Mexican Mathematicians in the World: Trends and Recent Contributions

Mexican Mathematicians in the World

Trends and Recent Contributions

Edited by Fernando Galaz-García, Cecilia González-Tokman, and Juan Carlos Pardo Millán. American Mathematical Society, 2021, 319 pp. https://bookstore.ams.org/conm-775

1. Mexican Mathematicians in the World

The series of conferences, “Matemáticos Mexicanos en el Mundo” (MMM, or Mexican Mathematicians in the World), has been held biennially since 2012. The first event was held in Guanajuato, México, to establish links among the community of Mexican mathematicians in the world. Since the first conference, it has fostered connections between Mexican mathematicians working outside of Mexico and those groups and institutions inside the country. The book in review consists of the memoirs of the fourth meeting of this series of conferences, held in CMO (Casa Matemática Oaxaca) during 2018. Memoirs for the first and second meetings have also been published in the Contemporary Mathematics collection of the AMS, jointly with the Aportaciones Matemáticas collection of the Sociedad Matemática Mexicana (SMM), see 12.

The conference and its memoirs highlight the work of Mexican mathematicians, but they serve slightly different purposes. After completing their undergraduate degree, students in Mexican institutions face the question of whether or not to go abroad for their graduate studies. On the one hand, it is possible to have a successful career in mathematics having done your entire education in a Mexican university, as one can attest by looking at faculty working at large Mexican universities. Moreover, by doing this, one gets immersed in the Mexican mathematical culture and traditions and has the chance to continue and enrich it. On the other hand, going outside the country exposes students and researchers to a whole array of areas of mathematics that are underrepresented in Mexico, and this, in turn, has the potential to expand the mathematics that are done in the country.

Going abroad to do graduate studies is, however, a difficult decision. As many students who have emigrated for their graduate studies can attest, it is challenging to do a PhD in a foreign country with a different culture and lifestyle. One of the most difficult challenges lies in establishing and maintaining strong ties to the mathematical community of their own country. Without these ties one can feel culturally and mathematically alienated, a feeling that becomes stronger the further one goes into the PhD, reaching its peak when it comes to the question of what to do after receiving a doctoral degree.

The MMM conference serves as a bridge between the Mexican mathematical community within the country and that abroad. This has helped many Mexican mathematicians abroad keep mathematical ties with their home country while encouraging collaboration between the two communities. By featuring research talks by Mexican mathematicians working abroad, it also highlights the work they are doing. Over twenty mathematicians who were speakers at one of the first four meetings (and who, therefore, were working outside of Mexico at the time) have successfully obtained tenure-track positions in leading Mexican mathematical institutions. There is still substantial work to be done in order to gradually reach more ex-pats whose work is not well represented in Mexico.

By contrast, the MMM memoirs celebrate and disseminate the research of Mexican mathematicians working abroad by highlighting a small portion of it in a way that is no longer restricted to a Mexican audience (the MMM conference is held in Spanish, while the MMM memoirs are published in English). The latest volume in the series consists of thirteen articles, of research or expository type, whose only common thread is having at least one Mexican author working outside of Mexico at the time of writing. The topics of the articles range from algebraic combinatorics to model theory, including analysis, number theory, and mathematical physics, amongst others. While the themes are varied and representative, they only cover a small portion of the work that Mexican mathematicians (abroad or not) are doing. To get a better perspective, one should look at all three MMM memoirs published thus far. Of course, it should be emphasized that it would be unreasonable to expect any conference memoirs or proceedings to cover the broad range of research topics of Mexican mathematicians, especially given the number of Mexican mathematicians working abroad or in Mexico.

Of the thirteen articles of this volume, nine are (co-)authored by Mexican mathematicians working in Europe, three by Mexican mathematicians in the US, and one by a Mexican mathematician working in South America. While geographically the authors in the proceedings skew heavily towards Europe, the editors have done a good job of having authors of all academic ages represented here, from postdoctoral researchers to well-established full professors. On the other hand, it is rather unfortunate that of the twenty-three authors (Mexican or not) in the proceedings, only 13% are women. This is lower than the share of female authors in the first (4/17, or about 24%) and second (5/14, around 36%) volumes of these proceedings. It is also lower than the share of female speakers at the fourth MMM conference; seven out of nineteen talks were given by women mathematicians, including a public outreach talk. For reference, the share of female professors in the Mexico City branch of the Instituto de Matemáticas of UNAM is about 25%.

2. The Papers

By its very nature, this is not a book designed to be read from beginning to end. It consists of several articles with different target audiences in mind. The better part of the papers here are surveys addressed to graduate students and beginners in the field, and only a few are research papers addressed to specialists. In what follows, we briefly summarize, by order of appearance in the text, the contents of each of the papers and describe their intended audience.

The proceedings open with a bit on the interaction between differential geometry and mathematical physics with a survey paper by Armando Cabrera Pacheco and Carla Cederbaum. The authors introduce techniques recently developed by Mantoulidis and Schoen for computing invariants of a Riemannian manifold together with a smooth function on it, that appear naturally in the context of mathematical relativity. While the paper assumes that the reader is familiar with fairly advanced topics in differential geometry, it also includes physical motivations for the problem at hand and copious references for the uninitiated. Armando received his PhD at the University of Miami and, at the time of writing this survey paper, is a postdoctoral fellow at the University of Tübingen.

The second work in this volume is a research paper on a particular type of Riemannian manifold called a “Ricci soliton,” defined in terms of the Ricci flow. The paper’s main result gives a characterization of four-dimensional Ricci solitons admitting a Kähler metric under certain homogeneity conditions. The authors of this paper, Esteban Calviño-Louzao, Eduardo García-Río, Ixchel Gutiérrez-Rodríguez, and Ramón Vázquez-Lorenzo already have several works on the geometry of four-dimensional homogeneous manifolds. Ixchel earned her PhD at the Universidad de Santiago de Compostela in Spain and now works at the University of Vigo (Spain).

Jorge Castillejos contributes a survey paper on noncommutative geometry. He reviews a notion of dimension for C*-algebras, which is an important class of algebras of operators on a Hilbert space, and argues that this notion of dimension (called nuclear dimension) is a non-commutative version of the more usual concept of dimension of a topological space. The paper builds the theory from the definition of a C*-algebra and carefully reviews its associated topological aspects. It is, therefore, well-suited reading for a graduate student looking to learn basic noncommutative geometry. It also contains references for those looking to go deeper into the topic. While this survey was written when Jorge was a postdoctoral researcher at KU Leuven, he presently works at the Cuernavaca Unit of the Instituto de Matemáticas of UNAM.

The following paper goes back to differential geometry, with a survey by Diego Corro and Jan-Bernhard Kordaß on a slice theorem for Riemannian metrics. Roughly speaking, let us recall that if $G$ is a topological group acting on a manifold $M$, a slice through a point $m \in M$ is a submanifold $m \in N \subseteq M$ with the property that there exists a neighborhood $U$ of the identity in $G$ such that, locally around $m$, $M$ looks like $U \times N$. Theorems guaranteeing the existence of a slice are important in differential and algebraic geometry, and this survey is a nice introduction to these, suitable for graduate students. Diego wrote this paper as a postdoctoral researcher at the Karlsruher Institute for Technology (KIT) in Germany. After this, he had a brief postdoctoral position at the Oaxaca Unit of the Instituto de Matemáticas of UNAM before returning to KIT.

Eduardo Duéñez and José Iovino then contribute a research paper on using model theory to study problems in analysis and ergodic theory. It continues the tradition initiated by Terence Tao of studying analytic problems by means of structural properties of underlying sets. It is the second paper of the authors applying model theory to analysis. It is one of the most technical papers contained in this volume, and it would be suitable for graduate students (and above) specializing in analysis who want to understand and apply model-theoretic techniques to analytic problems and vice versa. Eduardo is an associate professor at the University of Texas in San Antonio.

Hildeberto Jardón-Kojakhmetov and Christian Kuehn present a survey paper on dynamical systems and, more precisely, on a class of dynamical systems that involve multiple time scales to account for subprocesses that evolve at different time rates. These are known in the literature as fast-slow systems and, as the authors note, they frequenly appear in nature. Besides examining in detail a method to solve such systems, known as the blow-up method, the paper also contains several motivating examples that make it accessible to graduate students. Hildeberto is a tenure-track assistant professor at the University of Groningen.

The following paper deals with number theory. It is a survey by Luis Alberto Lomelí on specific aspects of the Langlands program that, roughly speaking, aims to study number-theoretic properties of number fields via the representation theory of their absolute Galois group. The Langlands program is a far-reaching subject that touches several areas of mathematics, including number theory, representation theory, algebraic geometry, analysis, and combinatorics. This paper is a technical introduction to the program. It is accessible to advanced graduate students with a fair grasp of Galois theory, representation theory, and algebraic geometry. Luis Alberto is an associate professor at the Pontificia Universidad Católica de Valparaíso, in Chile.

Continuing with number theory, albeit of a slightly different flavor, Kevin McGown and Enrique Treviño contribute a survey article on techniques to solve the following classical number-theoretic problem: given a positive integer $n$, find the least integer $a$ such that the equation $x^2 = a \mod n$ has no solution. The authors focus on the case when $n = p$, an odd prime number, and survey techniques inspired by algebraic geometry and functional analysis to find upper bounds on the solution to this problem. The paper is suitable for graduate students interested in number theory. Enrique is an associate professor at Lake Forest College in the USA.

Next, we have algebraic topology, with a survey by Claudio Meneses on the notion of thin homotopy and its applications to the study of smooth connections on principal $G$ bundles, on a smooth manifold $M$. The main idea is to reformulate these connections as homomorphisms from a group associated with the manifold $M$ and a base point $p$ to $G$. The first group that comes to mind is the fundamental group $\pi _1(M, p)$, which is too coarse for this purpose. Thus, one needs to replace the usual homotopy equivalence relation of loops with a weaker notion. This thin homotopy is the main subject of the survey, which contains several historical references and is suitable for graduate students that are already familiar with the usual notion of homotopy. Claudio is a DFG Principal Investigator at the University of Kiel, Germany.

Alberto Saldaña takes us to the world of PDEs with a survey paper on Dirichlet boundary problems for the bi-Laplacian. These problems aim to find a unique (sufficiently regular) solution $u$ for the following mixed boundary problem

$$\begin{align*} \Delta ^2u & = f \quad \text{on $U$} \\ u & = g \quad \text{on $\partial U$} \\ -\partial _{\nu }u &= h \quad \text{on $\partial U$} \end{align*}$$

on a domain $U \subseteq \mathbb{R}^d$ with outward normal $\nu$. Here, $\Delta = \partial _{11} + \cdots + \partial _{dd}$ is the Laplacian and $\Delta ^2 = \Delta \circ \Delta$ is the so-called bi-Laplacian operator. Unlike the Dirichlet boundary problem for the Laplacian, the solutions $u$ for the bi-Laplacian are less rigid and do not always satisfy crucial maximum principles or positivity properties. The paper analyzes methods to overcome this by interpolating the properties of $\Delta ^2$ with those of the Laplacian. The argument is based on the study of fractional Laplacians: these are nonlocal pseudodifferential operators $\Delta ^s$, which, for $s \in (1,2)$, pertain to a scale of operators lying between the Laplacian and bi-Laplacian operators. The survey is apt for graduate students specializing in PDEs. Alberto is now a tenure-track assistant professor at the Instituto de Matemáticas in Mexico City.

Yafet Sanchez Sanchez contributes a research paper at the intersection of Mathematical Physics and PDEs. The paper in question discusses the well-posedness (existence, uniqueness, and regularity) of solutions to the hyperbolic equation

$$\begin{equation*} \square _g u = f, \end{equation*}$$

where $g$ is a metric tensor—associated with a Lorentzian manifold—that belongs to a particular bi-Lipschitz subclass of Clarke’s curve integrable spacetimes. The main result of this paper is a higher-integrability estimate $\|u\|_{H^1} \le C( \|u\|_{L^2} + \|f\|_{L^2})$ for solutions $u$ of Clarke’s $\square _g$-hyperbolic system with source term $f$. Although this is a research paper, its content seems apt for graduate students with a solid background in the theory of PDEs. Yafet is now a postdoctoral researcher in Hannover.

Another research paper is provided by David Sher, Alejandro Uribe, and Carlos Villegas-Blas, who study the spectrum of a class of operators that arise by perturbing the Laplace–Beltrami operator on a special type of manifold. After this perturbation, the resulting operator is no longer self-adjoint, making studying its spectrum a challenging problem. Because of this, the authors study a modification called the “pseudospectrum.” This paper is addressed to specialists in the area. Alejandro is a full professor at the University of Michigan, while Carlos is a full professor at the Instituto de Matemáticas in Cuernavaca.

Finally, Jacinta Torres presents a survey paper, a companion or introductory piece to the author’s work with B. Schumann, that gives a combinatorial solution to the branching problem from the special linear to the symplectic Lie algebra, that is, the study of the decomposition of irreducible $\mathfrak{sl}(2n, \mathbb{C})$ representations when restricted to $\mathfrak{sp}(2n, \mathbb{C})$. The survey motivates, presents, and gives several examples of the combinatorial techniques that are necessary toward the solution of the branching problem. As such, its target audience is graduate students interested in combinatorial representation theory. Jacinta is now an assistant professor at the Jagiellonian University in Krakow, Poland.

3. Conclusions

As intended by the editors, the papers presented here give a fairly heterogeneous sample of the research work Mexican Mathematicians in the World are doing. As a constructive criticism, one could expect the range of subjects to broaden in the future volumes in this series. For example, the fields of applied mathematics appear to be rather underrepresented. As observed above, another area of opportunity is to increase the representation of female mathematicians and other underrepresented minorities in the proceedings.

Overall, the MMM series is a commendable starting point to bridge the work of Mexican mathematicians outside and inside the country. As noted in the Preface of the volume under review, the MMM series has been well-accepted by the Mexican mathematical community. Not only has it helped researchers to (re-)connect with Mexican mathematical institutions, but it has also served as a window for Mexican mathematics to look out to the world.