Student and Recent PhD Showcase

Oregon State University is hosting a winter number theory conference to showcase work of regional students and recent PhDs.  We aim to offer a friendly opportunity for junior researchers to give presentations on their research, engage in panel discussions for professional development, and connect with other number theorists throughout the Pacific Northwest.

Past Meeting:

February 18-19, 2023

Oregon State University

Location: Kelley Engineering Center 1001, 1003 

Presenters:

Postdoctoral Presenters:

Joshua Parker, Whitman College

Graduate Student Presenters:

Jordan Hardy, University of Idaho

Greg Knapp, University of Oregon

Leah Sturman, Oregon State University

Eric Williams, Portland State University

Undergraduate Presenters:

Liam Armstrong, Oregon State University 

Javier Rivera Romeu, Reed College (pending)

Jasmine Wetter, Western Oregon University

Schedule:

Saturday Feb. 18

9:30-10:00 Coffee, pastries

10:00-10:40 Joshua Parker: Prime Level Paramodular Hecke Algebras

10:55-11:35 Jordan Hardy: Abelian Surfaces with Complex Multiplication Carrying Nonprincipal Polarizations

11:45-1:15 Lunch Break

1:15-1:35 Jasmine Wetter: How to Find Integers that Cannot Divide Elusive Odd Perfect Numbers: Generalizing Sylvester's Proof

1:45-2:05 Liam Armstrong: Generalized Alder-Type Partition Inequalities

2:20-3:00 Leah Sturman: Resolving the Kang Park Conjecture

3:15-4:15 Panel Discussion: Conducting Research 

Panelists: Leanne Merrill, Clay Petsche, Holly Swisher, Jeff Vaaler

4:15-4:45 Reception/Snacks

Sunday Feb. 19

9:00-9:30 Coffee, pastries

9:30-10:10 Eric Williams: The Non-vanishing of Traces of Hecke Operators

10:25-11:05 Greg Knapp: On the Separation of Roots of Polynomials

11:20-11:35 Javier Rivera Romeu: On the Wall-Sun-Sun Conjecture

Abstracts: 

Liam Armstrong: Generalized Alder-Type Partition Inequalities

Abstract: In this talk we discuss recent developments in the study of partition inequalities generalizing the Andrews-Alder theorem.  These generalizations are rooted in original work of Kang and Park, were expanded upon by Duncan, Khunger, Swisher and Tamura, and then followed by work of Inagaki and Tamura.  Here, we discuss our contributions to this problem, including a proof of a conjecture of Inagaki and Tamura, its generalization, and implications for questions posed by Duncan et al.  This work is joint with Bryan Ducasse, Thomas Meyer, and Holly Swisher and was initiated as part of the 2022 NSF Number Theory REU at Oregon State University.

Jordan Hardy: Abelian Surfaces with Complex Multiplication Carrying Nonprincipal Polarizations

Abstract: An abelian variety is a projective surface which carries a group structure. Those abelian surfaces which are defined over a field of characteristic 0 are, over the complex numbers, isomorphic to a complex torus, that is, a quotient of a complex vector space V by a lattice L. Not every complex torus is isomorphic to an abelian surface. It is known that the torus V/L inherits the structure of an abelian surface if and only if the lattice L admits a Riemann form, which is an alternating Z-bilinear form that is the imaginary part of a positive definite Hermitian form. Such tori are called polarized tori. We are interested in the study of abelian surfaces whose endomorphism rings are as big as possible. Such surfaces are said to have complex multiplication, and in this case their endomorphism rings are isomorphic to certain subrings of totally imaginary number fields, called CM fields. Abelian surfaces with CM are useful for many applications. There is an invariant one can attach to the Riemann form defined on the lattice L called the type of its polarization. Much study has been devoted to those with a polarization of type (1,1), called principally polarized, but less attention has been paid to those surfaces with nonprincipal polarizations. In this talk we discuss under what circumstances we can find abelian surfaces with complex multiplication by a given quartic CM field K and a given polarization type (1, m). We then discuss further results and conjectures arising from this.

Greg Knapp: On the Separation of Roots of Polynomials

Abstract: In 1964, Mahler gave a lower bound on the distance between roots of polynomials in terms of the degree, height, and discriminant of that polynomial.  More recently, Rump, Bugeaud, Koiran, Pejkovic, and others have given improvements to Mahler’s lower bounds, but little attention has been given to upper bounds on the separation of a polynomial (the minimal distance between distinct roots of the polynomial).  In this talk, we will explore the historical developments in our understanding of the separation of polynomials and we will take a look at some new results, conjectures, and data concerning upper bounds on separation.

Joshua Parker: Prime Level Paramodular Hecke Algebras

Abstract: This talk will report on recent research concerning the local Hecke algebras acting on Siegel paramodular forms of square-free level. We offer a complete description of the structure of these noncommutative algebras, which for each prime includes a generating set, relations, and explicit formulas for computing the coefficients that occur in the multiplication of the Hecke operators in this setting. Using these generators and relations, we prove that the formal power series $\sum_{k=0}^\infty \chi (T(q^k)) t^k$ evaluates to a rational function for characters of the Hecke algebra.

Javier Rivera Romeu: On the Wall-Sun-Sun Conjecture

Abstract:  The Wall-Sun-Sun conjecture proposes the existence of exceptional primes p for which the period of the Fibonacci sequence modulo p is equal to modulo p^2. The Wall-Sun-Sun (or Fibonacci-Wieferich) primes have been extensively studied in connection with the ABC conjecture or Fermat's Last Theorem, but despite massive computational searches (up to 10^14) and attempts to prove it still remains open. In the talk, I will present the evolution in the study of the problem and some additional results.

Leah Sturman: Resolving the Kang Park Conjecture

Abstract:  In 1956, Alder conjectured that the number of d-distinct partitions of an integer n is greater than or equal to the number of partitions of n into parts congruent to plus or minus one modulo d+3. Alder’s conjecture can be thought of as a generalization of the first Rogers Ramanujan identity. Andrews and Yee proved Alder's conjecture for all but finitely many values of d using intricate combinatorial methods. Later Alfes, Jameson, and Lemke Oliver used asymptotic methods to prove Alder's conjecture for the remaining values of d. We discuss ongoing work on a conjecture of Kang and Park which analogously generalizes the second Rogers Ramanujan identity. This is joint work with Holly Swisher.

Jasmine Wetter: How to Find Integers that Cannot Divide Elusive Odd Perfect Numbers: Generalizing Sylvester's Proof

Abstract: Odd perfect numbers have been of interest to mathematicians for millennia. While they are generally believed not to exist, no one has been able to prove their nonexistence. Many discoveries have been made regarding the characteristics of odd perfect numbers, including James Sylvester's 1888 proof that no odd perfect number is divisible by 105. We will discuss a generalization of Sylvester's proof that allows us to prove that 2145 and at least 108 other integers aside from 105 are impossible odd perfect number divisors.

Eric Williams: The Non-vanishing of Traces of Hecke Operators

Abstract: A long-standing conjecture of Lehmer claims that the Fourier coefficients τ (n) of the ∆ function are non-vanishing. More generally, we can ask when the trace of the Hecke operator T_n acting on the space of weight k cusp forms of level N vanishes. For non-square n ≥ 1 and N coprime to n it is conjectured that the trace vanishes only when the space of cusp forms is trivial. In this talk we give a survey of the results in this direction and present a proof of the non-vanishing of T_3, based on joint work with Chiriac and Kurzenhauser.

Local Information:

Parking

Parking is free on the Oregon State University campus during weekends.  Please see here for location of parking lots.

Hotel

We have reserved a block of rooms at the Hilton Garden Inn for direct billing.  Please let the organizers know at registration if you expect to need lodging, and if you would like to use one of the reserved rooms, pending funding. 

Registration:

Please complete the following short registration form by Dec. 31 if you would like to present and/or request funding, and if possible by Jan. 31 if you are planning to attend.

Funding:

We have some funding available for travel and lodging.  Please indicate your funding requests with your registration.

We will prioritize funding for regional students and recent PhDs, particularly those who will be presenting, though we hope to provide funding for all regional participants.

Funding for mileage and travel will take the form of a reimbursement.  For hotel we can pay directly depending on the choice of hotel.  

For questions about financial support, please email Holly Swisher at swisherh (at) oregonstate (dot) edu.

Organizers:

• Leah Sturman, Oregon State University

• Holly Swisher, Oregon State University

Support has been provided by: