# Welcome to Antonio Montero's personal website

In this website you will find information about my work as mathematician as well as some of my non-academic interests. Scroll down or use the navigation bar to find out more.

I started my Ph.D. on July 2015 under the supervision of Dr. Daniel Pellicer . I expect to graduate by the spring of 2019.

### Contact info

Office $\infty$ (infinity)
Centro de Ciencias Matemáticas
UNAM Campus Morelia
Km. 8 Antigua Carretera a Pátzcuaro #8701.
Col. Ex-hacienda San José de la Huerta.
Apartado Postal 61-3 (Xangari) C.P. 58089.
Morelia, Michoacán. México.

### Research interests

I am interested in symmetry groups and symmetry properties of discrete and combinatorial objects such as polytopes, maps and graphs. My Ph. D. research project is related to chiral extensions of abstract polytopes.

• English: .
• Spanish: .

### Education

• Ph.D. in Mathematics (July 2015 - present)
• M. Sc. in Mathematics (February 2013 - June 2015)
• Defense: June 17, 2015.
• Thesis: Realizaciones regulares de poliedros en el 3-toro (Regular realizations of polyhedra in the 3-torus) PDF, Spanish.
• B. Sc. in Physics and Mathematics (August 2008 - February 2015)
• Defense: February 14, 2013.
• Thesis: Poliedros regulares en el 3-toro (Regular polyhedra in the 3-torus) PDF, Spanish.

### Publications and on-going projects

Here is a list of my publications and on-going research projects. I try to give as many access to them as possible (mostly through my author page on ArXiv). In any case, if you are interested in any of them you can always write me an email.

• Chiral extensions of abstract polytopes (on-going).
• Equivelar toroids with few flag-orbits (joint work with José Collins).
• Regular polyhedra in the $3$-torus.

### Talks

Here is a list of the talks I have given on several conferences. In many of them a pdf of the slides is available. This list is still under construction, so if you want to get the slides of talk and the slides are not available yet, just write me an email.

#### Equivelar toroids with few flag-orbits

Talk given at the conference Symmetries and covers of discrete objects 2016 at Queenstown, New Zealand in February 2016.

Abstract: An $(n+1)$-toroid is a quotient of a tessellation of the $n$-dimensional euclidean space with a lattice group. Toroids are generalizations of maps in the torus on higher dimensions and also provide examples of abstract polytopes. Equivelar maps in the $2$-torus ($3$-toroids) where classified by Brehem and Kühnel in 2008. In 2012 Hubard, Orbanić, Pellicer and Weiss classified equivelar $4$-toroids. However, a complete classification for any dimension seems to be a very hard problem. In the talk we will present a classification of equivelar $(n+1)$-toroids with less than $n$ flag-orbits; in particular, we will discuss a classification of $2$-orbit toroids of arbitrary dimension.

#### Regular polyhedra in the $3$-torus

Talk given at the workshop SIGMAP (Symmetries In Graphs, Maps And Polytopes) 2014 at West Malvern, United Kingdom in July 2014.

Abstract: Since Grünbaum's paper about regular polyhedra in ordinary $3$-space in the 70's, there has been a lot of work towards a classification symmetric polyhedra-like structures in several spaces. Coxeter, Schulte, McMullen and many other authors have studied symmetric tessellations of the $n$-dimensional torus. J. Bracho, in joint work with other authors found the regular polyhedra with planar faces in the projective space; a couple of years after, McMullen completed the list of regular projective polyhedra. In this talk we'll discuss the problem of classifying regular polyhedra in the $3$-dimensional torus.

## Outreach

As a mathematician and moreover, as a scientist I believe that it is important to take science close to people (at least, closer). As a result of that belief, during my education as a mathematician I am and have been involved in some outreach projects. I list some of them below. Currently I am working on my own project, but so far it is just an idea on my mind. Hopefully this section will be used to show this project.

I also have a couple of outreach papers that are intended to explain some aspects of my research work to non-mathematicians. You can find them here:

• "Simetrías: las Matemáticas detás de la belleza". Published on "Saber más" No.24, 2016 pp. 21-23. See article .
• "Simetrías de polígonos, poliedros y otras policosas". Published on "BUM, campus bulletin UNAM Morelia" No.62, 2016 p. 5. See article .

## Personal

So far I have only talked about my work (which sf fine, since this website is intended to do so). However in this section I hope you get to know a little bit about the man behind the mathematician

### The blogger

If you are interested in my publications as a blogger, you are welcome to visit the following blogs:

• SomeLittlePoints : This is my personal blog. There is not much there yet but hopefully the number of entries will increase.
• Fismateros : This is a blog that some of my friends and colleges use to maintain. It's been a while since the last entry, however, it is not officially dead and hopefully you will find a couple of interesting things there.

### The amateur musician

I play several musical instruments, however I am not particularly gifted in any of them. Recently I have focused my time on the bass guitar. Actually, I play on a band, but we haven't had many gigs nor even the opportunity of serious rehearsal, mostly because I haven't been in the same city than the other guys for a while. We are planning to become a blues band, but meanwhile you can check a couple of our rock covers in this play list

Reading is one of the activities that I enjoy the most. I usually read novels since I often read before bed and it helps me to rest. You can visit my GoodReads profile to see what I have read, what I am currently reading and what I want to read.

### Everybody else...

Of course I can continue adding columns and talking about other aspects of my life, however I will finish with these words.

I am an enthusiastic when it is about coffee, beer or mezcal (the gift that the gods gave to the Mexicans). I also enjoy playing football, frontenis and baseball. I have recently started practicing hiking and running, although I am far from being good in any of those.

## Other

This is more like a 'random' section. I will put here everything that I want to share with others but that is not directly related to me. This section is under constant construction, and hopefully it will eventually be full of interesting stuff.

• The Discrete Zoo : A very interesting project developed by Katja Berčič and probably others. This projects is intended to host computational libraries such as atlases, census and so on of discrete objects.
• vZome : A very interesting app developed by Scott Vorthmann, which intended to build virtual ZomeTool models. The features of this app are incredible and in constant improving. Some of the pictures I used for my master's thesis were created using this software.