KPPY 56

In this talk, I will give you a \"birds-eye\" to the theory of modular forms and some aspects for non-number theorists. Especially, my goal is to do an invitation to \"Maeda's conjecture\" that provides us important properties on modern number theory. This talk includes joint work with Yoshitaka Maeda (Hokkaido Univ).

The Eisenstein series is one of the famous example of modular forms, and important in the theory of the modular forms. We'll see a family of the explicit linear relations among two products of Eisenstein series which includes the classical known-resutls proved by Ramanujan, Liouville and many mathematicians. The theory of modular forms is able to prove these identities. Besides, these identities might be verified by using other approaches --a kind of additive formula for elliptic functions, the theory of multiple zeta values and combinatorics. In my talk, we mainly discuss the combinatorial aspects of the Eisenstein series identities.

In this talk I will present a convex set $K$ and a set of $n$ translation vectors $T$ such that the compliment of $K+T$ has asymptotically $cn^3$ connected components (voids). For one construction the complexity of $K$ is quadratic in $n$, and for another $K$ is independent of $n$. In the context of translating a convex body in 3-space that must avoid a point set of obstacles, this is equivalent to the existence of a point set and a body that can be placed in a cubic number of positions such that it cannot translate between these positions without intersecting any of the points. This also implies that for a distance function having quadratic complexity, the generalized Voronoi diagram of a point set can have cubic complexity. This is joint work with Boris Aronov, Otfried Cheong, and Xavier Goaoc.

The class of structures that admit near-unanimity functions is of interest in the field of computational complexity as they yield constraint satisfactions problems that are solvable in deterministic log-space. In the literature, there are diverse characterisations near-unanimity structures, but none that make the generation of all such graphs transparent. We present a new description of reflexive graphs and irreflexive symmetric graphs admitting near-unanimity functions. This description brings together many of the known descriptions, and provides a good picture of near unanimity graphs. This is joint work with Tomas Feder, Pavol Hell, Benoit Larose, Cindy Loten and Claude Tardif.

If you have any open problems that you can present in 5-15 minutes, please prepare them.