The paper, “Eisenstein polynomials defining cyclic p-adic fields with minimal wild ramification”, JP Journal of Algebra, Number Theory and Applications, Vol. 49, No. 1, 93-100, (2021), analyzes polynomials with p-adic numbers as coefficients whose symmetries correspond to the rotations of a regular polygon. The collection of symmetries is known as the Galois group of the polynomial, named for French mathematician Evariste Galois (1811-1832). Properties of the polynomial’s Galois group govern important arithmetic properties of the polynomial’s roots.

In their research, the authors consider special polynomials known as Eisenstein polynomials, where all coefficients, except the leading coefficient, are multiples of the prime number p. The goal was to determine necessary and sufficient conditions on the coefficients of these polynomials to guarantee their Galois groups are cyclic; that is, they correspond to the rotations of a regular polygon. Previous research by O. Ore and S. Amano studied Eisenstein polynomials of degree p. The authors extended this work to include degrees that are multiples of p but not p^2.

The paper — ““, Involve, a journal of mathematics, 13, no. 5, 747-758 (2020) — was coauthored with several students and faculty from Awtrey’s 2018 summer REU (research experiences for undergraduate) program, funded by the National Security Agency. The authors explored properties of the discriminants of polynomials with p-adic coefficients, where p is a prime number. The discriminant of a polynomial is a special number, computed using the polynomial’s coefficients, that gives some arithmetic information about the polynomial and its roots. For example, the discriminant of the quadratic polynomial ax^2+bx+c is ����2-4����;��this number is 0 if and only if the quadratic polynomial has a repeated root.

By studying the base p representations of the discriminants, the authors were able to show that, under certain divisibility conditions, the first nonzero digit of the base p representation of a polynomial’s discriminant is a defining characteristic of that polynomial. In other words, if two polynomials of the same degree have roots that can be expressed algebraically in terms of each other’s roots, then the first nonzero digit of the base p representations of their respective discriminants are the same. As one application, they show this particular digit completely identifies the symmetry properties in the case the polynomial has degree p, thereby giving an elementary method for confirming previous research in this area.

The authors include: faculty members Sebastian Pauli (UNC-Greensboro) and Scott Zinzer (Aurora University); graduate student Sandi Rudzinski (UNC-Greensboro), and undergraduates Alex Gaura (Princeton University) and Ariel Uy (Carnegie Mellon University).

The paper, “Galois groups of even sextic polynomials”, Canadian Mathematical Bulletin, Vol. 63, No. 3, 670-676 (2020) discusses new theory and algorithmic methods to determine the symmetry properties of polynomials of the form x^6+ax^4+bx^2+c, which they call even sextic polynomials. For such a polynomial, the collection of symmetries is known as the polynomial’s Galois group and is named in honor of French mathematician Evariste Galois (1811-1832). The properties of the polynomial’s Galois group govern many important arithmetic properties of the polynomial’s roots. Therefore, a fundamental task in computational algebra is to determine the structure of a polynomial’s Galois group.

For a general degree 6 polynomial, there are 16 possibilities for the structure of its Galois group. However, only 8 of these groups actually occur as the Galois group of an even sextic polynomial. In their work, Awtrey and Jakes give a very fast algorithm for determining the structure of an even sextic polynomial’s Galois group. As an application, they provide new one-parameter families of polynomials for each possible Galois group. An implementation of their algorithm is available on Awtrey’s website.

Pi Mu Epsilon was founded at Syracuse University and incorporated in Albany, New York on May 25, 1914. Currently, there are 403 chapters at colleges and universities across the Unites States. The purpose of Pi Mu Epsilon is the promotion and recognition of mathematical scholarship among students in post-secondary institutions. It aims to do this by electing members on an honorary basis according to their proficiency in mathematics and by engaging in activities designed to promote the mathematical and scholarly development of its members.

Awtrey was first elected to the Pi Mu Epsilon National Council in 2014. He currently serves on the Executive Council, as secretary-treasurer for the organization. In this role Awtrey manages the financial and organizational records of Pi Mu Epsilon. This includes the induction of approximately 4,000 new members each year, managing the $150,000 budget, managing the contract worker, overseeing inventory and merchandise sales, and filing all state and federal taxes for the national office, the individual chapters across the nation, and the contract worker. Throughout his time on the national council, Awtrey has designed and implemented systems for efficient data collection and reporting that help organize aspects of Pi Mu Epsilon. This includes the registration and abstract submission process for the annual meeting, travel reimbursement for students and faculty members who attend the annual meeting, evaluating and judging the student presentations at the annual meeting, and developing the database that manages all subscribers to the Pi Mu Epsilon Journal, a high-quality, peer-reviewed research journal that is reviewed on MathSciNet and archived on JSTOR.

Since 2013, Awtrey has also served as the local faculty advisor for ������Ƶ’s chapter of Pi Mu Epsilon; the chapter is North Carolina Nu. ������Ƶ’s chapter was installed in 2003, making us the 13th chapter in the state of North Carolina and the 307th chapter overall. Since Awtrey began serving as advisor, there have been 102 ������Ƶ students inducted into the NC NU chapter, not including those who will be inducted later this spring.

In addition, Awtrey works with the student officers each year to run a monthly Colloquium Speaker Series where the chapter brings in national leaders in business, industry, government, and academia to showcase the beauty and application of mathematics, statistics, data science, and computer science. For this speaker series, the chapter works hard to recruit a diverse set of speakers. Usually, at least one speaker each year is a member of ������Ƶ’s Department of Mathematics and Statistics; this normally happens at the fall annual pizza party, which has been occurring since fall 2013. Since 2013, the chapter has hosted 40 speakers, not including those from this academic year.

The chapter also administers the Pi Mu Epsilon Research Award, which recognizes outstanding dissemination of research by ������Ƶ undergraduates, as evidenced by national and regional conference presentations and publications in professionally-refereed research journals. The chapter initiated this award in spring 2014 to promote and increase the dissemination of research in professional venues that ������Ƶ students and faculty produce. The Research Award recipients and their mentors are honored each spring at the Department of Mathematics and Statistics Celebration Dinner. Since 2014, there have been a total of 17 recipients of the Research Award. Collectively, these ������Ƶ students have published 30 research papers and given 72 presentations on their research; this does not include the recipients for this year, which will be announced in May 2020.

The paper, “Galois groups of doubly even octic polynomials”, Journal of Algebra and Its Applications, Vol. 19, No. 1, 1-15, (2020), was based on work done during the 2017-18 academic year, when Shannon was finishing her honor’s thesis and Altmann, Cryan, and Touchette were students at Williams High School (WHS).

The collaboration was forged when Awtrey and ������Ƶ alumni Robin French ’15 and Dee Sizemore ’86, both of whom were teachers at WHS at the time, secured a grant from the Mathematical Association of America to provide enrichment activities for mathematically gifted students at WHS. One such activity was a research project, led by Awtrey, that produced this publication.

In their work, which was also presented at SURF in 2018, the authors studied a special class of polynomials, which they call “doubly even octic polynomials”, that are of the form x^8+ax^4+b where a and b are integers. Among the most important properties of such polynomials is the collection of symmetries of the polynomial’s roots; such symmetries encode all arithmetic properties of the polynomial. Building on previous work of even quartic polynomials (which are of the form x^4+ax^2+b), the authors discovered, in some circumstances, that simple algebraic relationships between a and b completely determine the collection of symmetries.

For example, one of their main results was the following: If b is not a perfect square and the number b(a^2-4b) is a perfect square, then the symmetries of the doubly even octic polynomial x^8+ax^4+b is always the same collection of 32 permutations of the roots. Since the collection of symmetries of a polynomial is, in general, very difficult to determine, the authors’ work provides an important contribution to the discipline by showing that such a determination can often be reduced to simple algebraic relationships among the coefficients. Moreover, their techniques for proving their results pave the way for future similar discoveries.

Altmann, Cryan and Touchette are children of ������Ƶ faculty; namely, Kyle Altmann (Associate Professor of Physics), Mark Cryan (Assistant Professor of Sport Management), and Brant Touchette (Professor of Biology and Environmental Studies). Anna Altmann ’23 is now a first-year student at ������Ƶ, Sam Cryan is a sophomore at Princeton University, and Madeleine Touchette is a first-year student at Xavier University. The other student coauthor, Kiley Shannon ’18, is working as a Certified Public Accountant at Ernst & Young in Seattle.

The grant from the Mathematical Association of America is ongoing. Awtrey managed the grant for the first two years, with assistance from Associate Professor of Mathematics Jim Beuerle and Lecturer in Statistics Ryne Vankrevelen. Currently, the grant activities are led by Beuerle and Vankrevelen.

Ryu presented her research talk, “Stability, Bifurcation, and the Emergence of Synchronization and Clusters in Time-Delayed Neural Networks,” as an invited speaker to a joint session between American Mathematical Society (AMS) and Association for Women in Mathematics in Women in Mathematical Biology. She discussed how coupling delays in synaptically coupled neuronal networks affect the network’s synchronized and clustering behaviors.

Ryu also co-organized an AMS Special Session on “Utilizing Mathematical Models to Understand Tumor Heterogeneity and Drug Resistance” in which seven invited speakers shared ideas related to both understanding cancer heterogeneity and its impact on the outcome of treatment.

In addition to research-related activities, Ryu attended a mini-course on Inquiry-Based Learning, supported by a CATL Teaching and Learning Grant, to learn how to implement project-based learning practice into an upper-level course, MTH 445 Numerical Analysis during Spring 2020.

Weaver primarily participated in sessions sponsored by the Mathematical Association of America’s (MAA) Project NExT, a professional development program for new teachers of mathematics and statistics. Weaver worked with three other NExT fellows from his cohort to organize a session on Fostering an Equitable Classroom. Lee was an invited panelist, bringing the view of an equitable classroom being promoted through the explicit use of evidenced-based learning-to-learn techniques. Lee also presented in a special session on SoTL in Statistics and Probability.

Clark chaired a meeting of the MAA’s Committee on Minicourses and monitored three minicourses of the 10 minicourses offered for quality control. Clark also attended a meeting of the MAA’s Section Officers as past chair of the Southeastern Section as well as a meeting of the MAA’s Council on Meetings as chair of the Committee on Minicourses.

Awtrey presented “Automorphisms of 2-adic fields of degree twice an odd number”, which was based on joint work with mathematics major Briana Brady ’19. The talk gave an overview of the new theory and computational techniques Awtrey and Brady have recently published. In addition, Awtrey conducted official business in his role on the Executive Council of Pi Mu Epsilon, the national collegiate math honor society. Awtrey currently serves as the organization’s secretary-treasurer, and at the conference he presented a plethora of reports at their semiannual business, staffed an exhibition booth for the organization, and helped set up and judge the very large and popular undergraduate student poster session (approximately 550 student presenters). The poster session is currently organized by the MAA, but this organizational role will soon transfer to Pi Mu Epsilon; Awtrey has been and continues to be extensively involved in design, implementation, and assessment logistics of the poster session.

In their paper, “Automorphisms of 2-adic Fields of Degree Twice an Odd Number”, JP Journal of Algebra, Number Theory, and Applications, 44, no. 2, 201-210, (2019), Awtrey and Brady generalize previous results in the field of p-adic number theory concerning arithmetic properties of polynomials with 2-adic coefficients.

To understand the context of this work, we need to consider binary numbers, which are strings of 0s and 1s. Since there are only two possibilities for each digit of a binary number, we say a binary number is a “base 2” representation of a number. Not only do binary numbers play an important role in how information is stored on a computer, but they also have many fascinating applications in number theory and cryptography, since the representation of numbers in different bases reveals important properties for that number. Expanding the collection of binary numbers to include infinite strings of 0s and 1s yields the set of 2-adic numbers. One area of current mathematical research focuses on studying arithmetic properties of polynomials with 2-adic coefficients, where the goal is to understand how many such “distinct” polynomials exist and whether defining properties of the distinct classes of polynomials can be systematically characterized.

The most important classes of polynomials with 2-adic coefficients are the so-called Eisenstein polynomials, which simply have the property that every coefficient is a multiple of two and the constant term is not a multiple of four. A classical result from the last century states that there are 6 distinct Eisenstein degree two polynomials with 2-adic coefficients. Recent research from J. Jones and D. Roberts in 2006 has shown that the number of distinct Eisenstein polynomials of degree six is 30, and for degree ten the number is 126. Recent work from Awtrey and ������Ƶ alumna Erin Strosnider ‘14 showed that this number is 510 when the degree is fourteen.

Awtrey and Brady were able to generalize these results, in several ways. For example, if n is an odd number, they showed the number of distinct Eisenstein polynomials of degree 2n is equal to 2^(n+2)-2. Moreover, they determined several important characteristics of each polynomial, including a systematic representation of each distinct class of polynomials. Brady presented her work at ������Ƶ’s Spring Undergraduate Research Forum in April 2019, and she is currently working as an Implementation Consultant for Fast Enterprises in the state of Washington.

In the paper, “Totally Ramified Degree-p Extensions Over the Unramified Quadratic p-Adic Field”, International Journal of Algebra, 13, no. 7, 297-306, (2019), the authors develop both theory and computational results related to classifying arithmetic properties of certain polynomials with p-adic coefficients whose degrees are twice the odd prime number p. This extends previous work by S. Amano in 1971, who studied the related case of p-adic polynomials whose degrees are an odd prime number p.

For a prime number p, the p-adic numbers are essentially infinite strings of numbers whose digits range from 0 to p-1. These numbers have their own notion of geometry, which is different from the traditional Euclidean geometry that is studied in pre-college mathematics courses. As such, they have found applications in many diverse areas. In physics, there are applications to the p-adic theory of strings, quantum mechanics, and spin glasses. In the theory of dynamical systems, there are applications to ergodicity, structures of cycles/attractors, and cryptographic applications like p-adic stream ciphers. In biology, there are applications to genetics, molecular motors, and cognitive models.

In their paper, Awtrey and Pritchard gave a formula for the total number of distinct p-adic polynomials of degree twice the odd prime number p; this number is known to be finite by a result of M. Krasner from the 1960s. They went on to determine further arithmetic characteristics of distinct classes of polynomials. These invariants are interesting in their own right, but they also aid in the computation of the symmetries of polynomial roots, which play an important role in the research area of computational algebra. As an application to their newly-developed theory, Awtrey and Pritchard were able to compute a complete description of all symmetry properties of degree 22 polynomials with 11-adic coefficients. This is a notoriously difficult task, and their work extends previous results, which were only known up to degree 14.

It is common to study properties of numbers by writing them in different bases.   If the final digit of the representation is 0, we know immediately the number is divisibly by that base. For example, the decimal number 40 can be written as 101000 in base 2 (also known as binary), 130 in base 5, and 37 in base 11. This shows 40 is divisible by 2 and 5 but not 11.

Similarly, it is natural to study arithmetic properties of roots of polynomials in terms of different "polynomial bases". These properties are encoded in what mathematicians call "extension fields". For a given extension field, there are many ways of representing it by polynomials; just as there are many ways to represent a decimal number in different bases.

If we are interested in prime number properties of these roots, then we can isolate our attention on one fixed prime number and ask, how many different extensions fields exist whose prime-number properties are fundamentally different?  Research from the 1960s shows that this number is finite, and so mathematicians are interested in completely classifying various properties of these extension fields, such as: the number of different fields, one polynomial that defines each field, and symmetry properties of that polynomial's roots.

The paper by Awtrey, Komlofske, Reese, and Williams focuses on polynomials whose degree is a power of the prime number p.  Their main result gives exactly one polynomial per field extension of degree p^2 and p^3 when the symmetries form the rotations of a regular polygon whose number of sides equals the degree of the polynomial. Their classification extends previous work from the 1970s that focused on the case where the degree was equal to the prime number p.

Clark attended a Section Officers Meeting in his role as chair of the Southeastern Section of the Mathematical Association of America. In his role as chair of the MAA's Committee on Minicourses he also chaired a meeting of that committee and attended a meeting of the MAA's Council on Meetings and Professional Development, in addition to serving as a monitor for three minicourse sessions.

Yokley was invited to the Society of Industrial and Applied Mathematics (SIAM) Education Committee meeting to provide an update on the establishment of the Math Modeling Hub (MMHub). The MMHub is virtual space for both community and resources to support mathematical modeling at all levels from pre-kindergarten through college. Yokley has worked as a part of the MMHub Steering Committee for the past two years and has recently become a part of the NCTM/SIAM/COMAP Joint Committee on Modeling across the Curriculum.

Arangala, Awtrey and Trocki each delivered oral presentations. Arangala's talk, "Teaching  Linear Algebra with Matlab Apps." featured Matlab apps published on MathWorks File Exchange that can use used to enhance the Inquiry Based Learning experience in the classroom. Trocki presented research conducted in collaboration with math major Ryan Bernardi '19. In his talk, "Assessing Student Engagement and Responsibility through Inquiry-Based Learning", Trocki shared findings on student perceptions of incorporating inquiry-based learning strategies in an Applied Calculus course.

Awtrey presented his research talk "On Galois p-adic fields of p-power degree", which was joint work with Peter Komlofske '19, Christian Reese '18 and Janae Williams '18. He discussed a classification of all Eisenstein polynomials with p-adic coefficients whose symmetries correspond to the rotations of a polygon with p, p^2, or p^3 sides; where p is an odd prime number. A corresponding paper has been accepted for publication in the Journal of Algebra, Number Theory, and Applications.

Awtrey also presented his research talk "On Galois groups of doubly even octic polynomials", which was joint work with Kiley Shannon '18 and Williams High School students Sam Cryan, Anna Altmann and Madeleine Touchette. Considering polynomials of the form x^8+ax^4+b, this work presents theoretical results that describe the symmetries of said polynomials given only minimal information on the coefficients. A corresponding paper has been accepted for publication in the  Journal of Algebra and Its Applications.