Fall 2022
After a long hiatus, we are resuming inperson talks in our newly renovated mathematics building. The seminar typically runs on Fridays from 2:304:30 in LC 315. For more information, contact Alex Duncan .
Date  Location  Speaker  Title  Host 
Friday, Oct 14 2:30PM 

Fall Break  
Friday, Oct 21 3:30PM 
LC 315  Tugba Yesin Sabancı University 
Divisibility by 2 on quartic models of elliptic curves and rational Diophantine $D(q)$quintuples  (local) 
Friday, Oct 28 2:30PM 
LC 315  Charlotte Ure University of Virginia 
A moduli interpretation of binary cubic forms  Alex Duncan 
Friday, Nov 4 2:30PM 
LC 315  Brooke Ullery Emory University 
Secants, Gorenstein ideals, and stable complexes  Alex Duncan 
Friday, Nov 11 4:30PM 
Petigru 108  Ken Ono University of Virginia 
Variants of Lehmer’s Conjecture on Ramunajan’s taufunction  Phi Beta Kappa 
Friday, Nov 25 2:30PM 

Thanksgiving  
Friday, Dec 2 12:00PM 

Reserved (Palmetto Number Theory Series) 
Abstracts
 Fall Break
Tugba Yesin  Divisibility by 2 on quartic models of elliptic curves and rational Diophantine $D(q)$quintuples
Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $Q$ with $P \in C(Q)$. We give an explicit criterion for the divisibilityby$2$ of a rational point on the elliptic curve $(C,P)$. This generalizes the classical criterion of the divisibilityby$2$ on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational $D(q)$quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that although the rational $D(16t+9)$quadruple $\{t,16t+8,225t+14,36t+20\}$ can not be extended to a polynomial $D(16t+9)$quintuple using a linear polynomial, there are infinitely many rational values of $t$ for which the aforementioned rational $D(16t + 9)$quadruple can be extended to a rational $D(16t + 9)$quintuple. Moreover, these infinitely many values of $t$ are parametrized by the rational points on a certain elliptic curve of positive MordellWeil rank.
Charlotte Ure  A moduli interpretation of binary cubic forms
The subject of classifying homogeneous forms has fascinated mathematicians for centuries. In this talk, I will discuss joint work with Rajesh Kulkarni, giving a moduli interpretation to the quotient of binary cubic forms with respect to linear transformation of the variables. In particular, I will explain how these orbits correspond to certain coverings, and Brauer classes, on elliptic curves.
Brooke Ullery  Secants, Gorenstein ideals, and stable complexes
As Bertram describes in his thesis, certain rank two vector bundles on curves can be parametrized by a projective space into which the curve itself embeds. The idea is that the least socalled "stable" bundles correspond to points on the embedded curve, the next most unstable lie on the first secant variety, then on the second secant variety, and so on. In this talk, I'll describe joint work with Bertram in which we generalize this to $\mathbb{P}^2$ by replacing vector bundles with complexes of vector bundles. This leads to a surprising connection between height three Gorenstein ideals, secant varieties (and their generalizations), and stability conditions on the bounded derived category of coherent sheaves on $\mathbb{P}^2$.
Ken Ono  Variants of Lehmer’s Conjecture on Ramunajan’s taufunction
The theory of modular forms enjoys some of the most significant recent advances in number theory. This includes the discovery of ideas that underly the proof of Fermat’s Last Theorem, provide the framework for much of the Langlands Program, and gives glimpses of deep connections between geometry, number theory and physics. Despite these deep advances, the innocent (in appearance) Conjecture of D. H. Lehmer on the nonvanishing of Ramanujan's taufunction remains open. In this talk the speaker will tell the story of this important function, and describe recent results on variants of Lehmer’s Conjecture, where much progress has been made in the last few years. (USC Colloquium)
 Thanksgiving
 Reserved (Palmetto Number Theory Series)