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Manuel Rivera

I am an Assistant Professor in the Department of Mathematics at Purdue University.

From September 2016 to August 2019 I held a joint Research Assistant Professorship between University of Miami and CINVESTAV (Mexico City). From September 2015 to August 2016 I was a postdoc at Institut de Mathématiques de Jussieu- Paris Rive Gauche. I completed my PhD in mathematics at CUNY Graduate Center in 2015 under the supervision of Dennis Sullivan. From 2006-2010 I was an undergraduate at MIT. I am originally from San Juan, Puerto Rico.

My research is in algebraic topology. I am interested in understanding different kinds of structures from the lens of homotopy theory. This had led me to think about homological and homotopical algebra, category theory, geometric topology, string topology, non-commutative geometry, symplectic topology, among other topics.

I am also interested in sharing, discussing, and developing initiatives and ideas that promote research and higher education, as well as general mathematical culture, in Latin America and among underrepresented groups in academia.

You can contact me at manuelr at purdue.edu.

My research is supported by the NSF Grant DMS 210554 Algebraic structures in topology and geometry.

CV

Publications and preprints

An algebraic model for the free loop space, M. Rivera, submitted, arXiv:2210.10096

The simplicial coalgebra of chains under three different notions of weak equivalence, G. Raptis, M. Rivera, submitted, arXiv:2206.06335

Adams’ cobar construction as a monoidal E-infinity coalgebra model of the based loop space, A. Medina-Mardones, M. Rivera, submitted, arXiv:2108.02790

Categorical models for path spaces, E. Minichello, M. Rivera, M. Zeinalian, accepted in Advances in Mathematics, arXiv:2201.03046

String topology in three flavours, F. Naef, M. Rivera, N. Wahl, to appear in EMS Surveys in Mathematical Sciences [Dennis Sullivan’s 80th birthday volume] arXiv:2203.02429

Adams’ cobar construction revisited, M. Rivera, Doc. Math. 27, 1213-1223 (2022), arXiv:1910.08455

Invariance of the Goresky-Hingston algebra on reduced Hochschild homology, M. Rivera, Z. Wang, Proc. London Math. Soc., 125 (2022), 219-257. arXiv:1912.13267

The simplicial coalgebra of chains determines homotopy types rationally and one prime at a time, M. Rivera, F. Wierstra, M. Zeinalian, Trans. Amer. Math. Soc. 375 (2022), no. 5, 3267–3303. arXiv:2006.05362

Rational homotopy equivalences and singular chains, M. Rivera, F. Wierstra, M. Zeinalian, Algebr. Geom. Topol. 21 (2021), no. 3, 1535-1552. arXiv:1906.03655

Singular chains and the fundamental group, M. Rivera, M. Zeinalian, Fund. Math. 253 (2021), no. 3, 297-316. arXiv:1801.06410

The colimit of an infinity local system as a twisted tensor product, M. Rivera, M. Zeinalian, High. Struct. 4 (2020), no. 1, 33-56. arXiv:1805.01264

The functor of singular chains detects weak homotopy equivalences, M. Rivera, F. Wierstra, M. Zeinalian, Proc. Amer. Math. Soc. 147 (2019), no. 11, 4987-4998. arXiv:1808.10237

Singular Hochschild cohomology and algebraic string operations, M. Rivera, Z. Wang, J. Noncommut. Geom. 13 (2019), no. 1, 297-361. arXiv:1703.03899

A combinatorial model for the path fibration, M. Rivera, S. Saneblidze, J. Homotopy Relat. Struct. 14 (2019), no. 2, 393-410. arXiv:1706.00983

Cubical rigidification, the cobar construction, and the based loop space, M. Rivera, M. Zeinalian, Algebr. Geom. Topol. 18 (2018), no. 7, 3789-3820. arXiv:1612.04801

A combinatorial model for the free loop space fibration, M. Rivera, S. Saneblidze, Bull. Lond. Math. Soc. 50 (2018), no. 6, 1085-1101. arXiv:1712.02644

Automorphism groups of countably categorical linear orders are extremely amenable, F. Dorais, S. Gubkin, D. McDonald, M. Rivera, Order 30 (2013), 415–426. arXiv:1202.4092

Expository notes

Chen’s Power Series Connection
Summary of Chen’s theory of iterated integrals and the (co)homology of the based loop space
Toronto Spaces, The Harvard College Mathematics Review 3 (2011) 24-27

Videos and slides

Talk at the four decades celebration of Sullivan’s Einstein Chair Seminar, at CUNY GC January 2023
Slides for a talk at Stasheff’s Deformation Theory Seminar at UPenn about algebraic models for the free loop space, November 2021
Talk at the Higher Homotopical Structures program at CRM, Barcelona, February 2021
Colloquium talk at Stony Brook University January 2019
Talk at the IAS, Princeton, NJ May 2017
Talk at “Algebraic Geometry in Mexico 2017”
Talk at “Algebraic Geometry in Mexico 2016”

Conferences

I organized the conference Algebraic structures in topology which took place in San Juan, Puerto Rico from May 30th – June 3rd 2022.

Seminars

I am currently one of the organizers for:

Purdue Topology Seminar
Cibercoloquio Latinoamericano de Matemáticas: See the following article at the Notices for more on this initiative: Cibercoloquio Latinoamericano de Matemáticas, Campos, Daniel; Rivera, Manuel; Salazar, María Amelia; Samper, José Alejandro; Simental, José; Villar, Soledad. Notices Amer. Math. Soc. 68 (2021), no. 5, 793–797

Links to other seminars:

Teaching

Category theory and simplicial methods – Spring 2023- MA598
Foundations of Real Analysis – Fall 2022 – MA341-202 Purdue University
Foundations of Real Analysis – Fall 2022 – MA341-100 Purdue University
Linear Algebra – Spring 2022- MA265 Purdue University
Hochschild and cyclic homology – Fall 2021 – MA 697 Purdue University
Elementary Topology – Fall 2021 – MA571 Purdue University
Introduction to Algebraic Topology – Spring 2021 – MA 572 Purdue University
Elementary Topology- MA571- Fall 2020 Purdue University
Introduction to higher category theory- Spring 2019 Purdue University