482, 264-302, 2017.

A blueprinted view on F1-geometry. In Absolute Arithmetic and F1-geometry (edited by Koen Thas). European Mathematical Society Publishing House, 2016.

Schemes as functors on topological rings (with Cecilia Salgado). J. Number Theory 159, 193-201, 2016.

Schubert decompositions for quiver Grassmannians of tree modules (with an appendix by Thorsten Weist). Algebra Number Theory 9(6), 1337-1362, 2015.

Blueprints - towards absolute arithmetic?. J. Number Theory 144, 408-421, 2014.

On Schubert decompositions of quiver Grassmannians. J. Geom. Phys. 76, 169-191, 2014.

Toroidal automorphic forms for function fields. Israel J. Math. 194, no. 2, 555-596, 2013.

Graphs of Hecke operators. Algebra Number Theory 7, no. 1, 19-61, 2013.

Automorphic forms for elliptic function fields. Math. Z. 272, no. 3-4, 885-911, 2012.

Projective geometry for blueprints (with Javier Lopez Pena). C. R. Math. Acad. Sci. Paris 350, no. 9-10, 455-458, 2012.

Toroidal automorphic forms, Waldspurger periods and double Dirichlet series (with Gunther Cornelissen). Multiple Dirichlet series, L-functions and automorphic forms, 131-146, Progr. Math., 300, Birkhäuser/Springer, New York, 2012.

Algebraic groups over the field with one element, Math. Z. 271, no. 1-2, 117-138, 2012.

Sheaves and K-theory for F1-schemes (with Chenghao Chu and Rekha Santhanam). Adv. Math. 229, no. 4,2239-2286, 2012.

The geometry of blueprints. Part I. Adv. Math. 229, no. 3, 1804-1846, 2012.

Mapping F1-land: an overview of geometries over the field with one element (with Javier Lopez Pena). Noncommutative geometry, arithmetic, and related topics, 241-265, Johns Hopkins Univ. Press, Baltimore, MD, 2011.

Torified varieties and their geometries over F1 (with Javier Lopez Pena). Math. Z. 267, no. 3-4, 605-643, 2011.

Functional equations for zeta functions of F1-schemes. C. R. Acad. Sci. Paris, Ser. I 348, 1143-1146, 2010.

Toroidal automorphic forms for some function fields (with Gunther Cornelissen). Journal of Number Theory 129 (6), 1456-1463, 2009.

Toroidal automorphic forms for function fields. Thesis 2008.

On the K-theory of graph C*-algebras (with Gunther Cornelissen and Matilde Marcolli). Acta Appl. Math. 102 (1), 57-69, 2008.

Completeness and compactness for varieties over a local field. Arch. Math. 88 (4), 344-348, 2007.

Lecture notes

Link to the lecture notes on Blueprints and Tropical Scheme Theory, 2018.

Lecture notes of a course on Galois theory, 2018.

Lecture notes of a series of 4 lectures on F1-geometry, held at IMPA, Rio de Janeiro, in January 2014.

Lecture notes of the talk F1-geometry, given at the Mathematische Arbeitstagung, Bonn 2009.


Slides of the talk The moduli space of matroids, held in Tuebingen, 2019.

Slides of the talk What is a Tropical Scheme?, held in Regensburg, 2018.

Slides of the talk From Quadratic Reciprocity to the Langlands Programme, in honor of Langlands receipt of the Abel prize, held in Rio de Janeiro, 2018.

Slides of the talk Matroids in Tropical Geometry, held at ALGA, Rio de Janeiro 2017.

Slides of the talk F1-geometry and Its Applications, held at the 6th European Congress of Mathematics, Krakow 2012.