This colloquium takes place every other Wednesday afternoon, 16:00-17:00, in room WN-M143 . For more information, please contact one of the organizers Joost Hulshof and Dennis Dobler.

A database of earlier years' talks can be found here.

## Upcoming talks in 2018:

Wed 19 September: **Jan Bouwe van den Berg** (VU), Room WN-S623, 16:00-17:00

*Title:* Computer-assisted theorems in dynamics

*Abstract:* In nonlinear analysis we often simulate dynamics on a computer, or calculate a numerical solution to a partial differential equation. This gives very detailed, stimulating information. However, it would be even better if we can be sure that what we see on the screen genuinely represents a solution of the problem. In particular, rigorous validation of the computations would allow such objects to be used as ingredients of theorems. In this talk we explore an approach based on a Newton-Kantorovich type argument in a suitable neighborhood of a numerically computed candidate. This method has been applied successfully for various problems in ordinary differential equations, delay differential equations and partial differential equations. We will illustrate the general setup using an example stemming from the Navier-Stokes equations in two dimensions. The latter is joint work in progress with Maxime Breden, Jean-Philippe Lessard and Lennaert van Veen.

## Previous talks in 2018:

Wed 13 Juni: **Paola Gori-Giorgi** (VU), Room WN-M143, 16:00-17:00

*Title:* Multi-marginal Optimal Transport and Density Functional Theory: A mathematical setting for physical ideas

*Abstract:* Electronic structure calculations are at the very heart of predictive computational materials science, chemistry and biochemistry. Their goal is to solve, in a reliable and computationally affordable way, the many-electron problem, a complex combination of quantum-mechanical and many-body effects. The most widely used approach, which achieves a reasonable compromise between accuracy and computational cost, is Kohn-Sham (KS) density-functional theory (DFT). Although exact in principle, practical implementations of KS-DFT must heavily rely on approximations for the so-called exchange-correlation (XC) functional. Empirical approximations (e.g., fitted on several data sets) are successful in normal cases, but typically lack predictive power for systems outside the training set. For this reason, exact mathematical conditions and rigorous guiding principles to build the XC functional have always played a key role in the field. In the recent years, it has been shown that there is a special semiclassical limit of the XC functional, relevant for the most challenging cases in KS DFT, which can be reformulated as a multi-marginal optimal transport problem, linking two rather distant research fields. In this talk I will review this reformulation, providing an overview of the key results from the optimal transport community, and discussing some of the open questions and conjectures that still need a rigorous proof.

Wed 30 Mei: **Michiel Bertsch** (University of Rome Tor Vergata), Room WN-M143, 16:00-17:00

*Title:***Mathematical modelling of Alzheimer's disease**

*Abstract:* Up to now there is no effective cure for Alzheimer's disease (AD). One of the major reasons is its complexity. Although the biomedical knowledge about AD is rapidly increasing, there is not yet a clear picture available about the major causes and the evolution of the disease. In such circumstances, can mathematical modelling be useful at all? In the colloquium I propose a modelling approach which, in a certain sense, is characterized by flexibility. I present a "toy model", which deliberately takes into account only a very limited amount of aspects of the disease (in this case the role of beta-amyloid and the existence of different time scales). The toy model seems to be flexible enough to include other aspects (such as the role of the tau protein) or novel biomedical insight. Surprisingly, the toy model itself suggests the possible importance of a very specific biomedical process, which is also discussed in the biomedical literature.BA

**Viresh Patel**(UvA), Room WN-M143, 16:00-17:00

*Title:* Quasi Ramsey problems

*Abstract:* Ramsey theory is currently one of the most active areas of research in combinatorics. The seminal question in the area, raised by Ramsey in 1930 can be formulated as follows: how large does n have to be to guarantee that in any room with n people we can find a set S of k people such that either every pair in S is acquainted or every pair in S is not acquainted. It is not immediately clear that such an n exists, although it is not hard to show. On the other hand the known bounds for n as a function of k are quite poor. I will discuss the Ramsey problem as well as variants of it. In particular I will discuss a relaxation of the problem above for which we are able to give quite precise bounds. This is based on joint work with Janos Pach, Ross Kang, Eoin Long and Guus Regts.

Wed 02 Mei: **Joris Mooij** (UvA), Room WN-M143, 16:00-17:00

*Title:* Joint Causal Inference from Observational and Experimental Data

*Abstract:*The standard method to discover causal relations is by using experimentation. Over the last decades, alternative methods have been proposed: constraint-based causal discovery methods can sometimes infer causal relations from certain statistical patterns in purely observational data. We introduce Joint Causal Inference (JCI), a novel constraint-based approach to causal discovery from multiple data sets that elegantly unifies both approaches. Compared with existing constraint-based approaches for causal discovery from multiple data sets, JCI offers several advantages: it deals with several different types of interventions in a unified fashion, it can learn intervention targets, it systematically pools data across different datasets which improves the statistical power of independence tests, and most importantly, it improves on the accuracy and identifiability of the predicted causal relations.

Wed 18 April: **Sjoerd Verduyn Lunel** (UU), Room WN-M143, 16:00-17:00

*Title:* Transfer operators, Hausdorff dimension and the spectral theory of positive operators

*Abstract:* In this talk we present a new approach to compute the Hausdorff dimension of conformally self-similar invariant sets using an elementary direct spectral analysis of a transfer operator associated with the problem. We start from scratch, introduce the notion of transfer operator and combine ideas from the theory of positive operators and from the theory of trace class operators and their determinants. Our approach is illustrated with examples from dynamical systems and number theory via Diophantine approximations.

Wed 21 Maart: **Peter Grunwald** (CWI, Leiden), Room WN-M143, 16:00-17:00

*Title:* Safe Testing

*Abstract:*A large fraction (some claim > 1/2) of published research in top journals in applied sciences such as medicine and psychology is irreproduceable. In light of this 'replicability crisis', standard p-value based hypothesis testing has come under intense scrutiny. One of its many problems is the following: if our test result is promising but nonconclusive (say, p = 0.07) we cannot simply decide to gather a few more data points. While this practice is ubiquitous in science, it invalidates p-values and error guarantees. Here we propose an alternative hypothesis testing methodology based on supermartingales - it has both a gambling and a data compression interpretation. This method allows us to consider additional data and freely combine results from different tests by multiplication (which would be a mortal sin for p-values!), and avoids many other pitfalls of traditional testing as well. If the null hypothesis is simple (a singleton), it also has a Bayesian interpretation, and essentially coincides with a proposal by Vovk (1993). We work out the case of composite null hypotheses, which allows us to formulate safe, nonasymptotic versions of the most popular tests such as the t-test and the chi square tests. Safe tests for composite H0 are not always Bayesian, but rather based on the 'reverse information projection', an elegant concept with roots in information theory rather than statistics.

Wed 07 Maart: **Nelly Litvak**, Room WN-M143, 16:00-17:00

*Title:* Power-law hypothesis for PageRank

*Abstract:* PageRank is a well-known algorithm, which has been proposed by Google for ranking pages in the World-Wide Web. PageRank can be interpreted as a stationary distribution of a random walk of a user that hops from one web page to another. Beyond the web search, PageRank has many applications in network of different kinds, for example, discovering communities in social networks, or finding endangered species in ecological networks. Most of these real-life networks have so-called power-law degree distribution: if a network is represented as a graph, then the fraction of vertices with degree k is approximately proportional to a negative power of k. Moreover, many empirical studies confirm that PageRank also has a power law distribution, with the same negative power as in-degree. In this talk I will discuss to which extend we can formalize this empirical observations analytically. Formally, we will model networks as random graphs and investigate the limiting behavior of PageRank as the graph size goes to infinity. I will present results for some specific random graph models, and very recent general limiting results for a large class of random graphs. This talk is based on joint works with Remco van der Hofstand and Alessandro Garavaglia (Eindhoven University of Technology) and Mariana Olvera-Cravioto (Univerity of California at Berkley).

Wed 21 Februari: **Gijs Heuts** (UU), Room WN-M143, 16:00-17:00

*Title:* Lie algebras and periodicity in homotopy theory

*Abstract:* Homotopy theory is the study of continuous deformations of spaces. The general problem of classifying such deformations is notoriously hard. However, if one is only interested in rational invariants of spaces then there are good algebraic tools available: Quillen constructed for every space a Lie algebra from which such invariants can be calculated, whereas Sullivan built a commutative algebra (much like the algebra of differential forms on a manifold) that retains essentially the same information. I will discuss a modern viewpoint of homotopy theory called the "chromatic perspective": much like a ray of white light is broken into different colours through a prism, a space can be decomposed into pieces corresponding to various "frequencies". The rational invariants correspond to one of these pieces. It turns out that Lie algebras may also be used to give models for the others.

Wed 07 Februari: **Damaris Schindler** (UU), Room WN-M143, 16:00-17:00

*Title:***Systems of quadratic forms
**

*Abstract:* In this talk we discuss some aspects concerning the arithmetic of systems of quadratic forms. Our focus will be on the local-global principle for the existence of rational or integral solutions and we will discuss some failures of this principle.