The famous no-three-in-line problem by Dudeney more than a century ago asks whether one can select 2n points from the grid $[n]^2$ such that no three are collinear. We present two results related to this problem. First, we give a non-trivial upper bound for the maximum size of a set in $[n]^4$ such that no four are coplanar. Second, we characterize the behavior of the maximum size of a subset such that no three are collinear in a random set of $\mathbb{F}_q^2$, that is, the plane over the finite field of order q. We discuss their proofs and related open problems.

We construct nonunique solutions of the transport equation in the class $L^\infty$ in time and $L^r$ in space, for divergence free Sobolev vector fields from $W^{1,p}$. We achieve this by introducing two novel ideas: (1) in the construction, we interweave scaled copies of the vector field itself, and (2) asynchronous translation of cubes, which makes the construction heterogeneous in space. These new ideas allow us to prove nonuniqueness in the range of exponents going beyond what is available using the method of convex integration, and sharply match with the range of uniqueness of solutions from Bruè, Colombo, De Lellis ’21.

We propose a quantum analog of the Mycielski transformation of graphs and iwe study relations between quantum automorphism groups of a quantum graph and its Mycielskian. Based on joint work with A. Bochniak (arXiv:2306.09994) and work in progress with A. Bochniak, P.M. Sołtan, and I. Chełstowski.

The List Colouring Conjecture posits that the list edge chromatic number of any graph is equal to the edge chromatic number, and thus is at most D+1 where D is the maximum degree.  This means that if each edge is assigned a list of $D+1$ colours then it is always possible to obtain a proper edge colouring by choosing one colour from each list.

In the 1990's, Khan proved that one can always obtain a proper edge colouring from lists of size $D+o(D)$, then Molloy and Reed obtained $D+D^{1/2}\mathrm{poly}(\log D)$.  We improve that bound to $D+D^{2/5}\mathrm{poly}(\log D)$
 

Saddle connections on a translation surface generalize both diagonals in a polygon and primitive vectors in a 2-dimensional lattice. Their slopes thus contain geometric and algebraic information about the surface. Slopes of saddle connections can be studied using the action of a horocycle subgroup of $\mathrm{SL}_2(\mathbb{R})$ on the moduli space of all translation surfaces. In particular, gaps between slopes are directly related to the return time function of a Poincaré section for the horocycle flow.

In this talk, we will describe a Poincaré section for horocycle flow in the smallest nontrivial stratum $\mathcal{H}(2)$ and see how to compute the return time function. Then we will examine some consequences for gap distributions. This is joint work with Anthony Sanchez.

In this talk, I will explain why fractional (or nonlocal) minimal surfaces are ideal objects to which min-max methods can be applied on Riemannian manifolds. After a short introduction about these objects and how they approximate minimal surfaces, I will present a vision for the future on how to generalize this setting to higher codimension. 

A large area of modern number theory (the Langlands program) studies a deep correspondence between the representation theory of Galois groups, algebraic varieties and certain analytic objects (automorphic forms). Many spectacular theorems have come from this area, for example the key insight in Wiles' proof of Fermat's last theorem was a connection between elliptic curves, modular forms and Galois representations.

The goal of this talk is to explain how geometric constructions, particularly related to Shimura varieties, arise naturally in the Langlands program. I will then talk about joint work with Stefan Patrikis, stating that Galois representations arising from certain Shimura varieties satisfy the properties predicted by the correspondence introduced above.

Classic mechanisms of spatial pattern formation in developmental biology are characterized by high degrees of multistability and sensitivity to initial conditions. These traits are commonly seen as undermining the capacity of these processes to exhibit robust morphogenesis. However, a growing body of experimental evidence suggests developing organisms can accomplish robust pattern selection in reaction-diffusion processes with relatively simple spatiotemporal forcings. To better understand this phenomenon, we perform a series of systematic investigations into the optimal controllability of a minimal pattern-forming system. Using machine-learning-inspired techniques, we generate simple optimal control protocols to drive an underactuated system to a desired steady state.  We numerically demonstrate the effectiveness of control in two universal scenarios of pattern formation: within a weakly nonlinear regime associated with a supercritical Turing instability and for localized states associated with homoclinic snaking.