Department of Mathematics

# Current Undergraduate Research Projects

**Student:** Grace Burton**Mentor:** Oscar Vega**Dates:** AY 2020-2022**Project:** Multiplier sequences over F_{2}, F_{3}, and F_{4}

**Student:** Raul Reynoso and Luis Vega**Mentor:** Oscar Vega**Dates:** AY 2021-2022**Project:** The mathematics of Tsuro

**Students:** Manuel Chacón and Gobind Puniani **Mentor: **Mario Banuelos**Dates**: AY 2019-current**Project:** The Role of Activation Functions in Deep Learning

## Past Undergraduate Research Projects

**Students:** Dhan Bautista, Anthony Cortez**Mentor:** Michael Bishop**Dates:** AY 2020-2021**Project:** Basics of Quantum Computation

**Students:** Joseph White**Mentor:** Michael Bishop**Dates:** AY 2020-2021**Project:** Explorations into Quantum Gravity

**Student:** Amanda Liles**Mentor: **Carmen Caprau**Dates: **Fall 2020**Project:** Braid Groups and Temperley-Lieb Algebra

** Student: **Jack Luong

**Khang Tran**

**Mentor:****2019-2020 AY**

**Dates:****Zeros of polynomials generated from rook paths**

**Project:**Main Outcome: Submitted for publication,https://arxiv.org/abs/2008.08707.

**William Reimer**

Student:Student:

**Michael Bishop**

**Mentor:****AY 2019-2020**

**Dates**:**Fractal Geometry**

**Project:****Juhoon Chung**

**Student:****Stefaan Delcroix**

**Mentor:****AY 2019-2020**

**Dates**:**$n=a^2+k \cdot b^2$**

**Project:****Jessica Yearton**

**Student:****Stefaan Delcroix**

**Mentor:****Spring 2020**

**Dates**:**$n=x^2+3y^2$**

**Project:****Students:** Andres Zumba and Libby Farrell**Mentor:** Khang Tran**Dates:** 2018-19 AY**Project:** Zero Distribution of a Sequence of Polynomials with a three-term recurrence**Main Outcome:** Presentation at the JMM 2018.

**Student: **Maria Diaz and Yuliana Segura**Mentor: **Oscar Vega**Dates: **2018-19 AY**Project: **The commuting graph of a group and its connections to the automorphism group of a
graph

**Student: **Haylee Harris**Mentor: **Oscar Vega**Dates: **2018-19 AY

**The coloring graph of strongly regular graphs**

Project:

Project:

**Main Outcome:**H. Harris. The Coloring Graph of Complete Graphs.

*The PUMP Journal of Undergraduate Research*. 2 (2019). 150-160.

**Student:** Andres Zumba**Mentor:** Khang Tran**Dates:** 2018-19 AY**Project:** Zeros of polynomials with four-term recurrence and linear coefficients**Main Outcome: **K. Tran, A. Zumba, Ramanujan J (2020). https://doi.org/10.1007/s11139-020-00263-0

**Student:** Michael Dunia**Mentor:** Michael Bishop**Dates:** 2018-19 AY**Project:** Minimum Length Scales from Quantum Gravity

**Student:** Robert Duyst**Mentor:** Agnes Tuska**Dates:** 2018-19 AY**Project:** Philosophy of Science and Mathematics Education**Main Outcome:** Smittcamp Honors Program Research Paper, presentation in Math 191T Junior Seminar
in Mathematics Education (Spring, 2019).

**Student: **Erick Gonzalez **Mentor: **Oscar Vega**Dates: **Spring 2018**Project: **The commuting graph of a group**Main Outcome:** E. Gonzalez. Commuting Graphs of Split Metacyclic Groups. *The PUMP Journal of Undergraduate Research*. 1 (2018). 62-67.

**Student: **Jack Luong and Josh Williamson**Mentor: **Tamás Forgács**Dates: **2017-18 AY

**The convergence of series with recursively defined term**

Project:

Project:

**Main Outcome:**T. Forgács, J. Luong and J. Williamson, A note on infinite series with recursively defined terms

*.*

*Amer. Math. Monthly*,

**126**(3) (2019), pp. 269-274. DOI: 10.1080/00029890.2019.1555411

**Students:** Abigayle Dirdak and Daniel Hooker**Mentor:** Khang Tran**Dates:** 2017-18 AY**Project:** Zeros of polynomials generated by multiple quadratic factors**Main Outcome:** Poster presentation at the JMM 2018.

**Student: **Miguel Baza **Mentor: **Oscar Vega**Dates: **2017-18 AY

**Multiplier sequences over finite fields**

Project:

Project:

**Students: **Bradley ScottOscar Vega

Mentor:

**2017-18 AY**

Dates:

Dates:

**Blocking sets of polygons and complements of hyperplane arrangements in finite projective planes**

Project:

Project:

**Student: **Jagdeep BasiOscar Vega

Mentor:

**2017-18 AY**

Dates:

Dates:

**Grim on complete multipartite graphs**

Project:

Project:

**Main Outcome:**Barretto, S.; Basi, J.; Miyake, M. Grim on Complete Multipartite Graphs.

*The PUMP Journal of Undergraduate Research*

**1**(2018). 48-61.

**Student: **Ariana Cavazos**Mentor:** Carmen Caprau**Dates:** 2017-18 AY**Project: **An invariant of colored singular links**Main Outcome:** Student's Presentations at Northern California Undergraduate Math Conference 2018
and Central California Research Symposium 2018.

**Student: **Abigayle Dirdak**Mentor:** Carmen Caprau**Dates: **Summer 2017**Project:** Alexander- and Markov-type theorems for virtual trivalent braids **Main Outcome:** C. Caprau, A. Dirdak, R. Post and E. Sawyer. Alexander- and Markov-type theorems
for virtual trivalent braids. *J. of Knot Theory and its Ramifications ***28**, No. 1 (2019) 195003 (33 pages).

**Student: **Samuel Barretto and Matthew Miyake **Mentor: **Oscar Vega**Dates: **2016-17 AY

**The study of Grim on complete multipartite graphs having a singleton as one of their partitions**

Project:

Project:

**Main Outcome:**Barretto, S.; Basi, J.; Miyake, M. Grim on Complete Multipartite Graphs.

*The PUMP Journal of Undergraduate Research*

**1**(2018). 48-61.

**Students: **Summer Al-Hamdani, Alexandra Leon**Mentor: **Tamas Forgacs

**Dates:**2016-17 AY

**Project:**Generating classical multiplier sequences using products of exponential factors

**Main Outcome:**AL-HAMDANI, Summer; LEON, Alexandra. On Classical Multiplier Sequences.

**The PUMP Journal of Undergraduate Research**, [S.l.], v. 1, p. 14-29, Jan. 2018. ISSN 2576-3725.

**Student: **Andres ZumbaKhang Tran

Mentor:

**2016-17 AY**

Dates:

Dates:

**Zeros of polynomials with four-term recurrence**

Project:

Project:

**Main Outcome:**K. Tran, A. Zumba. Zeros of polynomials with four-term recurrence.

*Involve, a Journal of Mathematics*Vol. 11 (2018), No. 3, 501-518.

**Students: **Andres Zumba** Mentor:** Khang Tran

**2015-16 AY**

Dates:

Dates:

**Zeros of polynomial generated by the reciprocal of 1+t^2+zt^3**

Project:

Project:

**Student's presentation in 37th Annual Central California Research Symposium**

Main Outcome:

Main Outcome:

**Student: **Brennen Fagan Doreen De Leon

Mentor:

**2015-16 AY**

Dates:

Dates:

**On the Immunization of Small Computer Networks**

Project:

Project:

**Student: **Aramayis OrkusyanDoreen De Leon

Mentor:

**2015-16 AY**

Dates:

Dates:

**On the Immunization of Large Computer Networks**

Project:

Project:

**Students: **Andrew De La Pena and Sarah McGahanCarmen Caprau

Mentor:

**2014-15 AY**

Dates:

Dates:

**Virtual singular braids and links**

Project:

Project:

**Main Outcome:**C. Caprau, A. De La Pena and S. McGahan. Virtual singular braids and links.

*Manuscripta Mathematica*(September 2016) 151, Issue 1, 147-175.

**Students:**Brennen Fagan and Aramayis Orkusyan

**Doreen De Leon**

**Mentor:****Dates:**2014-15 AY

**Project:**Optimal inoculation schemes to prevent the spread of computer viruses in an organization

**Students: **Manuel Lopez and Majerle ReevesDoreen De Leon

Mentor:

**2014-15 AY**

Dates:

Dates:

**Optimal immunization schemes for the flu vaccine**

Project:

Project:

**Students: **Richard Adams and Janae Dixon** Mentor:** Oscar Vega

**2014-15 AY**

Dates:

Dates:

**Grim on large graphs**

Project:

Project:

**Adams, R.;Dixon, J.; Elder, J.; Peabody, J.; Vega, O.; Willis, K. Combinatorial Analysis of a Subtraction Game on Graphs. International Journal of Combinatorics vol. 2016, Article ID 1476359, 8 pages, 2016. https://www.hindawi.com/journals/ijcom/2016/1476359/**

Main Outcome:

Main Outcome:

**Students: **Elaina Aceves and Jennifer Elder.** Mentor:** Carmen Caprau

**2013-14 AY**

Dates:

Dates:

**A Yang-Baxter model for specializations of the Kauffman polynomial**

Project:

Project:

**Presentation at the JMM in Baltimore, January 2014.**

Main Outcome:

Main Outcome:

**Students:** Elaina Aceves and Jennifer Elder.**Mentor:** Carmen Caprau**Dates:** 2013-14 AY**Project:** On invariants for spatial graphs**Main Outcome:** E. Aceves and J. Elder. On invariants for spatial graphs. *Rose-Hulman Undergraduate Mathematics Journal ***16**, No 2, Fall 2015, 17 pages.

** Students: **Kelsey Friesen and Thoa Tran.

**Carmen Caprau**

**Mentor:****2013-14 AY**

Dates:Dates:

**Invariants for virtual singular links**

Project:Project:

**Main Outcome:**Presentations (talk and poster) at the Central California Research Symposium, April 2014.

Best Oral Presentation by an Undergraduate Student Award.

Honorable Mention recognition for an Outstanding Poster Presentation.

**Student:** Serena McTeer **Mentor:** Carmen Caprau **Dates:** 2012-13 AY **Project:** The category of sl(3) webs **Main Outcome:** Presentation at the Central California Research Symposium, April 2013.

** Students: **Elaina Aceves and David Heywood

**Oscar Vega**

**Mentor:****Summer 2012**

Dates:Dates:

**Embedding of cycles in PG(n,q)**

**Project:****Aceves, Elaina; Heywood, David; Klahr, Ashley; Vega, Oscar. Cycles in projective spaces.**

**Main Outcome:***J. Geom.*

**105**(2014), no. 1, 111-117.

** Student: **Jamie Peabody

**Oscar Vega**

**Mentor:****Summer 2012**

Dates:Dates:

**Embeddings of cycle-related graphs in finite projective planes**

**Project:**

**Peabody, Jamie; Vega, Oscar; White, Jordan. Cycles, wheels, and gears in finite planes.**

**Main Outcome:***Contrib. Discrete Math.*

**8**(2013), no. 2, 75-87.

Poster presentation at the JMM in San Diego (Outstanding Poster Presentation Award winner), January 2013.

** Student: **James Stinecipher

**Ke Wu**

**Mentor:****Summer 2012**

Dates:Dates:

**Assessing the risk of liver cancer in Hepatitis B patients using Survival Analysis.**

**Project:**

**Poster presentation at the Central California Research Symposium (Davin Youngclarke Award for Undergraduate Research Poster).**

**Main Outcome:**** Student: **Joel Smith

**Carmen Caprau**

**Mentor:****2012-13 AY**

Dates:Dates:

**The singular Temperley-Lieb category**

**Project:**

**Caprau, Carmen; Smith, Joel. The singular Temperley-Lieb category.**

**Main Outcome:***ISRN Geom.*2014, Art. ID 321509, 9 pp.

** Students: **Elaina Aceves and Jennifer Elder.

**Carmen Caprau**

**Mentor:****2012-13 AY**

Dates:Dates:

**Invariants for spatial graphs**

**Project:**

**Poster presentation at the JMM in San Diego, January 2013. Talk at the JMM in Baltimore, January 2014.**

**Main Outcome:**** Student: **Benjamin Wright

**Oscar Vega**

**Mentor:****Fall 2012**

Dates:Dates:

**Representation of numbers and their powers in base b**

**Project:**

**Poster presentation at the JMM in San Diego, January 2013.**

**Main Outcome:**** Students: **Megan Kuneli, Robyn McDonald, and Katherine Urabe

**Oscar Vega**

**Mentor:****2011-12 AY**

Dates:Dates:

**The well-covered dimension of Cartesian products of graphs.**

**Project:**

**Birnbaum, Isaac; Kuneli, Megan; McDonald, Robyn; Urabe, Katherine; Vega, Oscar. The well-covered dimension of products of graphs.**

**Main Outcome:***Discuss. Math. Graph Theory*

**34**(2014), no. 4, 811-827.

** Students: **Jamie Peabody and Karen Willis

**Oscar Vega**

**Mentor:****2011-12 AY**

Dates:Dates:

**A subtraction game on graphs**

**Project:**

**Poster presentation at the JMM in Boston, January 2012.**

**Main Outcome:**** Student: **Katherine Urabe

**Tamas Forgacs**

**Mentor:****2011-12 AY**

Dates:Dates:

**Investigations of Legendre multiplier sequences**

**Project:**

**Blakeman, Kelly; Davis, Emily; Forgacs, Tamas; Urabe, Katherine. On Legendre Multiplier Sequences,**

**Main Outcome:***Missouri J. Math. Sci.,*

**24**(2012), 7-23.

Poster presentation at the JMM in Boston (Outstanding Poster Presentation Award winner), January 2012.

**David Heywood**

**Student:****Carmen Caprau**

**Mentor:****Spring and Fall 2012**

Dates:Dates:

**The SO(2n) Kauffman polynomial**

**Project:**

**Caprau, Carmen; Heywood, David; Ibarra, Dionne. On a state model for the SO(2n) Kauffman polynomial.**

**Main Outcome:***Involve*

**7**(2014), no. 4, 547-563.

** Students: **James Tipton and Benjamin Wright

**Tamas Forgacs**

**Mentor:****2010-11 AY**

Dates:Dates:

**Multiplier sequences for non-orthogonal bases and simple sets in general**

**Project:**

**Forgacs, Tamas; Tipton, James, Wright, Benjamin. Multiplier Sequences for Simple Sets of Polynomials,**

**Main Outcome:***Acta Math. Hungar.*

**137**(2012), 282-295.

** Student: **James Tipton

**Carmen Caprau**

**Mentor:****2010-11 AY**

Dates:Dates:

**Connections between graph theory and knot theory**

**Project:**

**Caprau, Carmen; Tipton, James. The Kauffman polynomial and trivalent graphs. To appear in Kyungpook Mathematical Journal.**

**Main Outcomes:**Talk at the Central California Research Symposium (Outstanding Undergraduate Student Talk in Mathematical Sciences Award winner)

** Students: **Mario Banuelos and Michelle Hoshiko

**Tamas Forgacs**

**Mentor:****2010-11 AY**

Dates:Dates:

**Multiplier sequences in finite fields**

**Project:**

**Talks at the CURM spring research conference, the Central California Research Symposium, the Pacific Coast Undergraduate Mathematics Conference, and the JMM in New Orleans. All given in 2011.**

**Main Outcomes:**** Student: **Isaac Birnbaum

**Oscar Vega**

**Mentor:****2010-11 AY**

Dates:Dates:

**The well-covered dimension of graphs.**

**Project:**

**Birnbaum, Isaac; Kuneli, Megan; McDonald, Robyn; Urabe, Katherine; Vega, Oscar. The well-covered dimension of products of graphs.**

**Main Outcome:***Discuss. Math. Graph Theory*

**34**(2014), no. 4, 811-827.

** Students: **Michelle Hoshiko and Lucy Montalvo

**Tamas Forgacs**

**Mentor:****Summer and Fall of 2009**

Dates:Dates:

**Highway speeding and policing**

**Project:**

**Poster presentation at the JMM in San Francisco, January 2010.**

**Main Outcome:**** Student: **Ana-Cristina Jimenez

**Carmen Caprau**

**Mentor:****2009-10 AY**

Dates:Dates:

**Studying knots via braids**

**Project:**

**Poster presentation at the JMM in New Orleans (Outstanding Poster Presentation Award winner), January 2011.**

**Main Outcome:**** Student: **Dhiraj Holden

**Stefaan Delcroix and Oscar Vega**

**Mentors:****Summer and Fall 2010**

Dates:Dates:

**The Collatz conjecture**

**Project:**

**Holden, Dhiraj. Results on the 3x+1 and 3x+d conjectures.**

**Main Outcome:***Fibonacci Quart.*

**49**(2011), no. 2, 131-133

** Student: **Iliana Perez

**Tamas Forgacs**

**Mentor:****2008-09 AY**

Dates:Dates:

**The cultural costs of international trade**

**Project:**

**Talk at the annual meeting of the Missouri Valley Economic Association in St. Louis, MO in October, 2008.**

**Main Outcome:****Student: **Ana-Cristina Jimenez**Mentor: **Larry Cusick

**2008-09 AY**

Dates:

Dates:

**Three strands, three crossings, one p-colorability**

Project:

Project:

**Poster presentation at the JMM in San Francisco, January 2010.**

Main Outcome:

Main Outcome: