Department Colloquium

Abstract: An n x n-matrix P is called a projection matrix if P is an idempotent, i.e., P^2 = P; in this context the notion seems anodyne.  If our matrices have coefficients in a more complicated algebraic structure, say a commutative ring R, then the image of a projection matrix is…

Abstract: In 1900, in the second ICM, Hilbert proposed a list of 23 problems that shaped much of mathematics throughout the past century. The tenth problem asked for an algorithm capable of deciding the solvability over the integer of any polynomial equation, in any number…

Abstract: A Besicovitch set is a compact subset of R^n that contains a unit line segment pointing in every direction. The Kakeya set conjecture asserts that every Besicovitch set in R^n has Minkowski and Hausdorff dimension n. I will discuss some recent progress on this conjecture, leading to…

There are situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects – i.e. via the finite quotients of the group. But how much understanding can one really gain about an infinite group by examining its…

If a Hilbert space is built from qubits, we may define complexity of a global unitary U by the minimal number of one- and two-qubit unitaries that comprise the global U. I will discuss three questions that can be posed using this complexity. (i) What quantum error correcting codes are there on D…

We discuss recent improved bounds for Szemerédi’s Theorem. The talk will seek to provide a gentle introduction to what is meant by higher order Fourier analysis, motivate the statement of the inverse theorem for the Gowers norm and discuss the high level strategy underlying the proof. Based on…

We review the rudiments of a diophantinetheory of affine Markoff cubics. These enjoy an action ofthe mapping class group on p-adic integral points thanks to their realization as the character varietyof the once punctured torus. This provides a powerful tool making them one of the few…

Since Szemeredi's Theorem and Furstenberg's proof thereof using ergodic theory, dynamical methods have been used to show the existence of numerous patterns in sets of positive upper density. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but…

PDEs with randomness are ubiquitous in physical sciences (lack of knowledge, thermal noise, etc.). The presence of randomness may have a drastic impact on existence and regularity of solutions. In this colloquium I will discuss both questions on some specific examples and give some insight on…

By analogy with Langlands's conjectures in arithmetic, Beilinson and Drinfeld conjectured that D-modules on the space of G-bundles on an algebraic curve are the same as (certain) coherent sheaves on the space of local systems on the same curve, but for the Langlands dual group. We will discuss…