Abstract:
What can mathematics contribute to biology? It can elucidate principles, develop models and provide tools for the analysis of biological data. In turn, this can inspire new mathematical research. The examples presented will include fundamentals of life, neurobiological models and biological (and other) networks. The new mathematical tools will include stochastic dynamical systems, combinatorial schemes, hypergraph analysis and geometric principles of machine learning.

Abstract:
Artificial intelligence (AI) based Molecular Sciences have begun to gain momentum due to the great advancement in experimental data, computational power and learning models. However, a major issue that remains for all these AI-based learning models is the efficient molecular representations and featurization. Here we propose advanced mathematics-based molecular representations and featurization. Molecular structures and their interactions are represented by high-order topological and algebraic models (including Rips complex, Alpha complex, Neighborhood complex, Dowker complex, Hom-complex, Tor-algebra, Rhombille tiling, etc). Mathematical invariants (from persistent homology, Ricci curvature, persistent spectral, Analytic torsion, algebraic variety, etc) are used as molecular descriptors for learning models. Further, we develop geometric and topological deep learning models that can systematically incorporate molecular high-order, multiscale, and periodic information, and use them for analysing molecular data from chemistry, biology, and materials.

Abstract:
The geometry of a complex network shapes the way the network processes, transmits, and organizes information. But what exactly is the geometry of a network, and how can we uncover it? In this talk, I will show how foundational concepts originated from Riemannian manifolds provide powerful tools for analysing and understanding the structure of complex data. By leveraging geometric insights, we can uncover hidden community structures in social networks, identify bottlenecks in chemical reactions, and explain phenomena such as over-squashing in (simplicial) neural networks.

Abstract:
Mathematical modeling has served as a cornerstone of biology over the past century, transforming it from a largely descriptive science into a predictive one. From Mendelian genetics and population dynamics to epidemic spreading, neural excitability, and pattern formation, mathematical frameworks have provided mechanistic insight and predictive power. Today, however, the rapid rise of artificial intelligence and data-driven approaches raises the question: is mathematical modeling still relevant? This talk argues that the future lies in combining modeling and learning, where hypotheses about data structure complement predictive algorithms, leading to deeper insights into biological systems.