Thermodynamic chemical reaction networks provide a unique class of physico-chemical systems that naturally integrate a wide range of mathematical disciplines, including dynamical systems, discrete and combinatorial mathematics, large deviation theory, and real algebraic geometry.
The collection of chemical reactions defines a hypergraph structure, which imposes intrinsic topological and algebraic constraints on the state space. Thermodynamic laws variationally characterize equilibrium states, while kinetic descriptions encode a duality between forces and fluxes together with the associated dissipative structures.
Taken together, these ingredients furnish a rich mathematical framework underlying discrete geometric analysis.
In this talk, we discuss how insights from chemical reaction network theory can be systematically combined with geometric analysis to explore new frontiers in discrete geometric analysis.
This talk will be a review on how nonlinear Laplacian shows up in geometric analysis. Specifically, we discuss heat flow on Finsler manifolds and hypergraphs, along with its connection with optimal transport theory and Ricci curvature.
Given two nonnegative densities with equal mass, the optimal transport problem seeks to rearrange one to look like the other with the least amount of “effort.” The minimal effort required can be used to quantify how similar or different the two densities are and define a notion of distance on the space of densities. Over the past 25 years, optimal transport techniques have led to several breakthroughs in the analysis of continuous data structures, in which distributions of data can be modeled as densities (or, more generally, measures) on a smooth manifold. Examples include image classification problems, in which 2d greyscale images are represented as densities on a rectangle, and gene trajectory inference, in which gene dynamics in a single cell are modeled by evolving measures in a high dimensional Euclidean space.
In spite of this progress, many data structures are fundamentally discrete and cannot naturally be modeled as measures on smooth manifold. Furthermore, data structures can also possess both continuous and discrete elements, such as a distribution of 2d images of clothing (continuous), coupled with the labels of the types of clothing (discrete).
In today’s talk, I will introduce the notion of vector valued optimal transport, which provides a notion of distance to compare data structures with mixed continuous and discrete elements. I will describe the connection between our approach and the four existing notions of vector valued optimal transport and discuss applications in image classification.
The immune repertoire responds to a wide variety of pathogenic threats. Immune repertoire sequencing experiments give us insight into the composition of these repertoires. Since the functioning of the repertoire relies on statistical properties, statistical analysis is needed to identify responding clones. Using statistical inference and machine learning methods, I will show how we can identify responding clonotypes and their dynamics.
Understanding how brain-wide neural circuits are genetically wired is a central challenge in neuroscience. Sperry’s chemoaffinity theory famously proposed that molecular gradients provide positional information for axonal targeting, but its relevance has been demonstrated mainly in relatively simple sensory systems. Whether similar genetic principles organize wiring across the entire brain has remained unclear. Here, we introduce SPERRFY (Spatial Positional Encoding for Reconstructing Rules of axonal Fiber connectivitY), a data-driven framework designed to uncover genetic rules of brain-wide wiring. By integrating whole-brain connectomic data with spatial gene-expression maps from the Allen Mouse Brain Atlas, SPERRFY identifies latent positional gradients that shape axonal connectivity. Using canonical correlation analysis, we extract pairs of spatial gradients that systematically align with observed connectivity patterns, revealing both global inter-regional structure and local intra-regional organization. Remarkably, connectivity reconstructed solely from these gradients predicts empirical wiring with high accuracy (AUC ≈ 0.88), and permutation-based controls confirm that the extracted structure cannot be explained by chance. In addition, SPERRFY highlights candidate genes whose spatial expression patterns are strongly associated with wiring gradients, offering molecular clues to the developmental logic of large-scale circuit formation. Together, these results extend Sperry’s original idea from local sensory maps to the entire brain, demonstrating that genetically encoded positional information provides a unifying principle for brain-wide connectivity. SPERRFY offers a general framework for discovering how genetic programs shape complex neural networks at scale.
The presenter proposes a chemical sensing system allowing identification of chemical information in mixtures qualitatively and quantitatively. Herein, artificial receptors are designed based on molecular recognition chemistry, which enables both selective and cross-reactive detection of analytes.
We develop models that digitalize and predict odors based on odorant molecular structures and olfactory receptor activity profiles. By integrating OR responses and 3D molecular similarity, we quantitatively predict perceptual odor similarity, contributing to a systems-level understanding of olfaction.
Most natural smells, whether in food, perfume, or the environment, are complex mixtures of many odor molecules. These mixtures are often assumed to interact in ways that create new emergent qualities, distinct from their components, making them difficult to model. Yet some recent studies suggest that mixtures behave more simply, with each component contributing independently to the overall perception. To test how often odor mixtures exhibit emergent or suppressed qualities, we collected descriptive ratings for 453 mixtures and 218 component odorants. Remarkably, the perceptual profiles of all mixtures fell within the perceptual space bounded by their components after accounting for session effects. Even mixtures previously thought to produce new percepts, for example a mixture of caramel and strawberry smelling like pineapple, were well explained by a simple model averaging component profiles, achieving high predictive accuracy. These findings suggest that the main challenge in modeling odor perception is characterizing individual components, rather than exhaustively mapping odor-odor interactions. By focusing on modeling single-component perception, we may move closer to a quantitative predictive framework for olfaction with wide scientific and applied impact.