Nikos Alikakos. Multi-phase minimizers for the Allen-Cahn system on the plane.

In this talk we investigate multi-phase minimizers for the Allen-Cahn system on the plane. Our emphasis is on distinct surface tension coefficients. The proofs do not rely on symmetry. Coexistence of an arbitrary number of phases is related to the existence of the relevant minimizing cones for the minimal partition problem. For example, the orthogonal cross with four phases is minimized for a certain class of surface voltage coefficients. We focus on two examples: the entire solution for the triple junction, and a four-phase minimizer with three-phase Dirichlet data (the triangle). The results presented in the talk are based on joint work with Zhiyuan Geng (the triple junction), and with Dimitrios Gazoulis (the triangle).

Ioannis Anapolitanos. How molecules get in shape.

Chemical properties of a molecule are determined by its shape. For example, the energy needed to initiate chemical reactions depends highly on the shape and structure of the involved molecules. It has also been known for a long time that the biological properties of protein molecules are determined by their structure. This naturally leads to several deep questions: How to determine the structure of molecules, is there a way to optimize the energy needed for chemical reactions, and so on. I will explain how to mathematically formulate this type of question for a specific group of chemical reactions, which are called isomerizations. This involves a single molecule, which changes its shape. Often, the initial and final shape of a molecule are given and one is seeking a path in the configuration space of the molecule, which describes how it changes its shape. The calculation of such reaction paths is very standard in quantum chemistry. While in quantum chemistry it is tentatively assumed that an optimal reaction path exists, this is far from obvious from a mathematical point of view. In fact, it is still an open problem! I will discuss recent results which provide a first step in this direction. In particular, I will describe some fundamental properties of molecules, concerning their structure and the forces between them. While our personal experience of the world involves mostly classical terms, the properties of molecules and matter at a microscopic scale necessarily involve quantum mechanics. However, for the purpose of this talk, I will not assume any previous knowledge of quantum mechanics. The talk is based on two recent papers, one with Mathieu Lewin, which is published, the other one being work in arxiv with Marco Olivieri and Sylvain Zalczer.

Dimitra Antonopoulou. Discontinuous Galerkin Methods for the $\varepsilon$-Stochastic Allen-Cahn Equation.

We consider the $\varepsilon$-dependent stochastic Allen-Cahn equation with mild space-time noise, which is posed on a bounded domain in $\mathbb{R}^d$, $d\geq 1$. The small positive parameter $\varepsilon\ll 1$ stands as a measure for the width of the transition layers. The noise depends on the parameter and tends to rough on the sharp interface limit $\varepsilon\rightarrow 0^+$. We introduce a space-time discontinuous in time nonlinear Galerkin scheme for which we prove existence and uniqueness for all dimensions $d\geq 1$. A priori error estimates of optimal order are proven when $d=2$, and for $d\leq 4$ we apply an a posteriori error analysis. The noise, when defined through a $\varepsilon$-convolution of a Gaussian process, is numerically approximated for very small values of $\varepsilon$ by the composite trapezoidal rule on the a.s. continuous paths of the $\varepsilon$-compactly supported integrand.

Anton Arnold. All relative entropies for general nonlinear Fokker-Planck equations.

We shall revisit the entropy method for quasilinear Fokker-Planck equations with confinement to deduce exponential convergence to the equilibrium. Even for prototypical examples like the porous-medium equation, only one relative entropy has been known so far - the Ralston-Newman entropy, which is the analog of the logarithmic entropy in the linear case.
We shall give a complete characterization of all admissible relative entropies for each quasilinear Fokker-Planck equation. In particular we find that fast-diffusion equations with power-law nonlinearities admit only one entropy, while porous medium equations give rise to a whole family of admissible relative entropies (similar to linear Fokker-Planck equations). These additional entropies then imply also new moment-control estimates on the porous-medium solution. (joint work with Jose Carrillo, Daniel Matthes)

Volker Bach. Convergent Renormalization Group Flow of Spectral Problems in Quantum Field Theory.

The renormalization group based on the isospectral Feshbach-Schur map introduced about 25 years ago is an important tool for the spectral analysis of Hamiltonians of quantum field theory, especially, for models of nonrelativistic quantum electrodynamics (Pauli-Fierz Hamiltonians). The renormalization transformation R is defined on a closed subset D of an infinite-dimensional Banach space W of sequences of coupling functions which parametrize the Hamiltonians. One of its key properties is its codimension-one contractivity, i.e., the contractivity of R on D up to a single coupling function. The new result presented in the talk is that, if W is replaced by a Banach (sub-)space W' of coupling functions of higher regularity, then there is a closed subset D' of W' such that R is a (genuine) contraction on D'. Its unique fixed point is the effective Hamiltonian presenting the essential spectral properties of the original model at small momenta, i.e., in the infrared limit. All physically relevant models possess the required higher regularity, and their Hamiltonians belong to D', for small magnitudes of the coupling constant, indeed. This is joint work with Sorour Karimi.

Andreas Buchheit. On the computation of multidimensional lattice sums without translational invariance.

This work introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics and have immediate applications in condensed matter and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools like Ewald summation ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method's runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient and precise simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with $3\times 10^{23}$ particles. Our method's accuracy is demonstrated through extensive numerical experiments.

Adolfo del Campo. Quantum Dynamics in Krylov Space.

The dynamics of quantum systems typically unfolds within a subspace of the state or operator space, known as the Krylov space. Krylov subspace methods provide a compact and computationally efficient description of quantum evolution, which is particularly useful for describing nonequilibrium phenomena of many-body systems with a large Hilbert space. In this talk, I will explore the notion of Krylov complexity as a probe for operator growth, quantum chaos, and scrambling. I will discuss the generalized quantum speed limits in Krylov space and the formulation of shortcuts to adiabaticity in Krylov space for quantum control and quantum optimization.

Scipio Cuccagna. The asymptotic stability on the line of ground states of the pure power NLS with $0<|p-3|\ll 1$.

For exponents $p$ satisfying $0<|p-3|\ll 1$ and only in the context of spatially even solutions we prove that the ground states of the Nonlinear Schrödinger equation with pure power nonlinearity of exponent $p$ in the line are asymptotically stable. The proof is similar to a related recent result of Martel for a cubic quintic NLS. Here we modify the second part of Martel's argument, replacing the second virial inequality for a transformed problem with a smoothing estimate on the initial problem, appropriately tamed by multiplying the initial variables and equations by a cutoff. This is joint work with Masaya Maeda.

Markus Garst. Instability of magnetic skyrmion strings induced by longitudinal spin currents.

It is well established that spin-transfer torques exerted by in-plane spin currents give rise to a motion of magnetic skyrmions resulting in a skyrmion Hall effect. In films of finite thickness or in three-dimensional bulk samples the skyrmions extend in the third direction forming a string. We demonstrate theoretically that a spin current flowing longitudinally along the skyrmion string instead induces a Goldstone spin wave instability. Our analytical results are confirmed by micromagnetic simulations of both a single string as well as string lattices, suggesting that the instability eventually breaks the strings. A longitudinal current is thus able to melt the skyrmion string lattice via a nonequilibrium phase transition. For films of finite thickness or in the presence of disorder a threshold current will be required, and we estimate the latter assuming weak collective pinning. This work is done in collaboration with Shun Okumura, and Volodymyr P. Kravchuk. Reference: Rev. Lett. 131, 066702 (2023).

Christof Melcher. Well-posedness and blow-up of deterministic and stochastic Landau-Lifshitz-Gilbert equations.

We shall discuss solvability of LLG in the framework scaling critical Sobolev spaces that feature a threshold conditions for the occurrence of topological singularities. For the stochastic equation, we focus on the energy-critical 2D case and prove the existence and pathwise uniqueness of strong solutions and a probabilistic version of the bubbling scenario at the first time of blow-up. The method is based on a Doss-Sussmann transformation that yields a covariant form of the LLG with random coefficients. This is joint work with B. Goldys and C. Jiao.

Spyros Kamvissis. Nonlinear Steepest Descent, Integrability & Relativity.

I will briefly present some old results on asymptotics of "completely integrable" equations, and also some speculations on the applicability to completely integrable reductions of the Einstein field equations.

Gian Michele Graf. Adiabatic charge pumps and Galilei covariance.

The Thouless theory of quantum pumps establishes quantized transport per cycle and determines the conditions for that. When the description is shifted to a moving frame, transported and residing charges mix. That transformation is encoded in Galilean space and time, but underlying it is one of vector bundles that may be described in a number of ways, that may or may not rely on Bloch theory. In one of them, the transformation mixes strong and weak indices of a bundle on a 2-torus; in another one, a 3-torus is at center stage; in a last, and somewhat informal way, the transformation is related to a paradox that astonished seafarers of centuries past. (Joint work with Tilman Esslinger and Filippo Santi.)

Garpard Kemlin. Numerical simulation of the Gross–Pitaevskii equation via vortex tracking.

In this talk, I will present a new method for the numerical simulation of the Gross–Pitaevskii (GP) equation, for which a well-known feature is the appearance of quantized vortices with core size of the order of a small parameter ε. Without a magnetic field and with suitable initial conditions, these vortices interact, in the singular limit $\epsilon\to 0$, through an explicit Hamiltonian dynamics. Using this analytical framework, we develop and analyze a numerical strategy based on the reduced-order Hamiltonian system to efficiently simulate the infinite dimensional GP equation for small, but finite, $\epsilon$. This method allows us to avoid numerical stability issues in solving the GP equation, where small values of ε typically require very fine meshes and time steps. We also provide a mathematical justification of our method in terms of rigorous error estimates of the error in the supercurrent, together with numerical illustrations.

Marius Lemm. Enhanced Lieb-Robinson bounds for a class of Bose-Hubbard type Hamiltonians.

Several recent works have considered Lieb-Robinson bounds (LRBs) for Bose-Hubbard-type Hamiltonians. For certain special classes of initial states (e.g., states with particle-free regions or perturbations of stationary states), the velocity of information propagation was bounded by a constant in time, $v\leq C$, similarly to quantum spin systems. However, for the more general class of bounded-density initial states, the first-named author together with Vu and Saito derived the velocity bound $v\leq C t^{D-1}$, where $D$ is the spatial lattice dimension. For $D\geq 2$, this bound allows for accelerated information propagation. It has been known since the work of Eisert and Gross that some systems of lattice bosons are capable of accelerated information propagation. It is therefore a central question to understand under what conditions the bound $v\leq C t^{D-1}$ can be enhanced. Here, we prove that additional physical constraints, translation-invariance and a $p$-body repulsion of the form $n_x^p$ with $p>D+1$, lead to a LRB with $v\leq C t^{\frac{D}{p-D-1}}$ for any initial state of bounded energy density. We also identify examples of quantum states which show that no further enhancement is possible without using additional dynamical constraints. Joint work with Tomotaka Kuwahara.

Sung-Jin Oh. Late time tail of waves on dynamic asymptotically flat spacetimes of odd space dimensions.

I will present recent joint work with Jonathan Luk (Stanford), where we develop a general method for understanding the precise late time asymptotic behavior of solutions to linear and nonlinear wave equations in odd spatial dimensions. In the setting of stationary linear equations, we recover and generalize the Price law decay rates. However, in the presence of a nonlinearity and/or a dynamical background, we prove that the late time tails are in general different(!) from the better-understood case of linear equations on stationary backgrounds. I will explain how this problem is related to the problem of the singularity structure in the interior of generic dynamical vacuum black holes.

Bernd Schroers. Geometry and Dynamics of Chiral Magnetic Skyrmions.

n essential and interesting part of the any model of chiral magnetic skrymions is the Dzyaloshinskii-Moriya interaction (DMI). In this talk I’ll review the gauge-theoretical interpretation of this term, and how this viewpoint allows one to construct and solve integrable models of magnetic skyrmions. The main part of the talk is about first-order dynamics of magnetic skyrmions according to the Landau-Lifshitz-Gilbert equation. I will discuss methods for constructing finite-dimensional models for approximating the dynamics, and illustrate it with the case of the skyrmion and skyrmionium breather.

Arick Shao. Bulk-boundary correspondence for vacuum asymptotically Anti-de Sitter spacetimes.

The AdS/CFT conjecture in physics posits the existence of a correspondence between gravitational theories in asymptotically Anti-de Sitter (aAdS) spacetimes and field theories on their conformal boundary. In this presentation, we prove a rigorous mathematical statement toward this conjecture in the classical relativistic setting. In particular, we show there is a one-to-one correspondence between aAdS solutions of the Einstein-vacuum equations and a suitable space of data on the conformal boundary (consisting of the boundary metric and the boundary stress-energy tensor), provided the boundary satisfies a geometric condition. This is joint work with Gustav Holzegel, and builds upon joint works with Athanasios Chatzikaleas, Simon Guisset, and Alex McGill.

Christos Sourdis. Quantitative linear nondegeneracy of approximate solutions to strongly competitive Gross-Pitaevskii systems in general domains in $N\ge 1$ dimensions.

We consider strongly coupled competitive elliptic systems of Gross-Pitaevskii type that arise in the study of two-component Bose-Einstein condensates, in general smooth bounded domains of $\mathbb{R}^N$, $N\geq 1$. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE (see [dancer]). Under natural non-degeneracy assumptions on a solution of the limit problem, we show that the linearization of the Gross-Pitaevskii system around a 'sufficiently good' approximate solution does not have a kernel and obtain an estimate for its inverse with respect to carefully chosen weighted norms. Our motivation is the study of the persistence of solutions of the limit scalar problem for large values of the coupling parameter which is known only in two dimensions [kowalczyk] or if the domain has radial symmetry [sourdis1].
[sourdis1] J.-B. Casteras and C. Sourdis, Construction of a solution for the two-component radial Gross-Pitaevskii system with a large coupling parameter, J. Funct. Anal. 279, 108674 (2020). [Dancer] E.N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal. 262 (2012), 1087--1131. [kowalczyk] M. Kowalczyk, A. Pistoia and G. Vaira, Phase separating solutions for two component systems in general planar domains, Calc. Var. 62, 142 (2023). [sourdis2] C. Sourdis, Quantitative linear nondegeneracy of approximate solutions to strongly competitive Gross-Pitaevskii systems in general domains in $N\geq 1$ dimensions, Arxiv preprint (2024).

Benjamin Stamm. Grassmann extrapolation and multi-point perturbation theory for Schrödinger eigenvalue problems.

This talk begins with an introduction to an extrapolation technique designed for generating accurate initial guesses of the density matrix in ab-initio Molecular Dynamics simulations. After detailing the methodology, we present numerical results demonstrating the technique's effectiveness in accuracy and maintaining long-term energy conservation. Building on these findings, we introduce a theory named multi-point perturbation theory for linear eigenvalue problems. This theory may provide a foundational explanation for the performance of the Grassmann extrapolation scheme and holds broader applicability. Finally, we showcase numerical results using a one-dimensional Schrödinger equation to illustrate the key features of this new theory.

Yakov Shlapentokh-Rothman. Polynomial Decay for the Klein-Gordon Equation on the Schwarzschild Black Hole.

We will start with a quick review of previous instability results concerning solutions to the Klein-Gordon equation on rotating Kerr black holes and the corresponding conjectural consequences for the dynamics of the Einstein-Klein-Gordon system. Then we will discuss recent work where we show that, despite the presence of stably trapped timelike geodesics on Schwarzschild, solutions to the corresponding Klein-Gordon equation arising from strongly localized initial data nevertheless decay polynomially. Time permitting we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums. The talk is based on joint work(s) with Federico Pasqualotto and Maxime Van de Moortel.

Martin Taylor. Future stability of spatially homogeneous FLRW solutions of the Einstein--massless Vlasov system.

Spatially homogeneous Friedmann–Lemaitre–Robertson–Walker (FLRW) solutions constitute an infinite dimensional family of cosmological solutions of the Einstein--massless Vlasov system. Each member describes a spatially homogenous universe, filled with massless particles, evolving from a big bang singularity and expanding towards the future at a decelerated rate. I will present a theorem on the future stability of this family to spherically symmetric perturbations.