Differential Equations seminars

The Korteweg–de Vries equation is a nonlinear third-order partial differential equation central to wave theory and commonly studied through inverse scattering. This seminar addresses the less extensively developed stochastic setting, focusing on the Cauchy problem for a generalized equation with noise in its nonlinear term. The noise is expressed through a symmetric integral with an arbitrary continuous integrator. This construction generalizes both Stieltjes integration and Stratonovich integration against a Wiener process. The proposed numerical methods extend Strang splitting and BCH schemes to stochastic Korteweg–de Vries equations. The presentation also examines existence, uniqueness and stability of solutions, and formulates a change-point problem for detecting shifts in the underlying regime from observations.

The one-phase Muskat problem models the boundary between wet and dry regions in a porous material under Darcy’s law. Including surface tension is physically important but makes the analysis more difficult. Hyunwoo Kwon presents a proof of global well-posedness for small initial data in the subcritical regime, together with convergence of the solution to zero in Lipschitz norm at long times. The work is presented as the first global well-posedness result for this version of the problem with surface tension. Time permitting, the talk also includes a numerical experiment. The research is joint with Hongjie Dong.

When can an algebraic differential equation be solved by algebraic functions? Daniel Litt discusses a question dating to Fuchs in 1875 and the number-theoretic perspective supplied by conjectures of Grothendieck and Katz for linear equations, and of Ekedahl, Shepherd-Barron, Taylor and Bost more generally. Joint work with Josh Lam provides progress for isomonodromy differential equations, a class that includes Painlevé VI. These results also motivate the study of motivic structures in the space of algebraic differential equations.

Multicomplex numbers extend complex numbers by adjoining several commuting square roots of minus one. William Verreault shows how classifying certain canonical involutions reduces to studying a particular family of involutory matrices in characteristic two, giving an explicit enumeration of the involutions. Returning to analysis, he explains how these involutions define complex structures and corresponding Cauchy–Riemann equations for multicomplex-valued functions. Although the chosen involution changes the coordinates of those equations, the resulting class of holomorphic functions remains unchanged.

Blaschke–Santaló diagrams visualize optimal inequalities connecting specified functionals. This talk introduces their underlying principles and main analytical methods through a detailed theoretical study of the diagram relating perimeter, volume and the first eigenvalue of the Dirichlet Laplacian. It then considers numerical methods for approximating such diagrams. The presentation draws on joint work with Jimmy Lamboley and Eloi Martinet.

Accurate waveforms for a small black hole entering a much larger one remain difficult near merger. This talk presents a staged treatment of the first-order merger and ringdown that can isolate nonlinear contributions for later corrections. It also analyses the transition to plunge in rotating backgrounds, where catastrophe theory explains why equatorial and inclined trajectories share the same Painlevé equation. Together these approaches expose structure useful for extending self-force calculations.

A Nichols algebra can be associated with an object of a braided tensor category through braid-group combinatorics or a universal characterization. Besides reconstructing quantum groups from their Cartan components, these algebras can generate new braided tensor categories with rich root-system structures. Simon Lentner connects this construction to conformal field theory, where braidings arise from monodromy of differential equations such as the Knizhnik–Zamolodchikov equation. The Kazhdan–Lusztig correspondence relates the representation theories in these settings to quantum groups. The talk describes a nonsemisimple form of this correspondence, conjectured by Feigin and collaborators and proved using Nichols algebras and categorical methods. It also presents recent work identifying the braided tensor category of affine sl(2) modules at admissible level through a quantum group of type sl(2|1).

Resolving singular Spin(7) spaces links exceptional geometry with gauge-theory vacuum structure. This talk builds torsion-free structures on compact orbifold resolutions from local ALE data encoded by harmonic self-dual forms. It explains the relation to supersymmetric gauge equations and asks when local solutions extend globally. Gluing methods and Dirac operators identify analytic obstructions, with explicit examples attaining full Spin(7) holonomy.

This seminar investigates Anderson localization in subwavelength elastic periodic systems with random perturbations. For the unperturbed structure, layer-potential methods recast the eigenvalue problem as boundary integral equations, yielding asymptotic expressions for subwavelength eigenvalues and establishing a band gap above the subwavelength band. A Floquet transform then brings the perturbed problem into a periodic formulation and produces equations for resonant frequencies under general perturbations. Numerical experiments on monomer and dimer structures test agreement with the analysis. Increasing the strength and number of random perturbations demonstrates localization, providing a mathematical basis for elastic metamaterial design.

How do humans stabilize an inverted pendulum, and why does a balanced pole eventually fall? Drawing on 25 years of fingertip pole-balancing research, this talk examines neural correction delays: longer poles move more slowly relative to the nervous system’s response time. Delay-differential models can stabilize the upright position, yet skilled people still experience falls. The proposed explanation is microchaos arising from interactions among delay, sensory dead zones and frequency-dependent force encoding. A region of transient falling solutions lies next to stable microchaotic dynamics. Such microchaos is absent in virtual frontal-plane balancing tasks, while models of standing postural sway lack the corresponding transient regime. The comparison suggests that human falls, unlike pole falls, are more plausibly associated with medical events or slips and trips.

Navid Azizan presents methods that make neural models obey physical, safety and operational constraints at deployment. Hard-constrained neural networks, or HardNets, enforce input-dependent constraints by construction while preserving universal approximation within the feasible function class. Applications include models of chaotic dynamics with bounded trajectories, energy-constrained operator learning, safe reinforcement learning and control with formal guarantees. The talk then turns to enforcing constraints during sampling from pretrained diffusion and flow-matching models. Formulating generation as trajectory optimization allows receding-horizon control to guide outputs toward feasibility without retraining or imposing excessive restrictions on the sampling process. Examples from fluid dynamics, robot planning and control, PDE control and language-guided image editing illustrate how constraints can define admissible behavior while retaining expressive learning and generation.

Uddipan Banik derives a dynamical account of dark-matter halo density profiles using the Boltzmann–Poisson equations for an expanding Universe populated by subhalos of many masses. A self-similar solution links density slope to halo mass growth; including gravitational scattering and subhalo capture gives gamma = 6/(s + 2). Initial collapse with s = 2 produces a prompt cusp of slope 3/2. Hierarchical growth connects the slope to the primordial power spectrum and recovers the Syer–White relation during matter domination. Stable clustering selects an inner Navarro–Frenk–White cusp of slope one, while stalled accretion approaches the outer slope of three. The framework connects these profiles to successive stages of halo assembly.

Leonid Berlyand discusses mathematical models for the onset of cell motion driven by myosin contraction. A two-dimensional free-boundary PDE model links cell-shape evolution to diffusion and Keller–Segel-type transport. The talk examines linear stability, a stability-determining eigenvalue and the way nonlinear diffusion changes the bifurcation from supercritical to subcritical. It also considers the curvature of the bifurcation curve, connections to bistability, and the role of non-self-adjoint operators. An example illustrates why a spectral gap alone need not guarantee stability. This is an in-person Analysis of Fluids and Related Topics seminar at Princeton.

Antonio Pedro Ramos examines point-evaluation bounds for integrable functions of spherical exponential type in the multivariable Paley–Wiener space. Radial extremizers connect the problem to a weighted space in one variable. The talk establishes a third-order linear differential equation with polynomial coefficients in every dimension, extending the previously known odd-dimensional result. It also develops interpolation and reciprocal identities using the extremizer’s zeros. The research is joint work with Felipe Gonçalves and Danylo Radchenko.

Qihao Ye examines practical, affordable ways for individual mathematicians to use AI in literature searches, proof development and verification. The talk identifies seven recurring failure modes and introduces the open-source QED multi-agent system designed to address them. A case study on constructing Carleman weights for wave equations on the half-line shows how automated search can be combined with a researcher’s mathematical insight and rigorous checking. The organizer links both the implementation and the accompanying research paper. Online via Rutgers DIMACS Zoom. Wednesday 23 September 2026, 10:00–11:00 EDT (America/New_York; UTC−4). Use the Zoom link and passcode on the official event page. Organized by the DIMACS AI and Mathematics Seminar. This listing advertises the verified remote attendance option.

This seminar discusses a proof-of-concept gravitational-waveform model described in arXiv:2511.10522. It augments post-Newtonian equations using physics-informed neural-network methods and universal differential equations. The talk also covers ongoing development of a precessing surrogate model designed to evaluate in less than one millisecond while covering a broader portion of the LVK frequency band.

Matthias Bolten discusses the scalability of multigrid solvers for linear systems arising from discretized partial differential equations. On modern supercomputers, heterogeneous CPUs and GPUs and the widening gap between computation, network, and memory speeds complicate parallelization. Classical multigrid analysis relies on tightly coupled multiplicative components, whereas additive and asynchronous variants relax this coupling. The talk compares approaches to improving high-performance multigrid scalability, including asynchronous execution.

Optimal transport is a mathematical method to define a distance between probability distributions. This is particularly useful in various domains, including physics, biology, machine learning, and economics, among others. After introducing the Optimal Transport (OT) problem at finite temperature, we show how it can be formulated as a statistical physics problem. This approach allows us to derive very efficient algorithms to effectively compute the distance between two probability distributions. The a priori unrelated Schrödinger bridge (SB) problem is presented, and it is shown to be a dynamical version of the optimal transport problem. Indeed, the Schrodinger bridge looks for the most probable path in probability distribution space, which connects two given probabilities. The Schrodinger bridge problem, originally devised for freely diffusing particles, can be generalized to the case of interacting particles. It can be formulated in terms of functional integrals over bosonic fields, which allows us to derive partial differential equations that characterize the most probable paths in probability space. CARL VAN VREESWIJK MEMORIAL LECTURE 2026. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-04-15. Recording duration: 00:55:05.

Mikaela Iacobelli introduces kinetic theory and the mathematical description of charged particles in plasmas through Vlasov-type equations. The colloquium examines stability and instability, well-posedness, and how solutions behave in singular limits. It also presents a new family of Wasserstein-type distances that offers tools for studying the stability of kinetic equations. The exposition is intended for a broad mathematical audience.

Active matter describes a class of systems that are maintained far from equilibrium by driving forces acting on the constituent particles. Here I will focus on confined active nematics, which exhibit especially rich flow behavior, ranging from structured patterns in space and time to disordered turbulent flows. To understand this behavior, I will take a deterministic dynamical systems approach, beginning with the hydrodynamic equations for the active nematic. This approach reveals that the infinite-dimensional phase space of all possible flow configurations is populated by Exact Coherent Structures (ECS), which are exact solutions of the hydrodynamic equations with distinct and regular spatiotemporal structure; examples include unstable equilibria, periodic orbits, and traveling waves. The ECS are connected by dynamical pathways called invariant manifolds. The main hypothesis in this approach is that turbulence corresponds to a trajectory meandering in the phase space, transitioning between ECS by traveling on the invariant manifolds. Similar approaches have been successful in characterizing high Reynolds number turbulence of passive fluids. Here, I will present the first systematic study of active nematic ECS and their invariant manifolds and discuss their role in characterizing the phenomenon of active turbulence.