Titles/Abstracts

Titles and Abstracts will be shown here when available.

Speaker: Yunqing Tang

Title: Thin sets of primes from reductions of abelian varieties

Abstract: For an elliptic curve E over the rational numbers, the Sato-Tate conjecture and the Lang-Trotter philosophy provide heuristics for the behavior of the Frobenius endomorphisms of the reductions of E modulo primes. These heuristics predict that certain sets of primes of density zero ought to be infinite for most E. Similar heuristics apply to abelian varieties, which are higher dimensional generalizations of elliptic curves. 

In this talk, I will discuss recent results that establish the infinitude of such sets for certain abelian varieties. These results are in various joint work with Davesh Maulik, Ananth Shankar, Arul Shankar, and Salim Tayou. 

Speaker: Florian Sprung

Title: Hilbert's 10th Problem 

Abstract: A question that has been part of recent mathematical conversations is if AI will take over soon and replace human mathematicians. Surprisingly, this question has a long history. At the International Congress of Mathematics in Paris in the year 1900, Hilbert posed 23 problems to be solved in the 20th century. The 10th problem asked if there is a putative process that can answer whether a polynomial with integer coefficients has integer solutions. (In today's words: Is there an algorithm that can determine whether these solutions exist?) We will discuss the history of the problem (it was solved in 1970), and if time permits, describe some generalizations and open problems.

Speaker: Ellen Eischen

Title: An Introduction to the Bernoulli Numbers, from Pythagoras to Present 

Abstract: We'll explore several problems — elementary and sophisticated, ancient and modern — that appear to come from vastly different areas, ranging from arithmetic to geometry to calculus to abstract algebra and beyond. Is traversing such a diversity of fields in a 50- minute talk sensible? Yes, thanks to the "Bernoulli numbers!" This collection of numbers unifies a set of topics that seem at first to be unrelated (including polynomials, infinite series, factorization, the Riemann zeta function, Fermat's Last Theorem, and more). After beginning with questions accessible to any undergraduate, we will eventually encounter topics that play a major role in research today.

Speaker: Tony Feng

Title: On the origin of modularity

Abstract: I will give an introduction to theta functions, whose study has driven many developments in modern number theory. One of the reasons they are interesting is because they enjoy a property called modularity. In joint work with Zhiwei Yun and Wei Zhang, we construct generalizations of theta functions that we call “higher theta functions", which we conjecture to be modular. I will describe some recent progress towards this conjecture.