Manuel Rivera

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

From 2016- 2019 I held a joint Research Assistant Professorship between University of Miami and CINVESTAV (Mexico City). From 2015-2016 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 using homotopical and homological techniques to study problems in geometry, topology, and algebra.

I am also interested in discussing and developing initiatives 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**

**Preprints**

- A detailed look at the Szczarba map, E. Minichiello, M. Rivera, M. Zeinalian, arXiv:2305.13638

- 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

**Accepted**

- An algebraic model for the free loop space, M. Rivera, to appear in
*Higher Structures in Geometry, Topology and Physics, Contemp. Math. AMS (2023)*

- Categorical models for path spaces, E. Minichello, M. Rivera, M. Zeinalian,
*Adv*.*Math*.**415**(2023) arXiv:2201.03046

- String topology in three flavours, F. Naef, M. Rivera, N. Wahl, to appear in
*EMS Surveys in Mathematical Sciences*(2023) [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

**Miscellaneous**

- 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

**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 the Einstein Chair Seminar at CUNY GC, January 2023
- Slides for a talk at the 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”

**Events** **organized**

I am/was an organizer for:

- Midwest Topology Seminar Spring 2023 – April 19th- April 30th
- Algebraic structures in topology in San Juan, Puerto Rico, May 30th – June 3rd 2022.
- Purdue Topology Seminar
- Cibercoloquio Latinoamericano de Matemáticas. 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

**Some seminars**

- Purdue’s Mathematical Physics Seminar
- Purdue’s Geometry and Geometric Analysis Seminar
- Northwestern’s Geometry and Physics Seminar
- CUNY Topology, Geometry and Physics Seminar
- UPenn’s Deformation Theory Seminar
- Einstein Chair Seminar at CUNY Graduate Center, its video collection, and youtube channel.
- Paris Algebra Seminar
- A list of online seminars: researchseminars.org
- A list of conferences: mathmeetings.net
- Midwest Topology Seminar

**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