Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x²+y²=z²)? This is one of the simplest questions in arithmetic Ramsey theory which is still open. I will present a recent partial result, showing that in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x²+y²=z² for some integer z which may be in a different cell. 

The proof consists, after some initial maneuvers inspired by ergodic theory, in controlling the behavior of completely multiplicative functions along certain quadratic polynomials. Considering separately aperiodic and "pretentious" functions, the last major ingredient is a concentration estimate for functions in the latter class when evaluated along sums of two squares.

The talk is based on joint work with Frantzikinakis and Klurman.

In this talk, we present an algorithm to compute p-adic heights on hyperelliptic curves with good reduction. Our algorithm improves a previous algorithm of Balakrishnan and Besser by being considerably simpler and faster and allowing even degree models. We discuss two applications of our work: to apply the quadratic Chabauty method for rational and integral points on hyperelliptic curves and to test the p-adic Birch and Swinnerton-Dyer conjecture in examples numerically. This is joint work with Steffen Müller.

Deep learning has revolutionised many scientific fields and so it is no surprise that state-of-the-art solutions to several inverse problems also include this technology. However, for many inverse problems (e.g. in medical imaging) stability and reliability are particularly important.

Furthermore, unlike other image analysis tasks, usually only a fairly small amount of training data is available to train image reconstruction algorithms.

Thus, we require tailored solutions which maximise the potential of all ingredients: data, domain knowledge and mathematical analysis. In this talk we discuss a range of such hybrid approaches and will encounter along the way connections to various topics like generative models, convex optimization, differential equations and equivariance.

It was asked by E. Szemerédi if, for a finite set $A\subset \mathbf{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper we show that this maximum is at least of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\varepsilon}$ for some $\varepsilon>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$. Our proof consists of three parts: combinatorial, analytical and number theoretical.

How can we find and apply the best optimization algorithm for a given problem?   This question is as old as mathematical optimization itself, and is notoriously hard: even special cases such as finding the optimal learning rate for gradient descent is nonconvex in general. 

In this talk we will discuss a dynamical systems approach to this question. We start by discussing an emerging paradigm in differentiable reinforcement learning called “online nonstochastic control”. The new approach applies techniques from online convex optimization and convex relaxations to obtain new methods with provable guarantees for classical settings in optimal and robust control. We then show how this methodology can yield global guarantees for learning the best algorithm in certain cases of stochastic and online optimization. 

No background is required for this talk, but relevant material can be found in this new text on online control and paper on meta optimization.

Prof. Elad's Bio

(Joint work with Kartik Prasanna)

Siegel modular forms are higher-dimensional analogues of modular forms. While each rational elliptic curve corresponds to a single holomorphic modular form, each abelian surface is expected to correspond to a pair of Siegel modular forms: a holomorphic and a generic one. We propose a conjecture that explains the appearance of these two forms (in the cohomology of vector bundles on Siegel modular threefolds) in terms of certain higher algebraic cycles on the self-product of the abelian surface. We then prove three results:
(1) The conjecture is implied by Beilinson's conjecture on special values of L-functions. Amongst others, this uses a recent analytic result of Radzwill-Yang about non-vanishing of twists of L-functions for GL(4).
(2) The conjecture holds for abelian surfaces associated with elliptic curves over real quadratic fields.
(3) The conjecture implies a conjecture of Prasanna-Venkatesh for abelian surfaces associated with elliptic curves over imaginary quadratic fields.

TBA