how many smooth structures are there on a sphere? for dimensions at least 5, kervaire--milnor solved this problem in terms of another problem in algebraic topology: the computations of stable homotopy groups of spheres. in this talk, i will discuss recent progress on this problem in algebraic topology and its applications on smooth structures, which includes the following result with guozhen wang, building up on moise, kervaire--milnor, browder, hill--hopkins--ravenel: among all odd dimensions, the n-sphere has a unique smooth structure if and only if n = 1, 3, 5, 61. i will also discuss some recent progress towards the kervaire invariant problem in dimension 126.

in applications to number theory, it is often desirable to
find effective dynamical results. we want to provide effective
information about averages of orbits of the horospherical subgroup
acting on a hyperbolic manifold of infinite volume. we will start by
presenting the setting and results for manifolds with finite volume.
then, we will discuss the difficulties that arise when studying the
infinite volume setting, and the measures that play a crucial role in
it. this is joint work with jacqueline warren.

based on the lagarias-odlyzko effectivization of the
chebotarev density theorem, kumar murty gave aneffective version of the
sato-tate conjecture for an elliptic curve conditional on the analytic
continuationand the riemann hypothesis for all the symmetric power
l-functions. using similar techniques, kedlaya and i obtained a
similar conditional effectivization of the generalized sato-tate
conjecture for an arbitrary motive. as an application, we obtained a
conditional upper bound of the form $o((\log n)^2(\log \log n)^2)$ for
the smallest prime at which two given rational elliptic curves with
conductor at most $n$ have frobenius traces of opposite sign. in this
talk, i will discuss how to improve this bound to the best possible in
terms of nand under slightly weaker assumptions. our new approach
extends to abelian varieties. this is joint work with kiran kedlaya and
francesc fit\'e.

circle packings in which all circles have integer curvature,
particularly apollonian circle packings, have in the last decade become
objects of great interest in number theory. in this talk, we explore
some of their most fascinating arithmetic features, from local to global
properties to prime components in the packings, going from theorems, to
widely believed conjectures, to wild guesses as to what might be true.

deligne's weight-monodromy conjecture gives control over the zeros of local factors of l-functions of varieties at places of bad reduction. his proof in characteristic p was a step in his proof of the generalized weil conjectures. scholze developed the theory of perfectoid spaces to transfer deligne's proof to characteristic 0, proving the conjecture for complete intersections in toric varieties. building on scholze's techniques, we prove the weight-monodromy conjecture for complete intersections in abelian varieties.

fix a smooth projective variety in projective space and project it to a linear subspace of the same dimension. the ramification divisor of this projection is classically known as the ``apparent boundary''. how does the apparent boundary move when we move the center of projection? i will discuss the geometry arising from this natural question. i will explain why the situation is most interesting for varieties of minimal degree, and how it is related to limit linear series for vector bundles.