School of Mathematics

A folklore belief in economics is that "simple auctions" should "perform well" if and only if the auction's participants view the items for sale as substitutes (as opposed to complements). How can this belief be formulated as a precise conjecture...

Given a hypergraph with edges of size at most 3, the 3-set cover problem asks to determine the minimum size of a family of edges that covers the vertex set. As the problem is NP-hard, it is natural to consider its fractional (linear programming)...

Since their inception, random graphs have been a central topic in probability and combinatorics, offering a remarkably rich landscape for studying how complex structures emerge from randomness. We will discuss the following fundamental questions...

The van der Waerden number w(k;r) is the minimum positive integer N such that every r-coloring of the positive integers up to N contains a monochromatic k-term arithmetic progression. Estimating these numbers has remained a challenging open problem...

We will give an introduction to Reeb vector fields on odd-dimensional manifolds; these are a special case of more general vector fields whose dynamics will be studied in the Special Year. Much more is known about Reeb vector fields in three...

I want to share discoveries made by GPT-6 Astra last week, providing an elegant new idea for questions I wondered about for years. All new theorems stated in this talk are fully formalised in Lean.

A self-similar measure is a stationary measure from...

Let $f: X \to S$ be a quasi-projective family of varieties defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. We show that the points of $S(\overline{\mathbb{Q}})$ that are Hodge generic for the variation of Hodge structures associated to $f$...

A symplectic manifold $M$ is called tame at infinity if it admits a compatible almost complex structure such that the corresponding Riemannian metric is complete and geometrically bounded. Some such condition is necessary to confine $J$-holomorphic...

The first construction of Ramanujan graphs is due to Lubotzky, Phillips, and Sarnak (STOC 1986, Combinatorica 1987). Their construction and analysis were deeply algebraic, and at the time it seemed that only algebraic methods were strong enough to...