Phone: | Ext.#6187 | |
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Office: | Young Hall 105 | |

Office Hours: | Fall 2018: Mondays and Wednesdays from 2:30pm to 4pm and by appointment. | |

Email: | trevino at lakeforest.edu |

I am an associate professor of Mathematics at Lake Forest College. I started working as an Assistant Professor in 2013 and was promoted in 2018. I got a baccalaureate of Science degree in Mathematics from the University of Texas at El Paso (UTEP) in 2006. I completed my PhD in Mathematics at Dartmouth College in June 2011, working with Carl Pomerance. My main research interest is Number Theory. My dissertation is titled Numerically Explicit Estimates for Character Sums; it involves a blend of analytic number theory and some serious computing. I continue to be interested in these kinds of problems, such as estimating the least quadratic non-residue, the least inert prime in a real quadratic field, and other concrete inequalities. Besides my interest in number theory, I also write recreational mathematics papers.

I am very fond of mathematical competitions such as the Mathematical Olympiad. I have been involved either as student or as a teacher in the Olympiad since the year 2000. Since 2015, I have been a member of the Mexican Mathematical Olympiad Committee. My love for problem solving extends to other competitions such as the Putnam, and to problem solving publications (such as The Riddler in FiveThirtyEight and the problem section in mathematics journals.)

Research Statement

Teaching Statement

Math 331: Modern Algebra II Lake Forest College Spring 2019.

Math 375: Combinatorics Lake Forest College Spring 2019.

Math 110: Calculus I Lake Forest College Fall 2018.

Math 330: Modern Algebra I Lake Forest College Fall 2018.

Math 499: Senior Seminar: Great Theorems of Mathematics Lake Forest College Fall 2018.

Math 110: Calculus I Lake Forest College Summer 2018.

CS 112: Computer Science I Lake Forest College Spring 2018.

Math 230: Abstract and Discrete Mathematics Lake Forest College Spring 2018.

Math 329: Number Theory Lake Forest College Spring 2018.

CS 112: Computer Science I Lake Forest College Fall 2017.

Math 311: Introduction to Real Analysiss Lake Forest College Fall 2017.

Math 499: Senior Seminar: Great Theorems of Mathematics Lake Forest College Fall 2017.

Math 210: Multivariable Calculus Lake Forest College Fall 2016.

Math 340: Geometry Lake Forest College Fall 2016.

Math 499: Senior Seminar: Great Theorems of Mathematics Lake Forest College Fall 2016.

Math 110: Calculus I Lake Forest College Summer 2016.

Math 150: Introduction to Probability and Statistics Lake Forest College Spring 2016.

Math 330: Modern Algebra I Lake Forest College Spring 2016.

Math 411: Real Analysis II Lake Forest College Spring 2016.

FIYS 169: Recreational Mathematics Lake Forest College Fall 2015.

Math 210: Multivariable Calculus Lake Forest College Fall 2015.

Math 230: Abstract and Discrete Mathematics Lake Forest College Fall 2015.

Math 110: Calculus I Lake Forest College Summer 2015.

Math 160: Math Methods with Applications Lake Forest College Summer 2015.

Math 150: Introduction to Probability and Statistics Lake Forest College Spring 2015.

Math 214: Differential Equations Lake Forest College Spring 2015.

Math 230: Abstract and Discrete Mathematics Lake Forest College Spring 2015.

FIYS 169: Recreational Mathematics Lake Forest College Fall 2014.

Math 330: Modern Algebra I Lake Forest College Fall 2014.

Math 160: Math Methods with Applications Lake Forest College Summer 2014.

Math 150: Introduction to Probability and Statistics Lake Forest College Spring 2014.

Math 230: Abstract and Discrete Mathematics Lake Forest College Spring 2014.

Math 160: Math Methods with Applications Lake Forest College Fall 2013.

Math 230: Abstract and Discrete Mathematics Lake Forest College Fall 2013.

Math 25: Further Topics Single Variable Calculus Swarthmore College Spring 2013.

Math 53: Analytic Number Theory Swarthmore College Spring 2013.

Math 28: Linear Algebra Honors Course Swarthmore College Fall 2012.

Math 58: Number Theory Swarthmore College Fall 2012.

Math 25: Further Topics Single Variable Calculus Swarthmore College Spring 2012.

Math 77: Advanced Topics in Algebra: Algebraic Number Theory Swarthmore College Spring 2012.

Math 27: Linear Algebra Swarthmore College Fall 2011.

Math 20: Discrete Probability, Dartmouth College Summer 2009.

Math 2: Calculus with Algebra and Trigonometry, Dartmouth College Winter 2009.

Clicking on the links below takes you to a preprint of the paper. Undergraduate (at the time) authors are shown with an asterisk.

- Triangular sums of consecutive triangular numbers (with P. Pollack and D. Subramaniam*).

Submitted. - On generalizing happy numbers to fractional base number systems (with M. Zhylinski*).

Submitted. - Introducción a la Teoría de Números Probabílistica.

Submitted. - Probabilistic Proof that 1+2+...+n = n(n+1)/2.

To Appear in American Mathematical Monthly. - Expected number of dice rolls for the sum to reach n.

To Appear in American Mathematical Monthly. - A birthday in St. Petersburg.

The College Mathematics Journal vol. 50, no. 1, pp. 36-40. - An Inclusion-Exclusion Proof of Wilson's Theorem.

The College Mathematics Journal vol. 49 (2018), no. 5, pp. 367-368. - Walking on rational numbers and a self-referential formula

with M. Fortman*, K. Kupiec*, and M. Rawlings*. Elemente der Mathematik 73 (2018), no. 4, 161-169. - A short proof of a sum of powers formula

American Mathematical Monthly 125 (2018), no. 7, 659-659. - Finding the four squares in Lagrange's Theorem

with Paul Pollack. Integers 18A (2018), article A15, 16 pages (electronic). - Counting Perfect Polynomials

with U.C. Cengiz* and P. Pollack. Finite Fields and Their Applications 47C (2017) pp. 242-255 DOI information: 10.1016/j.ffa.2017.05.006. - Resolving Grosswald's conjecture on GRH

with Kevin McGown and Timothy Trudgian. Functiones et Approximatio, Commentarii Mathematici 55.2 (2016), pp. 215-225. - The Burgess inequality and the least k-th power non-residue

International Journal of Number Theory Vol. 11, No. 5 (2015), pp. 1-26. - The smoothed Pólya-Vinogradov inequality

with K. Adamczewski*, Integers 15 (2015), article A20, 11 pages (electronic). - The least k-th power non residue

Journal of Number Theory 149 (2015), pp. 201-224. DOI information: 10.1016/j.jnt.2014.10.019 - The primes that Euclid forgot

with P. Pollack, American Mathematical Monthly 121 (2014), no. 5, 433-437. - Sets of monotonicity for Euler's totient function

with P. Pollack and C. Pomerance, The Ramanujan Journal 30 (2013), no. 3, 379-398. - On the maximum number of consecutive integers on which a character is constant

Moscow Journal of Combinatorics and Number Theory 2012, vol.2, iss. 1, pp. 56-72. Corrigendum appeared in Moscow Journal of Combinatorics and Number Theory 2017, vol.7, iss. 3. - The least inert prime in a real quadratic field

Mathematics of Computation, vol. 81, no. 279, July 2012, pp. 1777-1797. - Multi-dimensional Frobenius problem

with J. Amos*, I. Pascu*, V. Ponomarenko and Y. Zhang*, Involve, a Journal of Mathematics 4-2 (2011), 187--197. - On the counting function for the generalized Niven numbers

with R. Daileda, J. Jou*, R. Lemke-Oliver* and E. Rossolimo*, J. Théor. Nombres Bordeaux 21 (2009), no. 3, 503--515.

Clicking on the link takes you to a website where you can purchase the book.

Cuban Mathematical Olympiads written by Roberto Bosch and translated by yours truly. Published by XYZ Press in 2017. ISBN-13: 978-0-9968745-4-0.Kevin Kupiec and Marina Rawlings worked with me on a 3-week research project as Richter scholars in May 2014. The project was titled
**What's so rational about the alphabet?**. They presented the following poster at the Richter Symmposium in
June 2014. The following are the slides they made for a presentation at the Lake Forest student Symposium in
April 2015. This work together with Margaret Fortman's (mentioned below) is the content of the paper *Walking on rational numbers and a self-referential formula*.

Ugur Caner Cengiz worked with me on a 10-week research project as a Richter scholar in the Summer of 2014. He wrote the following paper:
Finding perfect polynomials mod 2. The following are the
slides for the presentation he gave at the Lake Forest Student Symposium. This work, together with some results by Paul Pollack formed the content of the paper *Counting perfect polynomials*.

Margaret Fortman worked with me on a 4-week research project as a Richter scholar in the Summer of 2015. She made the following poster on her work on the Tupper self-referential formula:

Poster. Margaret also wrote a paper about her work. You can find it here. This work together with the work done by Kevin Kupiec and Marina Rawlings is the content of the paper *Walking on rational numbers and a self-referential formula*.

Robert Mecham worked with me on a 10-week research project as a Richter scholar in the summer of 2015. Robert worked on the problem of placing first-year students to a first-year studies class that they rank highly while balancing constraints. His presentation "First-world solutions to First-year problems" was given at the Richter Symposium in July 2015.

Noel Orwothwun worked with me on a 4-week research project as a Richter scholar in the summer of 2016. Noel worked on Beatty sequences and prime number races on them. His talk Beatty Sequences and the Prime Race was given at the Richter Symposium in June 2016.

Dipika Subramaniam worked with me on a 4-week research project as a Richter scholar in the summer of 2018. Dipika worked on finding sums of triangular numbers that add to a triangular number. Her poster Playing with Triangular Numbers was presented at the Richter Symposium in June 2018.

Transcendental Numbers by Jacob Juillerat, Spring 2016.

A direct combinatorial approach to the sum of the kth powers of the first n integers This is a longer piece than the one I submitted for publication. It includes examples for the sum 1+2+...+n and 1^2+...+n^2.

A result about random variables inspired by the book "The Simpsons and their mathematical secrets". Note: Pages 3 and 4 are a proof of the same result as on the first 2 pages but without using integrals.

A bijective proof of a simple mutliplicative number theory result. Note: The theorem is easy to prove without the bijection but maybe not as insightful.

A pair of surprising identities. Note: I wrote these proofs after Walter Stromquist mentioned these identities to me. My proofs, as you can see, are not that well written. I apologize for that.

Facebook Friends. Note: I recommend reading Steven Strogatz NYTimes article on this phenomenon.

On the number of coin flips needed to get the same number of heads as tails. Note: My colleague Sugata Banerji mentioned this as a programming exercise and I liked the problem (specially with the appearance of Catalan numbers). I think it also has some similarities with the St. Petersburg paradox. I submitted a different version (appears above as "A birthday in St. Petersburg") for publication.

A proof of a stronger Law of Sines using the Law of Cosines. An unnatural use of the law of cosines to prove the law of sines.

I work at the Awesome Math summer program in the summers teaching fun mathematics to middle school and high school students from around the country (and the world). It is a great 3 week program (offered three times per summer) with classes of varying difficulty. The program was funded by Titu Andreescu, who has authored many problem-solving books and was the leader of the USA IMO (International Math Olympiad) team for many years.

Picture with my Geometry 1 students at the University of Puget Sound in 2016

Playing with Triangular Numbers given at the West Coast Number Theory Conference on December 16, 2018.

Playing with Triangular Numbers given at the California State University at Chico on December 13, 2018.

Finding the four squares in Lagrange's theorem given at the Joint Math Meetings at the AMS Special Session on A Showcase of Number Theory at Liberal Arts Colleges given on January 11, 2018.

Counting Perfect Polynomials given at the West Coast Number Theory Conference on December 18, 2017.

Gerrymandering and Math given at the Faculty Discussion Group at Lake Forest College on October 26, 2017.

Walking on numbers and a self-referential formula given at the Awesome Math Summer Camp at Cornell University on August 3, 2017.

The least quadratic non-residue and related problems given at the Séminaire Dynamique, Arithmétique, Combinatoire of the I2M Math Laboratory of Aix-Marseille Université on March 28, 2017.

Summer Research Projects for First Year Students given at the AMS Special Session on Open & Accessible Problems for Undergraduate Research at the Joint Math Meetings on January 7, 2017.

Resolving Grosswald's conjecture assuming GRH given at the West Coast Number Theory Conference on December 17, 2016.

Summer Research Projects for First Year Students given at the California State University at Chico on October 17, 2016.

Resolving Grosswald's conjecture on GRH given at the Carl Pomerance's 70th birthday Conference on June 11, 2015.

The least quadratic non-residue and related problems given at the California State University at Chico on April 17, 2015.

Prime gaps: a breakthrough in number theory given at the Faculty Discussion Group at Lake Forest College on September 16, 2014.

El mínimo no-rediduo cuadrático y otros problemas relacionados given at the Instituto de Ciencias Matemáticas in the Universidad Autónoma de Madrid on July 21, 2014.

The Burgess inequality and the least k-th power non-residue given at the Emerging New Faces in Analytic Number Theory (ENFANT) workshop at the Hausdorff Institute on July 12, 2014.

The least quadratic non-residue modulo a prime and related problems given at the Midwest Number Theory Conference for Graduate Students and Recent PhDs on June 3, 2014.

The primes that Euclid forgot given at the Oberlin Number Theory Seminar on March 10, 2014.

Prime Gaps: A breakthrough in number theory given at the Seminario Interuniversitario de Investigación en Ciencias de Matemáticas in Ponce, Puerto Rico on March 1, 2014.

The primes that Euclid forgot given at the AMS Special Session on Analytic Number Theory at the Joint Meetings on January 16, 2014..

Bounds on graphs with high girth and high chromatic number given at the INTEGERS Erdos Centennial Conference on Combinatorial Number Theory on October 26, 2013.

The primes that Euclid forgot given at the Underrepresented Students in Algebra and Topology Symposium on April 20, 2013.

On squares and non-squares modulo a prime given at Lake Forest College on February 12, 2013.

The Burgess inequality and the least k-th power non-residue. given at the Special Session on Arithmetic Statistics at the Joint Meetings of the AMS/MAA on January 10, 2013.

The numerically explicit Burgess inequality and an application to quadratic nonresidues. given at the AMS Sectional Meeting in Akron, OH on October 21, 2012.

El mínimo no-rediduo cuadrático y otros problemas relacionados given at a colloquium lecture at the Universidad de Colima on May 8, 2012

On the maximum number of consecutive integers on which a character is constant given at the Joint Math Meetings on January 5, 2012.

The smoothed Pólya-Vinogradov inequality given at the Integers Conference, October 28, 2011.

The least quadratic non-residue and related problems given at a colloquium lecture at Swarthmore College on March 1st, 2011, at University of North Texas on October 2011 and at the automorphic forms seminar at Purdue University in February 2012.

The least inert prime in a real quadratic field given at the Palmetto Number Theory Series XIV, December 4, 2010.

Burgess bounds for the Burgess inequality for character sums given at the Integers conference, October 16, 2009.

On the counting function of the generalized Niven numbers given at the Québec/Maine Number Theory Conference, October 2008.

The higher-dimensional Frobenius problem given with Yan Zhang at Young Mathematician's Conference (YMC), August 2005.

Mathematical Olympiads given at a colloquim lecture at Trinity University on July 2005.

*Last modified on January 30, 2019 by Enrique.
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