Carlos Alejandro Alfaro Montúfar

Banco de México

alfaromontufar [at] gmail [dot] com

carlos [dot] alfaro [at] banxico [dot] org [dot] mx

**Research Interests**

Mathematically, my research lie on Algebraic Combinatorics, Topological Graph Theory and Optimization. Furthermore, I have an ample experience dealing with economical and technological problems, ranging from implementing optimization models for the mexican central bank to creating video games for Android, as well as, simulating 3D RNA recombination.

**Short Bio**

I received my Ph.D. degree in 2014 from the Department of Mathematics at CINVESTAV under supervision of Carlos E. Valencia. Since 2014, I am researcher at BANXICO (the Central Bank of Mexico), where my principal activities consist in developing and implementing optimization models. Between 2011 and 2012, I visited the HP Labs at Palo Alto, CA to work with Burcu Aydin, Kemal Guler, Lyle Ramshaw, Pano Santos and Bob Tarjan. There, I developed algorithms to study tree-structure data, and supported the implementation of assignment algorithms for resource planning allocation.

CV (PDF)

**Journal Papers**

Algebraic combinatorics

- Small clique number graphs with three trivial critical ideals (with C. E. Valencia) submitted.
- Graph classes for critical ideals, minimum rank and zero forcing number, submitted.
- Distance ideals of graphs (with L. Taylor) submitted.
- Critical ideals, minimum rank and zero forcing number (with Jephian C.-H. Lin) submitted.
- On two-quotient strong starters for F_q (with Christian Rubio-Montiel and Adrián Vázquez-Ávila) accepted in Utilitas Mathematica [https://arxiv.org/abs/1609.05496].
- Digraphs with at most one trivial critical ideal (with Carlos E. Valencia & Adrián Vázquez-Ávila) in Linear and Multilinear Algebra, 2017. [https://arxiv.org/abs/1703.08621]
- Critical ideals of graphs with twin vertices (with H. H. Corrales & Carlos E. Valencia) in Advances in Applied Mathematics, 2017 [http://arxiv.org/abs/1504.06257].
- Graphs with two trivial critical ideals (with C. E. Valencia), in Discrete Applied Mathematics, 2014. [http://arxiv.org/abs/1304.4211]
- On the sandpile group of the cone of a graph (with C. E. Valencia), in Linear Algebra and its Applications, 2012. [http://arxiv.org/abs/1004.3321]

Operations Research

- Optimizing the production cost of minting with mixed integer programming (with J. Aguilera, C. Guadarrama, R. Martinez-Noriega & A. Sanchez-Flores) submitted.
- Dimension Reduction in Principal Component Analysis for Trees (with B. Aydin, E. Bullitt, A. Ladha, and C. E. Valencia), in Computational Statistics & Data Analysis, 2014. [http://arxiv.org/abs/1202.2371]

Graph theory

- The crossing number of the cone of a graph (with Alan Arroyo, Marek Derňár and Bojan Mohar) accepted in SIAM Journal of Discrete Mathematics [https://arxiv.org/abs/1608.07680].
- Covering and 2-packing numbers in graphs (with Christian Rubio-Montiel and Adrián Vázquez-Ávila) submitted.

Linear systems

- On linear systems in which the transversal number equals the 2–packing number (with G. Araujo-Pardo, C. Rubio-Montiel, A. Vázquez-Avila) submitted.
- On a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed (with A. Vázquez-Ávila) submitted.

- Graphs with few trivial critical ideals (with Carlos E. Valencia), in Electronic Notes in Discrete Mathematics, 2015. Presented in LAGOS'15.
- The crossing number of the cone of a graph (with Alan Arroyo, Marek Derňár and Bojan Mohar), in Lecture Notes in Computer Science vol. 9801, 2016. Presented in Graph Drawing'16 by Bojan Mohar (slides).
- Ciritcal ideals of digraphs (with Carlos E. Valencia and Adrián Vázquez-Ávila) accepted in Matemática Contemporânea. Presented in VII Latin American Workshop on Cliques in Graphs, November 8-11 2016.

**Patent**

Selection of data paths. US Patent Application. Ref. 83037389. (with B. Aydin, K. Guler, C. E. Valencia) Hewlett-Packard Company, 2012.

**Copyright**

Knotj3d. INDAUTOR Reg. 03-2007-100314141100-01 (with O. Gutú and R. Lopéz-Hérnandez) 2007.

JGraphs - (github) A code in Java that serves as a graphical interface for computing the critical group and the critical ideals of a digraph. It requires Sage and Emacs. But if wanted, the code can be modified to use an alternative Computer Algebra System (a previous version used Mathematica and Macaulay2) or Text Editor. This also can be used to export in Tikz format the rectilinear drawing of the graph.

CSandPile - (github) CSandPile is a tool developed in C++ for computing the group operations of the recurrent representatives of non-negative configurations of a graph. It helps to understand the combinatorial structure of the group operations of the recurrent configurations that generate the sandpile group of a graph.

RiceRocks - (github) (It doesn't work on Internet Explorer) Just push play. Enjoy and modify! I coded a version for Android watch it here.

KnotJ3D - (github) This is a Java applet, which draws 3D tangles. There are implemmented some operations like sum and closure. It was coded in Java3D, so you need to install it to run it.

Optimización de algunos costos de acuñación de moneda, presented in seminario de matemáticas at ITAM

A sandpile group characterization problem, presented in sandpile group workshop at BIRS Casa Matemática Oaxaca

Critical Ideals of Digraphs, presented in VII Latin American Workshop on Cliques in Graphs

## VII Latin American Workshop on Cliques in Graphs, November 8-11 2016

The books that inspiring me more are:

- The Sharing Economy by Arun Sundararajan

- In Pursuit of the Traveling Salesman by William J. Cook

- The Master Algorithm by Pedro Domingos

- The Poincare Conjecture: In Search of the Shape of the Universe by Donal O’Shea

More interesting readings:

- "Euler worked on coinage for the national mint" cited from Euler's Gem

- Las dos culturas de las matemáticas by W.T. Gowers

- The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them? by Antonio J. Durán, Mario Pérez, and Juan L. Varona

- How to build an economic model in your spare time by Hal R. Varian

**Some links**

Mathematics and Programming in the Cloud with CoCalc

Recent MX$20 coins

50 years of integer programming 1958-2008

** ** IFORS Developing Countries On-Line Resources page "Operational Research" (OR) is the discipline of applying advanced analytical methods to help make better decisions. By using techniques such as problem structuring methods and mathematical modelling to analyze complex situations, Operational Research gives executives the power to make more effective decisions and build more productive systems.

Hispanic Heritage: Pano Santos, a Beautiful Mind (watch an inspiring interview from FOX NEWS to Pano Santos, a scientist of HP Labs)