By Andrés R. Vindas Meléndez, graduate student at the University of Kentucky

“How do you do mathematics the Colombian way?” This was the question posed by Federico Ardila at the end of the first week of the 6^{th}Encuentro Colombiano de Combinatoria (ECCO 2018). The question my master’s thesis advisor asked motivated me to write a reflection of ECCO 2018 as a non-Colombian attempting to answer the question. I hope that this may also serve as an invitation/motivation for anyone interested in combinatorics to participate in future ECCOs.

Two years ago, in 2016 I participated in the 5^{th}Encuentro Colombiano de Combinatoria (ECCO 2016), an experience that changed my perspective and approach to doing mathematics and has motivated me to shape the inclusive mathematics community that I want to see. After her ECCO 2016 experience, Viviane Pons (Université Paris-Sud) wrote a wonderful blog post detailing and reflecting on many aspects of the conference. I encourage you to read the post here.

__What is ECCO?__

The Encuentro Colombiano de Combinatoria is a conference that runs every two years. More than a standard conference, it is a school and a gathering of mathematicians of all academic levels who bring their passion for learning and sharing combinatorics. In 2003, the first ECCO was organized by Federico Ardila as a resulting effort of his SFSU-Colombia Initiative. One goal was to build closer ties among students from Colombia, the USA, and other countries to provide a space for collaboration in person, especially for students who have limited access to such an experience. In 2016, Federico Ardila wrote an article for the Notices of the AMS, titled, “Todos Cuentan: Cultivating Diversity in Combinatorics,” where he shares more on the creation of ECCO and his approach to creating spaces for students to grow in their mathematical learning, primarily in combinatorics.

__ECCO 2018__

ECCO 2018 was hosted by the Universidad del Norte in Barranquilla, Colombia from June 5^{th}to the 16^{th}and also served as a CIMPA School. There were over 100 participants and about half were Colombian and the other half foreigners. This year’s theme was “Combinatorics meets Algebra, Geometry, and Optimization.” The two weeks consisted of four minicourses, which highlighted the theme. Two minicourses were delivered per week, one in the morning and one in the afternoon. The first week’s minicourses were led by Günter Ziegler (FU Berlin) and Vic Reiner (Univ. of Minnesota). For those of you familiar with polytopes, you have probably encountered Günter Ziegler’s book, “Lectures on Polytopes.” I fanboyed as he signed my copy of the book. Günter Ziegler’s lectures presented us with *extremal examples and combinatorial parameters of polytopes*. Vic Reiner delivered his lectures covering *q-counting and representation theory*. The second week’s minicourses were led by Lauren Williams (Berkeley/Harvard) and Rekha Thomas (Univ. of Washington). Lauren Williams’ lectures provided us with an *introduction to total positivity and cluster algebras*. Rekha Thomas’ lectures took us into the world of *polynomial optimization*.

The style and presentation of this ECCO was similar to the previous years, but one change was that this was the first year that Federico Ardila was not directly involved in the organizing committee. Now organized by a committee of former participants, this change highlights the sustainability of the event and the strong foundations that Federico has laid for ECCO’s success. Many joked that Federico was like the proud grandfather sitting around watching the fruition of his efforts.

After each lecture we took a coffee break where we had some great (and incredibly hot) Colombian coffee and pastries. Then we gathered again in the lecture hall where each day we had two graduate students or post-docs give a research talk. These students and post-docs were from institutions around the world (Chile, Colombia, Finland, Germany, Mexico, USA, etc.). After the talks, we then split into groups and participated in exercise sessions where we worked on problems related to the content of the minicourses that day. Each week also had a plenary talk and a session on open problems. The first week we had the pleasure of hearing from Mauricio Velasco (Universdad de los Andes) and the second week we were honored to have Sara Billey (Univ. of Washington). Viviane Pons and her team also led a SageMath tutorial per week. A new addition to ECCO was a poster session, where many undergraduate and graduate students (including myself), presented their work. The conference ended with a panel, which I had the pleasure of participating in, where motivations, struggles, and social issues within mathematics were discussed.

__Math the Colombian Way__

We often hear of the French, German, American, or Hungarian way of approaching mathematics, but how do you do mathematics the Colombian way? Home to Federico and many of the ECCO organizers, Colombia now has a strong combinatorics presence. The Colombian approach to doing mathematics (combinatorics), as I see it, places a strong emphasis in community building. To many mathematicians it is clear that mathematics is a collaborative effort, but there is more than just collaboration to building a community. In building a community, there needs to be a sense of friendship and accountability. That sense of friendship and accountability allows everyone to actively participate in mathematics in a comfortable setting, without a fear of mistakes and also acknowledges that as a group there is no success if any person is left without understanding the mathematics occurring.

At the beginning of ECCO, participants were asked to read and share their thoughts of the Community Agreement. By laying down the expectations of respect, we assured one another that this would be a rewarding and welcoming experience for all. Many of the things on the Community Agreement seem like “common sense,” but as present times have shown us, common sense is not so common. Many participants shared that the Community Agreement set mental expectations and reduced any anxiety they may have had in realizing interactions with other participants.

Part of the community building was also not in the classroom, rather it was centered around music and the dancefloor. The introduction of dance to a professional setting, such as a mathematics conference, may seem strange to many, but it is engrained in the Colombian way of mathematics. Dancing tears down the hierarchy and power dynamics that we often see in mathematics classrooms and conferences. At ECCO we had leading researchers seeing participants struggle with mathematical concepts, but we also had those participants see the leading mathematicians skirmish to learn some salsa moves on the dancefloor. Indeed, dancing with each other allowed for everyone to foster an inclusive and lively environment, as well as a cultural understanding of our host country and its people. The dancefloor was a space where many participants who were unfamiliar with the dances and music of Colombia soon were cheered on and supported by the local students to the point where there was a feeling of security. This spirit of support transferred to the conference, especially for the problem sessions.

At ECCO, the participants solidify their understanding of the concepts presented during the lectures by working on exercises that were written by the minicourse instructors. During the problem sessions, the groups consisted of participants from all levels, normally a professor/post-doc, a graduate student, and several undergraduates. Many of the problems were challenging yet rewarding once the groups worked together to reach a solution. As with many approaches to learning mathematical concepts, by working on complex problems there is a building of perseverance and reflection. The exercises were mathematically meaningful, but what is noteworthy is that all group members played an active role in reaching a solution and understanding of the concepts. I observed that the more experienced mathematicians went directly to thinking about the abstraction of the problems, where the younger students emphasized a more concrete approach to exemplify the theory occurring in the problem, of course both ways of thinking are valuable. An overarching outcome from every problem session is that everyone can engage, be excited about, and contribute to mathematics. After working on the problems for an hour or so, volunteers presented their group’s solutions. Here we saw wonders of how working in a supportive environment can contribute to mathematical understanding beyond what we have been exposed to. For example, we had students who had never had Abstract Algebra presenting deep results from Representation Theory. Experiences like these are a result from the wonderful exposition of the minicourse instructors, but also the patience and guidance of the more mathematically experienced group members.

Another aspect of the Colombian approach towards mathematics is the celebration of all successes and contributions, no matter how big or small. During the panel, one of the panelists joked that we clapped for everything to the point where it became second nature to clap after someone spoke. In spaces like ECCO, every participant had a role of being a motivator. By clapping after a solution was presented, after a lecture or research talk was delivered, we encouraged one another to be the best mathematician we can be, while showing our support and attention.

The Colombian approach, as I have interpreted it, may not seem all too different compared to other approaches towards doing mathematics. But I can assure you that when we have all done our best to build a community that is equally professional, welcoming, inclusive, and excited about mathematics we can see our potential as mathematicians and can observe that mathematics can provide a life that is both academically and socially fulfilling.

¡Bravo! Wonderful summary of a wonderful experience. We have a lot to learn from the Colombian way.