This will be an expository lecture intended for a general mathematical audience to illustrate, through examples, the theme of $p$-adic variation in the classical theory of modular forms.  Classically, modular forms are complex analytic objects, but because their Fourier coefficients are typically integral, it is possible to also do elementary arithmetic with them.   Early examples arose already in the work of Ramanujan.  Today one knows that modular forms encode deep arithmetic information about elliptic curves and Galois representations.  Our main goal will be to illustrate these ideas through simple concrete examples.   

A non-nilpotent graph Γ(G) of a finite group G has elements of G as vertices, with x and y joined by an edge iff a subgroup generated by these two elements is non-nilpotent. During the talk we will prove several (often unrelated) properties of this construction; for instance, any simple graph can be found as an induced subgraph of Γ(G) for some (solvable) group G. The talk is based on my article "A few remarks on the theory of non-nilpotent graphs" (May 2023).

 Many optimal control, estimation and design problems can be formulated as so-called dynamic optimization problems, which are optimization problems with differential equations and other constraints. State-of-the-art methods based on collocation, which enforce the differential equations at only a finite set of points, can struggle to solve certain dynamic optimization problems, such as those with high-index differential algebraic equations, consistent overdetermined constraints or problems with singular arcs. We show how numerical methods based on integrating the differential equation residuals can be used to solve dynamic optimization problems where collocation methods fail. Furthermore, we show that integrated residual methods can be computationally more efficient than direct collocation.

This seminar takes place at RAL (Rutherford Appleton Lab).