30th SouthEasternRegional Meeting On Numbers
(SERMON XXX)
March 31 - April 2, 2017
Jacksonville, FL
Message from the
Provost and Vice President of Academic Affairs
Dear Conference Attendees:
Welcome to the 30thannual Southeastern Regional Meeting on Numbers, and welcome to the University of North Florida.
UNF is very pleased that you have selected our campus as the venue for your annual SERMON meeting.
Professor Michelle DeDeo has worked hard to organize your conference and ensure that you have a comfortable setting in which to discuss your recent research in the areas of pure and applied number theory.
If your schedule permits, I hope that you will explore our beautiful campus as well as the surrounding area, including our beaches. UNF is an ideal location for an academic conference, and I am sure that Michelle and her colleagues will do everything they can to accommodate your needs.
We hope that you have a rewarding conference, that you enjoy your time at UNF, and that you will plan to return in the near future.
Earle Traynham
Message from the
Chair of the Mathematics & Statistics Department
Dear Conference Attendees:
On behalf of the Department of Mathematics and Statistics at the University of North Florida, I extend a warm welcome to the mathematicians attending theSouthEastern Regional Meeting on Numbers. SERMONhas a rich tradition that goes back about thirty years. Perhaps that isn’t as longstanding as other math conferences, but since UNF is only forty-five years old, thirty years is impressive. UNF is hosting this meeting because of the efforts of our colleague Dr. Michelle DeDeo.
As a mathematician I hope that a conference is filled with engaging talks and fruitful ideas. Additionally, when there are breaks or when the talks end, I like to be in an area that has something more to offer. I hope you find the UNF campus visually enticing, the nature trails inviting, and the river and beaches beckoning. I wish you well as you spend some time immersed in Number Theory at the University of North Florida.
Scott Hochwald
ACKNOWLEGEMENTS
A Note from the Hosts
This conference provides an exciting opportunity to share research ideas that will shape the future of number theory and it would not be possible without the contributions of several individuals.
The 30th SERMON Conference would like to acknowledge the support of:
- The National Science Foundation
- The National Security Agency
- Dr. Earle Traynham, UNF Provost & VP of Academic Affairs
- Dr. Daniel Moon, UNF Interim COAS Dean
- Dr. Scott Hochwald, UNF Chair – Mathematics & Statistics
- The PaNTS and SERMON Consortium
We appreciate your generous support and welcome you toSERMON.
Regards,
Michelle DeDeo & Matthew Boylan
SCHEDULE OVERVIEW
SATURDAY
8:30AMCoffee and Refreshments
9:00AMDorian GoldfeldColumbia University - INVITED ADDRESS
10:00AM Duc HuynhArmstrong State University
10:30AM Mary AmbrosinoNC State University
11:00AMFrank GarvanUniversity of Florida
11:30AMLUNCH
1:30PMJenny FuselierHigh Point University
2:00PMXiang-dong HouUniversity of South Florida
2:30PM Coffee & Refreshments
2:45PMKevin KeatingUniversity of FloridaUniversity
3:15PM Jackson Morrow Emory University
3:45PMCoffee & Refreshments
4:00PM Winnie LiPennsylvania State University - INVITED ADDRESS
SUNDAY
9:00AM Coffee & Refreshments
9:30AMAlex SmithHarvard University - INVITED ADDRESS
10:30AMJonathan GerhardJames Madison University
11:00AMAmod AgasheFlorida State University
11:30AMAli UncuUniversity of Florida
ATTENDEES
Ali Uncu / / University of Florida
Allen Pelley / / Bethune-Cookman University
Amod Agashe / / Florida State University
Blair Dietrich / / Georgia Military College
Cuyler Warnock / / Georgia College & State University
David Zureick-Brown / / Emory University
Dorian Goldfeld / / Columbia University
Dr. Hector N. Torres / / Bethune-Cookman University
Duc Huynh / / Armstrong State University
Elie Alhajjar / / George Mason University
Frank Garvan / / University of Florida
Harsh Mehta / / University of South Carolina
Hui Xue / / Clemson University
Jackson Morrow / / Emory University
Jenny Fuselier / / High Point University
Jonathan Gerhard / / James Madison University
Jorge Antonio Martinez-Soto / / Bethune Cookman University
Josaphat Uvah / / University of West Florida
Kelly Carey / / Bethune-Cookman University
Kevin James / / Clemson University
Kevin Keating / / University of Florida
Marie Jameson / / University of Tennessee, Knoxville
Martha Allen / / Georgia College
Martine Levy Nelson / / Bethune-Cookman University
Mary Ambrosino / / NC State University
Masood Poorandi / / Bethune-Cookman University
Matthew Boylan / / University of South Carolina
Michelle DeDeo / / University of North Florida
Rosina Campbell / / Armstrong State University
Sarah Ayoku / / Georgia Southern University
Seenith Sivasundaram / / Betghune Cookman University
Sungkon Chang / / Armstrong State University
Hector Torres / / Bethune-Cookman University
Tyler Melton / / Armstrong State University
Winnie Li / / Pennsylvania State University
Xiang-dong Hou / / University of South Florida
SERMON 30
Schedule & Abstracts
April 1-2, 2017
SATURDAY
8:30AM – Coffee and Refreshments
Invited Address
9:00AM
Dorian Goldfeld
Columbia University
Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self-dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value s=1/2 should be non-negative. They proved that the Riemann hypothesis implies super-positivity for self dual cuspidal automorphic L-functions on GL(n).
This talk is based on recent joint work with Bingrong Huang where we prove, for the first time, that there are infinitely many L-functions associated to modular forms for SL(2,ℤ) each of which has the super-positivity property.
10:00AM
Duc Huynh
Armstrong State University
Constructing elliptic curves of prime order
From complex multiplication, we know that the construction of an elliptic curve E of prescribed order depends almost entirely on finding a root of the Hilbert class polynomial H(X) with certain discriminant D - such a root is also the j-invariant of E. We will see how we can choose a minimal D to reduce the coefficients of the H(X). Furthermore, we will use isogeny volcanoes to find the j-invariant in a different way.
10:30AM
Mary Ambrosino
NC State University
Maximum Gap of Cyclotomic Polynomials
In this talk, we investigate the sparsity structure of cyclotomic polynomials, in particular, the maximum gap between two consecutive exponents that appear. We will present a lower bound for the maximum gap and observe that it is very often exact. We also conjecture an exact expression under a certain condition and prove that it is true for infinitely many cases.
11:00AM
Frank Garvan
University of Florida
New Mock Theta Function Identities
In his last letter to Hardy, Ramanujan defined ten mock theta functions of order 5 and three of order 7. He stated that the three mock theta functions of order 7 are not related. We give simple proofs of new Hecke double sum identities for two of the order 5 functions and all three of the order 7 functions. We find that the coefficients of Ramanujan's three mock theta functions of order 7 are surprisingly related.
11:30AMLUNCH
1:30PM
Jenny Fuselier
High Point University
Hypergeometric Functions over Finite Fields (Sat)
In this talk, we introduce hypergeometric functions over finite fields, originally due to Greene, Katz, and McCarthy. We study these functions in a way parallel to the classical hypergeometric functions and present a systematic approach for translating some classical hypergeometric identities and evaluations to the finite field setting by an explicit dictionary. This is joint work with Ling Long, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu.
2:00PM
Xiang-dong Hou
University of South Florida
Zeros of Polynomials over Finite Fields
Let be the finite field with elements. For, let be the set of zeros of in. Research on is a central theme in finite fields, number theory, and algebraic geometry. We survey various bounds for; most of them are well known, but some are not well publicized. In a few cases, it is possible to determine all the polynomials meeting a known bound.
2:30PM – Coffee & Refreshments
2:45PM
Kevin Keating
University of Florida
Trace, Norm, etc.
Let be a local field whose residue field has characteristic and let be a finite separable totally ramified extension of degree. Let be a separable closure of and let denote the -embeddings of into. For let denote the th elementary symmetric polynomial in variables, and for set.
In this talk we consider the problem of finding a lower bound for in terms of. The solution that we present depends on the indices of inseparability of the extension .
3:15PM
Jackson Morrow
Emory University
Unexpected quadratic points
On a hyperelliptic curve over, there are infinitely many points defined over quadratic fields: just pull back rational points of the projective line through the degree two map. But for a positive proportion of genus odd hyperelliptic curves over, when ordered by height, we give a bound on the number of quadratic points not arising in this way. The proof uses -adic methods, tropical geometry, and work of Bhargava and Gross on average ranks of hyperelliptic Jacobians. This is joint work with Joseph Gunther.
3:45PM – Coffee & Refreshments
Invited Address
4:00PM
Winnie Li
Pennsylvania State University
Zeta functions are counting functions in nature. For instance the Dedekind zeta function for a number field counts the number of integral ideals with a given norm, the zeta function for a variety defined over a finite field counts the number of solutions in finite extensions of the base field, while the Selberg zeta function counts the number of geodesics of a given length in a compact Riemann surface.
In this talk we shall discuss the analogue of these zeta functions in combinatorial setting, namely the zeta functions attached to graphs and higher dimensional simplicial complexes. It is interesting to compare the similarities and dissimilarities between number-theoretic and combinatorial zeta functions.
SUNDAY
9:00AM – Coffee & Refreshments
Invited Address
9:30AM
Alex Smith
Harvard University
-Selmer groups, -class groups, and Goldfeld's conjecture
Take to be an elliptic curve with full rational -torsion (that satisfies some extra technical assumptions). In this talk, we will show that of the quadratic twists of have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the -Selmer groups in these families to -Selmer groups for any. In addition, using the close analogy between -Selmer groups and -class groups, we will prove that the -class groups of the quadratic imaginary fields are distributed as predicted by the Cohen-Lenstra heuristics for all.
10:30AM
Jonathan Gerhard
James Madison University
An exact product formula for abelian varieties of odd prime dimension
Let be the characteristic polynomial of Frobenius of an abelian variety of odd prime dimension over a finite field; we use to relate three seemingly disjoint objects. First, we consider the factorizations of primes in, a degree number field. Second, we use a parameterization of Shinoda (1980) to describe certain conjugacy classes of the matrix group. Our main result (following Gekeler (2003) and Achter and Williams (2015)) is a product formula relating the class number of to the relative densities of conjugacy classes of. Finally, we give a (conjectural) application of our formula to the size of isogeny classes of abelian varieties of odd prime dimension.
11:00AM
Amod Agashe
Florida State University
A generalization of Kronecker's first limit formula to GL(n)
The classical Kronecker's first limit formula gives the polar and constant term in the Laurent expansion of a certain two variable Eisenstein series, which in turn gives the polar and constant term in the Laurent expansion of the zeta function of a quadratic imaginary field. We will recall this formula and sketch how it can be generalized to more general maximal parabolic Eisenstein series and to zeta functions of arbitrary number fields
11:30AM
Ali Uncu
University of Florida
New Partition Inequalities
We will discuss counting partitions with altering signs with respect to their smallest part where we also restrict the difference between the largest and the smallest parts. This study will lead us to inequalities between families of partitions. We will prove these inequalities and implied summation formulas using nothing but injective maps between sets.
THANK YOU FOR ATTENDING SERMON XXX
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