Let F_q be the field with q elements. We will consider the task of testing if a m-variate function f over F_q is close to being a polynomial of degree at most d. This task was originally considered in the setting where d was small compared to q and the resulting analyses found many applications in probabilistic checking of proofs. In this talk I will describe more recent results where we consider the setting where d > q. The natural test would be to pick a random subspace of dimension roughly d/q and see if the f restricted to this subspace is a polynomial of degree d. Earlier analyses had shown that this natural test rejects functions that are, say 1%, far from being a degree d polynomial, with probability at least q^{-d/q}. In this talk I will describe our improved analyses which show that this same test rejects such functions with constant probability for constant q. Time permitting I might also mention some applications where the setting of d > q is useful. Based on joint works with Arnab Bhattacharyya, Elad Haramaty, Swastik Kopparty, Noga Ron-Zewi, Grant Schoenebeck, Amir Shpilka and David Zuckerman.

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz proved that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets. So they posed the problem of determining or estimating the largest number $\N{k}$ of the points in such a family. They proved by means of an example that $N(k)\geq2k-2+\frac{1}{2}\binom{2k-2}{k-1}$. In 1985, Zsolt Tuza proved that the upper bound of $N(k)$ is best possible up to $2\binom{2k-2}{k-1}$. In this talk, we discuss the recent development of the upper bound on $N(k)$.

The vertices of the Tamari lattice are binary trees with an order relation defined by a rotation on binary trees. Such a lattice can be realized geometrically as a convex polyhedron, called the associahedron. I will begin with a survey of this active subject and then will give some extensions of the Tamari lattice in relation with the so-called rational Catalan combinatorics and higher diagonal coinvariants (joint work with L.-F. Préville-Ratelle).