We discuss ongoing work with Joseph Leung in which we obtain estimates for sums of Fourier coefficients of GL(2) and certain GL(3) automorphic forms along the values of irreducible binary cubics.

The lambda-calculus was invented to formalise arithmetic by encoding numbers and operations as abstract functions. We will introduce the lambda-calculus and present two encodings of modular arithmetic: the first is a recipe to quotient your favourite numeral system, and the second is purpose-built for modular arithmetic. A highlight of the second approach is that it does not require recursion i.e., it is defined without fixed-point operators. If time allows, we will also give an implementation of the Chinese remainder theorem which improves computational efficiency. 

We shall explain how to represent a couple of basic notions in model theory by standard simplicial diagrams from homotopy theory. Namely, we shall see that the notions of a {definable/invariant type}, {convergence}, and {contractibility} are defined by the same simplicial formula, and so are that of a {complete E-M type} and an {idempotent of an oo-category}.  The first reformulation makes precise Hrushovski's point of view that a definable/invariant type is an operation on types rather than a property of a type depending on the choice of a model, and suggests a notion of a type over a {space} of parameters. The second involves the nerve of the category with a single idempotent non-identity morphism, and leads to a reformulation of {non-dividing} somewhat similar to that of lifting idempotents in an oo-category. If time permits, I shall also present simplicial reformulations of distality, NIP, and simplicity.

We do so by associating with a theory the simplicial set of its n-types, n>0. This simplicial set, or rather its symmetrisation, appeared earlier in model theory under the names of {type structure}  (M.Morley. Applications of topology to Lw1w. 1974), {type category} (R.Knight, Topological Spaces and Scattered Theories. 2007), {type space functors} (Haykazyan. Spaces of Types in Positive Model Theory. 2019; M.Kamsma. Type space functors and interpretations in positive logic. 2022).