Department Colloquium
Organizers: Mohammed Abouzaid, Amir Dembo (Fall & Winter Quarters), and Kannan Soundararajan (Spring Quarter)
Upcoming Events
Abstract: 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…
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…
Past Events
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…
Self-expanders are a special class of solutions to the mean curvature flow, in which a later time slice is a scale-up copy of an earlier one. They are also critical points for a suitable weighted area functional. Self-expanders model the asymptotic behavior of a mean curvature flow when it…
In 1998, Smale published his `list of mathematical problems for the next century'. His 17th problem asked if a zero of d random complex polynomials in d unknowns can be found by an algorithm in polynomial time on average. Beltrán and Pardo proved the existence of an efficient randomized…
Modular forms are complex analytic functions with striking symmetries, which play fundamental role in number theory. In the last few decades there have been a series of astonishing predictions from theoretical physics that various basic mathematical numbers when put in a generating…
Computer theorem provers (which know the axioms of mathematics and can check proofs) have existed for decades, but it's only recently that they have been noticed by mainstream mathematicians. Modern work of Tao, Scholze and others has now been taught to Lean (one of these systems), and (…
Around 10 years ago, Donaldson and Sun discovered that metric limits of Ricci positive Kähler–Einstein manifolds are algebraic varieties, and their metric tangent cones also underlie some algebraic structure. I will talk about a general algebraic geometry theory behind this phenomenon. In…