Week 26.05.2025 – 01.06.2025

Addressing water scarcity, soil degradation, and shifting weather patterns is of primary importance to ensure long-term food security.
In this talk, I will describe intercropping-an ancient and complementary alternative to monoculture farming-known for its lower input
requirements and often comparable yields. For intercropping to be widely adopted, it's essential to evaluate its controllability,
effectiveness and resilience. Based on a dataset regrouping the results of thousands of experiments, this talk aims to lay quantitative
foundations of intercropping design and control.While further experiments are necessary before computational tools can reliably guide
this practice, the presented findings mark an initial step toward sustainable multi-species agriculture.

Abelian varieties are higher-dimensional generalisations of elliptic curves and are ubiquitous in algebraic geometry and number theory. Central to their theory is the concept of a polarisation. If
A
is an abelian variety over an algebraically closed field, then every polarisation is represented by an ample line bundle on
A
. However, such a line bundle may not exist if the field is not algebraically closed, or when it is replaced by a more general base scheme\DSEMIC in fact, this failure already occurs for Jacobians.

In 1999, Poonen and Stoll asked: can every polarisation be represented by a line bundle on some torsor under
A
? In this talk, I will expand on the background for this question and I will explain why the answer is often ``yes'', but not always. Along the way, we will encounter Mumford theta groups, Serre's notion of negligible group cohomology and moduli spaces of abelian varieties.

Species sampling processes have long provided a fundamental framework
for random discrete distributions and exchangeable sequences. However,
analyzing data from distinct, yet related, sources, requires a broader
notion of probabilistic invariance, with partial exchangeability as the
natural choice. Over the past two decades, numerous models for partially
exchangeable data, known as dependent nonparametric priors, have
emerged, including hierarchical, nested, and additive processes. Despite
their widespread use in Statistics and Machine Learning, a unifying
framework remains elusive, leaving key questions about their learning
mechanisms unanswered.

We fill this gap by introducing multivariate species sampling models, a
general class of nonparametric priors encompassing most existing
dependent nonparametric processes. These models are defined by a
partially exchangeable partition probability function, encoding the
induced multivariate clustering structure. We establish their core
distributional properties and dependence structure, showing that
borrowing of information across groups is entirely determined by shared
ties. This provides new insights into their learning mechanisms,
including a principled explanation for the correlation structure
observed in existing models.

Beyond offering a cohesive theoretical foundation, our approach serves
as a constructive tool for developing new models and opens new research
directions aimed at capturing even richer dependence structures.