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

- String topology in three flavours, F. Naef, M. Rivera, N. Wahl,
*submitted*[Dennis Sullivan’s 80th birthday], arXiv:2203.02429

- Categorical models for path spaces, E. Minichello, M. Rivera, M. Zeinalian,
*submitted*, arXiv:2201.03046

- The cobar construction as an E-infinity bialgebra model of the based loop space, A. Medina-Mardones, M. Rivera,
*submitted*, arXiv:2108.02790

- 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**

- Adams’ cobar theorem
- 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**

- 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 will 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

Other interesting seminars:

- Purdue’s Mathematical Physics Seminar
- Northwestern’s Geometry and Physics Seminar
- UPenn’s Deformation Theory Seminar
- Einstein Chair Seminar at CUNY Graduate Center and its video collection.
- A list of online seminars: researchseminars.org
- Midwest Topology Seminar
- CUNY Topology, Geometry and Physics Seminar

**Teaching**

- 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