In this presentation, I will give an overview of a collaborative research project on observer theory as a strategy for estimating evolution problems. For wave-equation-type problems, we show how this approach allows us to use measured data of various kinds to reconstruct the observed trajectory and estimate the initial conditions or parameters. We analyze this by combining observability conditions with regularization techniques. We will also discuss discretization, numerical analysis, and data sampling. Finally, we will illustrate the application of this approach in several practical cases, from the wave equation to elastodynamics, including applications in cardiac modeling and estimation – which originally inspired our approach.

We consider variational problems whose natural setting is some fractional Sobolev spaces. The two main types of energies that one encounters are

1) usual integral functionals depending on fractional gradients (given as the gradient of a convolution; e.g., Riesz potentials);

2) double integrals, possibly determined by a singular kernel, as for Gagliardo seminorms.

For such energies, interesting new problems arise regarding semicontinuity conditions, limits as the kernel degenerate, and interaction with oscillations and concentration (such as in homogenization, dimension reduction, and Ginzburg-Landau vortices, etc.). In the course I will recall classical results for local integral functionals, and discuss how these results must be modified for nonlocal energies.

References:

• R. Alicandro, A. Braides, M. Solci, and G. Stefani. Topological singularities arising from fractional-gradient energies. Math. Ann. 393 (2025), 71–111.
• A. Braides. A simplified counterexample to the integral representation of the relaxation of double integrals. Comptes Rendus Math. 362 (2024), 487–491.
• A. Braides, S. Mannella, and A. J. A. Schiavoni Piazza. Homogenization and averaging in the Bourgain–Brezis–Mironescu limit. arXiv:2609.21933.
• A. Braides, M. Picerni, and A. J. A. Schiavoni Piazza. A homogenized Maz’ya–Shaposhnikova formula. arXiv:2610.10935.
• A. Braides, A. Pinamonti, and M. Solci. Dimension reduction of fractional Sobolev seminorms on thin domains. arXiv:2603.13968.
• C. Kreisbeck and H. Schönberger. Quasiconvexity in the fractional calculus of variations: characterization of lower semicontinuity and relaxation. Nonlinear Anal. 215 (2022), Paper No. 112625.

We consider variational problems whose natural setting is some fractional Sobolev spaces. The two main types of energies that one encounters are

1) usual integral functionals depending on fractional gradients (given as the gradient of a convolution; e.g., Riesz potentials);

2) double integrals, possibly determined by a singular kernel, as for Gagliardo seminorms.

For such energies, interesting new problems arise regarding semicontinuity conditions, limits as the kernel degenerate, and interaction with oscillations and concentration (such as in homogenization, dimension reduction, and Ginzburg-Landau vortices, etc.). In the course I will recall classical results for local integral functionals, and discuss how these results must be modified for nonlocal energies.

References:

• R. Alicandro, A. Braides, M. Solci, and G. Stefani. Topological singularities arising from fractional-gradient energies. Math. Ann. 393 (2025), 71–111.
• A. Braides. A simplified counterexample to the integral representation of the relaxation of double integrals. Comptes Rendus Math. 362 (2024), 487–491.
• A. Braides, S. Mannella, and A. J. A. Schiavoni Piazza. Homogenization and averaging in the Bourgain–Brezis–Mironescu limit. arXiv:2609.21933.
• A. Braides, M. Picerni, and A. J. A. Schiavoni Piazza. A homogenized Maz’ya–Shaposhnikova formula. arXiv:2610.10935.
• A. Braides, A. Pinamonti, and M. Solci. Dimension reduction of fractional Sobolev seminorms on thin domains. arXiv:2603.13968.
• C. Kreisbeck and H. Schönberger. Quasiconvexity in the fractional calculus of variations: characterization of lower semicontinuity and relaxation. Nonlinear Anal. 215 (2022), Paper No. 112625.

We consider variational problems whose natural setting is some fractional Sobolev spaces. The two main types of energies that one encounters are

1) usual integral functionals depending on fractional gradients (given as the gradient of a convolution; e.g., Riesz potentials);

2) double integrals, possibly determined by a singular kernel, as for Gagliardo seminorms.

For such energies, interesting new problems arise regarding semicontinuity conditions, limits as the kernel degenerate, and interaction with oscillations and concentration (such as in homogenization, dimension reduction, and Ginzburg-Landau vortices, etc.). In the course I will recall classical results for local integral functionals, and discuss how these results must be modified for nonlocal energies.

References:

• R. Alicandro, A. Braides, M. Solci, and G. Stefani. Topological singularities arising from fractional-gradient energies. Math. Ann. 393 (2025), 71–111.
• A. Braides. A simplified counterexample to the integral representation of the relaxation of double integrals. Comptes Rendus Math. 362 (2024), 487–491.
• A. Braides, S. Mannella, and A. J. A. Schiavoni Piazza. Homogenization and averaging in the Bourgain–Brezis–Mironescu limit. arXiv:2609.21933.
• A. Braides, M. Picerni, and A. J. A. Schiavoni Piazza. A homogenized Maz’ya–Shaposhnikova formula. arXiv:2610.10935.
• A. Braides, A. Pinamonti, and M. Solci. Dimension reduction of fractional Sobolev seminorms on thin domains. arXiv:2603.13968.
• C. Kreisbeck and H. Schönberger. Quasiconvexity in the fractional calculus of variations: characterization of lower semicontinuity and relaxation. Nonlinear Anal. 215 (2022), Paper No. 112625.

We consider variational problems whose natural setting is some fractional Sobolev spaces. The two main types of energies that one encounters are

1) usual integral functionals depending on fractional gradients (given as the gradient of a convolution; e.g., Riesz potentials);

2) double integrals, possibly determined by a singular kernel, as for Gagliardo seminorms.

For such energies, interesting new problems arise regarding semicontinuity conditions, limits as the kernel degenerate, and interaction with oscillations and concentration (such as in homogenization, dimension reduction, and Ginzburg-Landau vortices, etc.). In the course I will recall classical results for local integral functionals, and discuss how these results must be modified for nonlocal energies.

References:

• R. Alicandro, A. Braides, M. Solci, and G. Stefani. Topological singularities arising from fractional-gradient energies. Math. Ann. 393 (2025), 71–111.
• A. Braides. A simplified counterexample to the integral representation of the relaxation of double integrals. Comptes Rendus Math. 362 (2024), 487–491.
• A. Braides, S. Mannella, and A. J. A. Schiavoni Piazza. Homogenization and averaging in the Bourgain–Brezis–Mironescu limit. arXiv:2609.21933.
• A. Braides, M. Picerni, and A. J. A. Schiavoni Piazza. A homogenized Maz’ya–Shaposhnikova formula. arXiv:2610.10935.
• A. Braides, A. Pinamonti, and M. Solci. Dimension reduction of fractional Sobolev seminorms on thin domains. arXiv:2603.13968.
• C. Kreisbeck and H. Schönberger. Quasiconvexity in the fractional calculus of variations: characterization of lower semicontinuity and relaxation. Nonlinear Anal. 215 (2022), Paper No. 112625.

Concrete is the most widely used manufactured material in the world and forms the foundation of modern infrastructure. However, the production of Portland cement, the key binding component of concrete, is responsible for approximately 8% of global anthropogenic CO2 emissions. Improving the performance and durability of concrete is therefore an important strategy for reducing its environmental footprint by extending service life and minimizing repair and replacement needs. Achieving this goal requires a fundamental understanding of how hydration products form, how microstructures evolve, and how these microscopic features control macroscopic mechanical properties. Concrete is also a highly porous and heterogeneous material that can deteriorate under harsh environmental conditions, such as freeze-thaw cycles and chemical attack. Understanding the mechanisms responsible for such degradation is essential for developing more durable and resilient cementitious materials. At the same time, cement-based materials can act as carbon sinks through the carbonation of hydration products, particularly portlandite and calcium silicate hydrate (C-S-H). The efficiency of CO2 sequestration depends on a complex interplay of dissolution, transport, nucleation, and precipitation processes occurring across multiple length scales. All these phenomena are rooted in molecular interactions among water, ions, hydrates, and mineral surfaces. Molecular simulation provides a powerful tool for probing these mechanisms and bridging microscopic processes with macroscopic behavior. In this seminar, I will present a multiscale perspective on cementitious materials, spanning molecular to mesoscale phenomena and covering both physical and chemical processes. I will begin with a brief introduction to molecular simulation methods. Then I will discuss several computational approaches, including classical molecular dynamics with non-reactive and reactive force fields, enhanced sampling techniques such as metadynamics, and coarse-grained modeling. Selected applications in C-S-H gelation, ice crystallization in confined space, and carbon mineralization will be presented to demonstrate how molecular simulations can provide fundamental insights and support the development of more sustainable and durable cement-based materials.

Geometric ideas have played a tangential role in statistics since the origins of mathematical statistics, and their role has grown increasingly central in modern methodologies— from sampling to model inference. In this talk, we describe how geometry not only motivates key tools in statistics, from Hamiltonian-based samplers to kernel distances, but also underpins the very foundations of probability. The talk will focus on drawing connection between fields and discuss the historical motivations, from a modern viewpoint.

We study the modeling of a compressible two-phase flow in a porous medium. The governing PDE system is known as the Verigin problem with phase transition, which is the compressible analogue to the Muskat problem. We prove the convergence of an implicit time discretization scheme using the Wasserstein distance, obtaining distributional solutions in the limit that satisfy an optimal energy-dissipation rate. This talk is based on a joint work with Anna Kubin and Tim Laux.

Alzheimer’s Disease (AD), the most common form of dementia, is characterized by the accumulation of misfolded proteins in the brain, ultimately leading to cortical atrophy and cognitive decline. Accurate measurement of the atrophy patterns is essential not only for diagnosis but also for early detection and potential prevention. However, due to the high variability in cortical geometry across individuals, population- based analyses typically rely on long, sophisticated neuroimaging pipelines for cortical feature computation and inter-subject alignment. These traditional approaches often require several days to weeks of computation for large cohorts, creating a major bottleneck for large-scale neuroimaging studies. Deep learning–based approaches have the potential to dramatically accelerate this process.

In this talk, I will present the recent work conducted in our team to streamline the cortical surface-based pipeline. I will review optimization-based approaches and end-to-end deep learning methods for cortical mesh reconstruction and surface registration. Some insights into the Spherical Demons algorithm, a classical diffeomorphic registration method on the sphere, will be provided. I will then introduce NC-Reg, our novel approach for rigid registration of spherical surfaces using a neural representation. Subsequently, I will outline our ongoing and future work, with a particular focus on CorticalFlow++, a deep learning model for fast cortical mesh reconstruction directly from MRI images.

We believe that this work lays a strong foundation for more efficient, scalable, and robust tools for cortical surface analysis. These developments have the potential to enable large- cohort neuroimaging studies and ultimately contribute to both computational neuroscience and clinical applications

Boundary control of first-order hyperbolic systems has been extensively studied in the spatially one-dimensional setting. In multi-dimensions, however, progress has been limited, largely due to the loss of Riemann invariant structure. This talk reports recent progress in my doctoral research on boundary stabilization, boundary controllability, and boundary optimal control for multi-dimensional first-order hyperbolic systems. I will also highlight how energy propagation and internal interaction may provide a unifying perspective across these three problems.

Narrow escape problems study the behavior of Brownian motion in bounded domains with reflecting boundaries, except for small escaping regions. Domains with bottlenecks arise naturally in applications such as the transmission of ions or molecules in dendritic spines. In this talk, I will discuss the expected escape time, the distribution of exit locations, and the (un)conditional distributions of escape times. A central technical difficulty is to prove convergence to a suitable quasi-stationary distribution for an associated discretized Markov process, which plays a key role in the analysis of these quantities.

This talk presents two settings where continuous optimization unlocks progress on fundamentally combinatorial problems. In the first part, we consider pruning of large language models, where state-of-the-art methods operate layer-wise, minimizing a per-layer objective on a small calibration dataset. The underlying NP-hard problem is that of finding a binary mask that determines which weights to keep — the so-called mask selection problem. We propose to relax these combinatorial constraints to their convex hull and solve the resulting convex program using the Frank-Wolfe algorithm, reducing per-layer error by up to 80% with consistent improvements in perplexity and downstream accuracy. In the second part, we turn to the Hadwiger-Nelson problem, a long-standing open problem from discrete geometry about coloring the Euclidean plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this geometric coloring problem as an optimization task with a probabilistic, differentiable loss function, enabling gradient-based exploration of admissible configurations. This leads to the discovery of two novel six-colorings providing the first improvements in thirty years to the off-diagonal variant of the original problem. Despite their different origins, both lines of work share a common denominator: relax a discrete problem into a continuous one, solve it with first-order methods, and extract combinatorial structure from the solution.

Motivated by the study of stochastic gradient descent for over-parameterized models, we consider Langevin dynamics with noise projected onto directions which are orthogonal to an isometric group action. Our main result identifies a novel form of implicit regularization: when the initial and target density are both invariant under the group action, projected Langevin is equivalent in law to Langevin with identity diffusion but with an additional drift term proportional to the negative log volume of the group orbit. The proof is based on constructing a certain stochastic process on the group itself, and identifies the extra drift term as the negative mean curvature of the group orbit. Based on ongoing joint work with Govind Menon and Adrien Vacher.

Single-cell omics methods provide high-throughput, molecular-scale measurements of cellular processes, offering insights into the machinery of cell fate decision-making. However, the high dimensionality, stochasticity, and cross-sectional nature of these measurements hinder their integration with biophysical models of gene regulation. My research aims to bridge this gap by combining generative AI with the biophysics of gene regulation to infer actionable and interpretable models of cell fate decision-making from single-cell omics data. In this direction, I recently developed probability flow inference (PFI), a computational approach to infer models specified by arbitrary stochastic differential equations from time-resolved single-cell RNA-seq data. I will discuss the mathematical underpinnings of the method, its relationship with the flow matching paradigm and the open mathematical questions it raises. I will show that such models not only infer better biological representations, but also generalize better to out-of-distribution data, outperforming state-of-the-art purely data-driven approaches.