Speaker:   Michal Shavit
Title:   Rigorously Certified Neural-Network Computation of Nonlinear Water Waves at Near-Machine Precision
Abstract:   I will present a computational approach to studying strongly nonlinear regimes of PDEs that combines physics-informed neural computation at near-machine precision with rigorous certification and formal verification.
We demonstrate the approach on high-amplitude periodic traveling water waves, recovering known families of solutions and revealing new high-amplitude gravity–capillary wave profiles. For a representative profile, we carry out the full pipeline, showing how staggered physics-informed neural computation can attain residuals approaching machine precision, at the (10^{-13}) level. We then use interval arithmetic to rigorously certify the residual while accounting for floating-point and numerical errors, and formally verify the resulting certificate in Lean.
This is joint work with Yongji Wang and Tristan Buckmaster.
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Speaker:   Anuj Kumar
Title:   Nonuniqueness of solutions to the two-dimensional Euler equations with integrable vorticity
Abstract:   Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in L^p ( p > 1). A celebrated open question is whether the uniqueness result can be generalized to solutions with L^p vorticity. In this talk, we resolve this question in negative for some p > 1. To prove nonuniqueness, we devise a new convex integration scheme that employs non-periodic, spatially-anisotropic perturbations, an idea that was inspired by our recent work on the transport equation. To construct the perturbation, we introduce a new family of building blocks based on the Lamb-Chaplygin dipole. This is a joint work with Elia Bruè and Maria Colombo.
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Speaker:   Sergey Nersisyan

Abstract:  
It is known that in the simply connected case, exotica in dimension four can be localized to contractible submanifolds. In this talk,
I will first review embeddings of contractible 4-manifolds in the 4-sphere and other 4-manifolds. Then I will present a kind of monotonicity result about
exotic embeddings in the 4-sphere, as well as a construction of interesting 2-component "links" of Mazur manifolds.

Speaker:   Dina Soltani Tehrani
Title:   Scale Locality of Kinetic Energy Transfer in Normal Shocks
Abstract:   We investigate the scale locality of kinetic energy (KE) transfer across scales in one-dimensional normal shocks using a coarse-graining approach. There is a common belief that shocks and other discontinuous structures in a flow necessarily imply a non-local scale-transfer of KE so that a finite portion of the KE at a given scale gets dissipated directly into heat at the viscous (molecular) scales without undergoing an inertial range scale-local cascade. Whether such a cascade exists and whether it is scale-local depends on how the governing KE equation is coarse-grained, and this choice can be informed by physical considerations. We propose a formulation for the coarse-grained KE budget of variable-density flows, which we use to demonstrate that for a one-dimensional normal shock, pressure-dilatation separates into a KE-to-internal energy (IE) conversion at the largest scales and a microphysical KE--IE conversion mechanism that we call "microphysical pressure-dilatation". In this formulation, we find that KE is transferred solely through a scale-local inertial-range cascade, similar to that in a Burgers shock. The cascade term is constant over the inertial range and matches the KE injected at the largest scales and the net KE sink at the smallest scales, which is the sum of viscous dissipation and microphysical pressure-dilatation. Microphysical pressure-dilatation changes sign with increasing Prandtl number and explains why velocity and temperature have equal front widths despite different values for viscosity and thermal conductivity. We discuss the implications of these results for interpreting energy transfer and anomalous dissipation in compressible flows; in particular, existing rigorous arguments establishing the limiting behavior of pressure-dilatation as $\ell \to 0$ in the inviscid limit can, with suitable modifications, be carried over to our formulation.
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Speaker:      Michele Dolce 
Title:   Instability of the 2D Taylor-Green vortex
Abstract:   The 2D Taylor-Green (TG) vortex (aka cellular flow) is the prototypical example of an Euler steady state on T^2 possessing truly two-dimensional features, like elliptic and hyperbolic stagnation points. Its streamfunction, sin(x)sin(y), lives on the second Fourier shell, making it susceptible to large-scale destabilizing mechanisms. Despite the apparent simplicity of the steady state, a proof of its spectral instability has long remained elusive, and was only recently observed numerically. To solve this problem, I will introduce a new criterion to detect unstable eigenvalues for a wide class of linear Hamiltonian operators. We apply this to prove the stability of the TG vortex with respect to odd perturbations. In the subspace of functions even in both variables, we combine our criterion with a rigorous computer-assisted argument to locate two unstable eigenvalues. This fully characterizes the unstable spectrum of the TG vortex and implies nonlinear instability in velocity. This is a joint work with G. Cao-Labora, M. Colombo and P. Ventura.
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Speaker:   Tommaso Rosati  
Title:   Lyapunov exponents for linear cocycles driven by stochastic Navier-Stokes
Abstract:   We provide the first convergence result for finite time Lyapunov exponents of Passive Scalar Advection and Linearized Stochastic Navier-Stokes (two linear cocycles driven by the stochastic Navier-Stokes equations). The proof is based on establishing unique ergodicity for a Markov process with values in an infinite-dimensional projective space. Uniqueness of the invariant measures requires an extension of asymptotic coupling tools, and proving a generic non-collinearity for passive scalar advection. Joint work with Sam Punshon-Smith.
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Speaker:   Gautam Iyer
Title:   Enhanced Dissipation and Mixing
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Speaker:   Takumi Matsuzawa
Title:   Nonlinear diffusion and decay of a turbulent blob
Abstract:   This talk is concerned with how turbulence spreads into the surrounding quiescent fluid and decays. Investigating these fundamental aspects is often hindered by mean flows and their interactions with material boundaries. Here, we overcome these challenges by creating a spatially localized “blob” of turbulence by colliding vortex loops at the center of a still water tank. Because the blob is initially unaffected by mean flow, this setup enables a clean investigation of the interplay between the decay and propagation of turbulence. Using particle image velocimetry and logarithmic time sampling, we observe the decay and expansion of the blob until it reaches the walls of the tank and eventually transitions to a regime of approximately spatially uniform decay. We confirm sharp, propagating fronts separating turbulent and quiescent regions, together with a dual power-law decay. We interpret these dynamics within the framework of the Kolmogorov–Barenblatt equation while accounting for the growth of the integral length scale over time. Our work provides a detailed picture of how turbulence evolves once forcing is removed and reveals the unexpectedly long time over which a turbulent cascade persists during decay. This talk reports on joint work with Minhui Zhu, Nigel Goldenfeld, and William Irvine.
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Speaker:   Camilla Nobili
Title:   From Boundary Layers to Ultimate Scaling: Covariance Design and Transport Bounds in Thermal Convection
Abstract:  A central question in turbulent thermal convection is how near-wall bound- ary layers and bulk turbulent fluctuations constrain the global heat transport (Nu). In this talk, we explore the scaling bounds of thermal transport using a stochastic framework that bridges kinematic limits with dynamical parameteri- sations.The presentation is divided into two parts. In the first part, we consider a generic, incompressible stochastic velocity field with a given energy budget. In the spirit of wall-to-wall optimal transport (e.g., Hassanzadeh, Chini & Doering, 2014), we show that near-wall geometric decay universally caps transport at an asymptotic limit of Nu ≲ Pe2/3 as Pe → ∞. In the second part, coupling the thermal field to Stokes dynamics, we investigate how specific covariance struc- tures of temperature fluctuations select between classical turbulent regimes. By tuning the spatial covariance matrix, we systematically reconstruct both the Malkus Ra1/3 scaling and the Kraichnan–Spiegel Ra1/2 ”ultimate” regime. This approach provides a direct geometric and statistical mechanism explaining how turbulent correlation lengths in the bulk versus the boundary layer dictate macroscale heat transfer.
This work is joint with Franco Flandoli and Theresa Lange.
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Speaker:   Kyle Liss
Title:   The Batchelor spectrum for a passive scalar driven by deterministic forcing
Abstract:   In 1959, Batchelor predicted that a passive scalar that is forced at large spatial scales and advected by an incompressible flow inducing exponential growth of scalar gradients develops a universal power-law energy spectrum, now known as Batchelor's law. Recently, there have been several works that rigorously establish a form of Batchelor's law when the scalar is driven by a white-in-time stochastic forcing. In this talk, I will discuss joint work with Jonathan Mattingly in which we study the setting of smooth, deterministic forcing. For a specific time-periodic velocity field, we prove that all sufficiently smooth initial scalars are attracted to a limiting solution that satisfies a cumulative form of Batchelor’s law.
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September 15, 2026, 
Coffee/Tea: 3:15 pm
Lecture: 3:45 pm, Della Pietra Family
Auditorium

Title:   My Memories of Frank Yang

Dr. Sheldon Glashow
Nobel Laureate 1979

Speaker:   Gilbert Weinstein

Abstract:  
Harmonic maps with prescribed singularities arise naturally in the study of stationary black holes and geometric inequalities in general relativity. This talk will address two aspects of their analysis. First, we will explain how model maps encode the prescribed singular behavior and serve as the starting point for an existence proof by exhaustion. Second, we will discuss finer asymptotic behavior near the singular set and at infinity, and its role in uniqueness arguments and the mass–angular momentum inequality.

Speaker:   Alexandros Alexakis
Title:   TBA
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Speaker:   Renzo Ricca
Title:   A Topological Approach to Turbulence Cascade
Abstract:   Superfluid turbulence is a phenomenon characterised by the production of large-scale structures made by complex tangles of vortex filaments that gradually decay to form small-scale loops. A topological analysis of superfluid tangles shows [1-3] that very complex knots and links upon continual interaction and reconnection tend to untie to form small, unknotted, unlinked vortex rings. Here we present a topological approach to turbulence cascade based on the idea that the dynamical process can be viewed in terms of geodesic flows in an appropriate knot polynomial space. This is done by using appropriately adapted polynomials of knot theory [4] to quantify the generic topological cascade of vortex knots observed in experiments. Optimal evolutionary unlinking paths can thus be determined and put in relation with observational data and energy cascade [5]. Further work in this direction is in progress [6].

This is joint work with Xin Liu (Beijing U. Technology, China).
[1] Cooper, R.G., Mesgarnezhad, M., Baggaley, A.W. & Barenghi, C.F. 2019 Knot spectrum of turbulence. Sci. Rep. 9, 10545.
[2] Barenghi, C.F. 2024 Tangled vortex lines: dynamics, geometry and topology of quantum turbulence. In Knotted Fields (ed. R.L. Ricca & X. Liu), pp. 243-279. Lecture Notes in Mathematics 2344, Springer-Nature, Switzerland.
[3] Guan, H., Zuccher, S. & Liu, X. 2025 Topological cascade of quantum Borromean rings. Phys. Fluids 37, 024126.
[4] Ricca, R.L. & Liu, X. 2026 A new framework for the Jones polynomial of fluid knots. JKTR 35, 2340024.
[5] Liu, X., Ricca, R.L., Li, X.-F. 2020 Minimal unlinking pathways as geodesics in knot polynomial space. Comm. Physics 3, 136.
[6] Liu, X., Ricca, R.L. & Zhuo, Y. 2026 Quantifying the hierarchical topological decay of fluid knots. Submitted.
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Speaker:   Victor Steinberg
Title:   Elastically driven turbulence in low-inertia polymer solution flow
Abstract:   I present a short review of the experimental results on polymer solution flow at Re<1 and large Weissenberg number Wi characterized elastic stress generated by polymer stretching. First, I shortly described viscoelastic flows with curved streamlines, where super- or subcritical (hysteretic) linear instabilities at Wic are observed and characterized and the instability mechanism has been explained and verified experimentally. Moreover, at Wi>>Wic a transition to chaotic flow, called elastic turbulence (ET), was discovered, and ET has been characterized experimentally and explained theoretically and numerically. Contrary to curvilinear viscoelastic flows, inertia-less viscoelastic straight channel flow, which is proved to be linearly stable similarly to Newtonian parallel shear flows, exhibits an instability directly to transition chaotic flow that at Wi>>Wic further a secondary instability to ET takes place, and then further instability reveals unexpected drag reduction
(DR) chaotic flow. Thus, three chaotic flow regimes are observed, and in all of them elastic waves predicted earlier have been discovered and characterized. Furthermore, in the three flow regimes, coherent structures (CSs) of self–organized streamwise velocity fluctuations are observed, and they display cycling self-sustained process synchronized by elastic waves. This surprising finding together with DR provides evidence that despite some similarity with CSs in Newtonian parallel shear flows the key role of the elastic waves in energy transfer from the main flow to wall-normal vorticity fluctuations that results in their amplification and promotion sustained chaotic flow. Finally, I shortly discuss the recent discovery of stochastic resonance (SR) observed as a peak in streamwise velocity spectrum,
Eu, when elastic waves are very weak in lower sub-region of transition regime at Wi> Wic and spanwise velocity power spectrum Ew remains flat. Thus, SR occurs at three
conditions: chaotic Eu, flat Ew and weak elastic waves.
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Title:   Optimizing ice cubes    

Abstract:   The Mpemba effect—the counterintuitive possibility that a hotter system can cool faster than a colder one—is part of a broader class of anomalous relaxation phenomena in which distance from equilibrium does not determine relaxation time. I will describe how such effects arise from the spectral structure of stochastic dynamics: different initial states excite the slow relaxation modes with different amplitudes, and an initially distant state may relax rapidly if its overlap with the dominant slow mode is small or vanishes. I will then introduce the idea of an “optimal ice cube”: a suitably prepared auxiliary system which, when briefly coupled to the system of interest, suppresses its slowest relaxation mode and thereby accelerates equilibration. This provides a simple framework for designing optimal thermalization protocols and illustrates how nonequilibrium resources can be used to control relaxation.

Speaker:   Anna Frishman
Title:   Closing the loop: bi-directional fluxes and large-scale two-dimensional flows in rotating turbulence
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Speaker:   Timo Schorlepp
Title:   Instanton calculus for turbulence: recent developments
Abstract:   Extreme events, such as intense intermittent bursts, are defining features of turbulent systems, yet they are notoriously difficult to capture theoretically. Pioneered in the 1990s, instanton calculus offers a powerful approach to address this challenge by isolating the most probable trajectories of extreme fluctuations. In this talk, I will review several recent developments and future perspectives regarding instantons in turbulence. On the technical side, I will briefly discuss the numerical computation of instantons using tools from optimization, as well as the calculation of Gaussian fluctuations around the instanton and zero modes for more accurate predictions. In terms of applications, I will first revisit the classical problem of Burgers turbulence and show how its structure function exponents can be obtained from a combination of instanton calculus and fusion rules at the onset of intermittency. Finally, I will consider instantons for vorticity and strain in the 3D incompressible Navier-Stokes equations, presenting numerical results alongside preliminary findings from the theoretical analysis of the vorticity instanton. References:
Schorlepp, Grauer, EPL (2026)
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Speaker:   Eric Jovinelly

Abstract:  
Rational curves are intricately linked to the birational geometry of varieties containing them. Certain curves, called free curves, have the nicest deformation properties. However, it is unknown whether mildly singular Fano varieties contain free rational curves in their smooth locus. In this talk, we discuss free curves of higher genus. Using recent results about tangent bundles, we prove that any klt Fano variety has higher genus free curves. We then use the existence of such free curves to get some applications: we prove the existence of free rational curves in terminal Fano threefolds; study the fundamental group of the smooth locus of a Fano variety; and obtain a new characterization of projective space via lengths of extremal rays. This is joint work with Brian Lehmann, Eric Riedl, and Osamu Fujino.

Speakers: Discussion Session
Title:   Discussion of the Millennium Problem
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Speaker:   Jiasheng Liu
Title:   Breaking and emergence of universality in spin wave turbulence
Abstract:   Turbulence of weakly interacting waves displays a great deal of universality: independence of the pumping and dissipation scales and of the details of the interaction. We discuss how an inverse cascade (from large to small wavenumbers) of spin waves evolves from weak to strong turbulence. The next- to-leading-order correction to the kinetic equation exhibits explicit dependence on the pumping scale, signaling UV nonlocality. As a result, the Kolmogorov-Zakharov description of spin-wave turbulence breaks down at a wavenumber parametrically larger than that at which spectrally local interaction becomes strong. We paraphrase this as: nonlocality enhances nonlinearity. We then discuss strong turbulence in a multi-component extension of the model. We argue that strong turbulence of spin waves realizes the state of critical balance between dispersion and nonlinearity. In such a state, UV nonlocality causes the turbulence level at large scales to decrease as the flux increases, culminating in a state independent of the flux but dependent on the pumping scale.
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Speaker:   Guido Boffetta
Title:   Butterfly effect and predictability in turbulence
Abstract:   I will review the predictability problem in fully developed turbulence. In particular I will discuss how the Lyapunov exponent depends on the Reynolds number and examine the role of intermittency, both in 3D Navier-Stokes turbulence and in Surface Quasi-Geostrophic flow, a model for mesoscale geophysical flows. I will also analyze the evolution of localized perturbations and their effects on flow predictability.
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Speaker:   Nir Navon
Title:   TBA
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Speaker:   Gilbert Weinstein

Abstract:  
The no-hair theorem asserts that stationary, asymptotically flat vacuum black holes are described by the Kerr family, parametrized by mass and angular momentum. Under an analyticity assumption, Hawking’s rigidity theorem yields axial symmetry. For axially symmetric solutions with a single connected event horizon, Robinson (1975) proved that the solution must be Kerr. The possibility of equilibrium configurations with multiple black holes, however, has remained largely open. In this talk, we show that stationary, axially symmetric multiple black holes cannot be in equilibrium. The proof uses harmonic maps with prescribed singularities. We will emphasize the geometric ideas behind this result and explain how a differential inequality and the maximum principle enter the argument. This is joint work with Qing Han, Marcus Khuri, and Jingang Xiong.

Speaker:   Alexei Maliybaev
Title:   Perturbative anomalous exponents from Kolmogorov multipliers
Abstract:   Intermittency, manifested through anomalous scaling, remains one of the central unresolved problems in turbulence theory, with few analytical approaches extending beyond idealized linear transport models. We introduce a perturbative framework for anomalous scaling in turbulent transport based on multiplier statistics, rather than zero-mode calculations. We demonstrate the approach using a shell model combining deterministic and Kraichnan-like stochastic components. We reduce the problem to the analysis of a stationary Fokker-Planck equation for Kolmogorov multipliers, defined as ratios of successive scalar amplitudes. Its solution yields the invariant measure through a perturbative expansion around a Gaussian distribution. Using the resulting multiplier statistics, we compute explicit anomalous scaling exponents for structure functions of arbitrary order, including odd, even, and non-integer moments. Although demonstrated here for a shell model, the framework suggests a systematic perturbative route toward analytical theories of intermittency in turbulence. This is a joint work with Simon Thalabard.
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Speaker:   Nigel Goldenfeld
Title:   Stochastic theory for laminar-turbulence transition, pattern formation and front propagation in pipe flow
Abstract:   How do fluids become turbulent as their flow velocity is increased? In recent years, careful experiments in pipes and Taylor-Couette systems have revealed that the lifetime of transient turbulent regions in a fluid appears to diverge with flow velocity just before the onset of turbulence, faster than any power law or exponential function. I show how this superexponential scaling of the turbulent lifetime in pipe flow is related to extreme value statistics, which I show is a manifestation of a mapping between transitional turbulence and the statistical mechanics model of directed percolation. This mapping itself arises from a further surprising and remarkable connection: turbulence and emergent mean flows in a pipe behave as a predator-prey ecosystem. In new work, I show that the "ecological" model of transitional turbulence, when extended to include streamwise interactions, not only describes the transitional phenomena, but also the series of four distinct states --- decaying puffs, splitting puffs, weak slugs, strong slugs --- that lead to fully-developed turbulence. Work performed in collaboration with: Hong-Yan Shih, Tsung-Lin Hsieh and Xueying Wang
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Speaker:   Luca Biferale
Title:   Data-driven Modeling of Eulerian and Lagrangian Turbulence
Abstract:   We present different stochastic generative frameworks for the generation and augmentation of complex, multiscale signals arising in Eulerian and Lagrangian turbulence. The approach builds on generative Diffusion Models, a class of probabilistic deep learning methods capable of learning high-dimensional data distributions beyond Gaussian approximations.
We demonstrate applications to both three-, two- and one-dimensional stochastic signals, including the temporal evolution of turbulent observables along Lagrangian trajectories. The proposed methodology is benchmarked against Gaussian Process Regression using complementary statistical diagnostics and pointwise error metrics. Particular emphasis is placed on the faithful reproduction of non-Gaussian statistics, intermittency, extreme events, and scale-dependent correlations.
We further present preliminary results on the generalization to memorization transition.
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Speaker:   Guru Jayasingh
Title:   TBA
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Speaker:   Snehanshu Maiti
Title:   Eulerian–Lagrangian Relations in Decaying Two-Dimensional Navier–Stokes Turbulence
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Speaker:   Hrant Hakobyan

Speaker:   Daniel Lecoanet
Title:   Internal Waves in Turbulent Convective / Stably-Stratified Systems
Abstract:   Most studies of turbulent convection consider a fluid confined between two solid horizontal boundaries. However, in nature, convection regions are often adjacent to stably-stratified regions. Here we study how turbulent convection can excite internal waves in an adjacent stably-stratified medium. I will present theoretical and numerical calculations showing that the wave energy spectrum is related to a fourth-order correlation function of the turbulent velocities. I will discuss applications of this theory to both non-rotating and rotating convection.
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Speaker:   Renzo Ricca

Title:   Topological Hydrodynamics