Title: A positive dichotomy between Diophantine geometry and Dynkin friezes

Abstract: General finiteness results for positive integer solutions to Diophantine equations are rare. In this talk, I will describe how the theory of cluster algebras can be a remarkable source of finiteness & infinitude in the form of a Siegel theorem for positive integral points on affine varieties. On the finiteness of positive integral points, I will highlight 7-dimensional and 8-dimensional affine varieties that arise in the Fontaine–Plamondon conjecture on Dynkin friezes. On the infinitude of positive integral points, I will highlight surfaces and threefolds that arise in the Mordell–Schinzel program on Diophantine equations xyz = G(x, y).

Prof Amod Agashe, Florida State University with title and abstract as below.

Title: The zeros of the Riemann zeta function and its generalization to modular forms

Abstract: The Riemann hypothesis, one of the most important open problems in mathematics, says that the Riemann zeta function should have zeros only at complex numbers with real part ⁠1/2 and at negative even integers. We will study the completed Riemann zeta function (the one with the Gamma factor) and discuss how it sheds some light on the location of the zeros. There is a generalization of the Riemann hypothesis to L-functions of modular forms, and we will discuss what can be said in this context. The talk should be accessible to advanced undergraduates and graduate students, including those in engineering.