We explore a connection between a weak topology on spaces of probability measures, a classical combinatorial problem in matching, and numerical schemes for the solution of PDEs by neural networks.

We prove a fluid limit for the coarsening phase of the condensing zero-range process on a finite number of sites. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is described by a piecewise linear trajectory in the simplex indexed by the sites. The […]

Evolutionary deep neural networks have emerged as a rapidly growing field of research. This talk discusses numerical integrators for such and other classes of nonlinear parametrizations u(t) = Φ(θ(t)) where the evolving parameters θ(t) are to be computed. The primary focus is on tackling the challenges posed by the combination of stiff evolution problems and […]

On Thursday, May 08, 2025, at 12:15 pm, Prof. Dr. Shi Jin (Shanghai Jiao Tong University) will give a talk on “Quantum Computation of partial differential equations and related problems” at TUM, Boltzmannstr. 3, 85748 Garching Forschungszentrum, room 03.06.011. Abstract: Quantum computers have the potential to gain algebraic and even up to exponential speed up compared with its classical counterparts, and can lead to […]

Quantum computing with discrete variable (DV, qubit) hardware is approaching the large scales necessary for computations beyond the reach of classical computers. Separately, hardware containing native continuous-variable (CV, oscillator) systems has received attention as an alternative approach, yet the universal control of such systems is non-trivial. In this talk, I will highlight novel quantum algorithms, […]

In this talk we consider discrete Gaussian free fields with ergodic random conductances on ℤd, d ≥ 2, where the conductances are possibly unbounded but satisfy a moment condition. As our main result, we show that, for almost every realisation of the environment, the rescaled field converges in law towards a continuous Gaussian field. We […]

Let X be a simple random walk in Znd with d≥3 and let tcov be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set Lα of points that have not been visited by time αtcov and prove that it exhibits a phase transition: there exists α∗ so that for all α>α∗ and all ϵ>0 there […]

In this talk, we address the full discretization of Friedrichs’ systems with a two-field structure, such as Maxwell’s equations or the acoustic wave equation in div-grad form. We follow a method of lines approach, where we first discretize in space via the discontinuous Galerkin method. Subsequently, we consider different second-order schemes for time integration, namely […]

In this talk, we address the full discretization of Friedrichs’ systems with a two-field structure, such as Maxwell’s equations or the acoustic wave equation in div-grad form. We follow a method of lines approach, where we first discretize in space via the discontinuous Galerkin method. Subsequently, we consider different second-order schemes for time integration, namely […]

State-sum constructions have numerous applications in both mathematics and physics. In mathematics, they yield invariants for knots and manifolds and serve as a powerful organizing principle in representation theory. To illustrate this principle, we discuss equivariant Frobenius-Schur indicators. In the context of physics, we explain how state-sum models offer a conceptual framework for tensor network […]

In this talk, we consider irreversible translation-invariant interacting particle systems on the d-dimensional hypercubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that […]

We consider a system of closed random walk trajectories interacting by mutual repulsion. This probabilistic model is motivated by its connections to statistical mechanics models, such as the Bose gas, the double dimer model, or the spin O(N) model. We prove the occurrence of macroscopic loops in dimension d>2 (joint with A. Quitmann, 2022) and […]

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter. These estimates are called local laws and they are the cornerstone in most of the recent results […]

Fix some graph or uniform hypergraph F and then consider the following simple process: start with a large complete (hyper)graph Kn on n vertices and iteratively remove (the edge set of) copies of F as long as possible; each selection is made uniformly among all present copies of F at that time. This process is […]

Phase transitions are natural phenomena in which a small change in an external parameter, like temperature or pressure, causes a dramatic change in the qualitative structure of the object. To study this, many scientists (such as Nobel laureates Pauling and Flory) proposed the abstract framework of lattice models. The focus of this talk is on […]

Of great interest in plasma physics is to determine whether excited charged particles in a non-equilibrium state will relax to neutrality or transition to a nontrivial coherent state. Due to the long range interaction between particles, the self-consistent generating electric field oscillates in time and disperses in space like a Klein-Gordon wave, known in the […]

Quantum systems typically reach thermal equilibrium when in weak contact with a large external bath. Understanding the speed of this thermalisation is a challenging problem, especially in the context of quantum many-body systems where direct calculations are intractable. The usual way of bounding the speed of this process is by estimating the spectral gap of […]

Quantum walks (QWs) can be viewed as quantum analogs of classical random walks. Mathematically, a QW is described as a unitary, local operator acting on a grid and can be written as a product of shift and coin operators. We highlight differences to classical random walks and stress their connection to quantum algorithms (see Grover’s […]

We study the generalised Lloyd model, that is, a random Schrödinger operator on the lattice Zd of the form H = −∆ + λV, λ > 0, with Vi = ∑j Tij Wj where Wj ∼ Cauchy(0, 1) are iid random variables. If all coeffi- cients Tij are non-negative one can find an exact formula […]

This talk will provide a partial overview of how to model molecular observables and spectral responses. We shall start with the essential quantum mechanical concepts and formalisms that are used in theoretical chemistry and molecular physics. This will be followed by an illustration of the tools that are involved in numerical simulations in this context.

The dilute Curie-Weiss model is the Ising model on a (dense) Erdös-Rényi graph G(N,p). It was introduced by Bovier and Gayrard in 1990s. There the authors showed that on the level of laws of large numbers the magnetization as well as the free energy behave as they do in the usual Curie-Weiss model (i.e. mean-field […]

Branching Brownian motion, branching random walks, and the F-KPP equation have been the subject of intensive research during the last couple of decades. By means of Feynman-Kac and McKean formulas, the understanding of the maximal particles of the former two Markov processes is related to insights into the position of the front of the solution […]

(Pseudo)spectral methods are popular for solving a wide variety of differential equations and generic optimization problems. Due to favourable approximation properties, such as rapid convergence for smooth functions, they are particularly popular and effective for solving time-independent Schrödinger equations. For example, in the domain of molecular quantum physics, spectral and pseudospectral methods are the building […]

We study the contact process on scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where the neighbourhood of each vertex is updated with a rate depending on its strength. We identify the full phase diagram of metastability exponents in dependence on the tail exponent of the degree distribution and the rate of updating. […]

Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges that will facilitate a transition from NISQ to the fault tolerance era. In this talk, I […]

In Bernoulli percolation, the incipient infinite cluster (IIC) is a version of the “open cluster of the origin at criticality conditioned to be infinite”. Since this event should have probability 0 on Zd, the IIC is constructed via a limiting procedure. For d > 6, several constructions have been given and shown to produce the […]

Regularisation by noise in the context of stochastic differential equations (SDEs) with coefficients of low regularity, known as singular SDEs, refers to the beneficial effect produced by noise so that the singularity from the coefficients is smoothed out yielding well-behaved equations. Kinetic SDEs, also sometimes called second order SDEs, as one typical type of stochastic […]

In this seminar, we will discuss the estimation of group action using the non-commutative Fourier transform. We’ll explore an interesting relationship between the non-commutative Fourier transform and this estimation problem. Next, we’ll optimize our estimation method by leveraging this connection under various conditions, including energy constraints. Finally, we’ll apply the obtained results to uncertainty relations […]

Although many-body quantum simulations have greatly benefited from high-perfor- mance computing facilities, large molecular systems continue to pose formidable challenges. Mixed quantum-classical models, such as Born-Oppenheimer molecular dynamics or Ehren- fest dynamics, have been proposed to overcome the computational costs of fully quantum ap- proaches. However, current mixed quantum-classical models typically suffer from long- standing […]

The Fröhlich polaron models a charged quantum particle interactiong with a polar cystal. Since the moving particle has to drag along a ‘cloud’ of polarization, it appears heavier than it would be without the interaction. An old conjecture of Landau and Pekar states that this so-called effective mass scales as the fourth power of the […]

It is known through classical works of Kac, Salem, Zygmund, Erdös and Gal that lacunary sums behave in several ways like sums of independent random variables, satisfying, for instance, a central limit theorem or a law of the iterated logarithm. We present some recent results on their large deviation behavior, which show that on this […]

We consider a random walk in a truncated cone K_N , which is obtained by slicing cone K by a hyperplane at a growing level of order N. We study the behaviour of the Green function in this truncated cone as N increases. Using these results we also obtain the asymptotic behaviour of the harmonic […]

In this talk, we investigate random convex interfaces which are generated as convex hulls of random point sets. We are interested in their asymptotic behavior when the size of the input goes to infinity. In a first part, we mainly identify average and maximal fluctuations in the radial and longitudinal directions through precise convergence results. […]

Interacting Bose gas at zero temperature is often described by the Bogoliubov approximation. It involves quasiparticles, called phonons, with a rather curious dispersion relation responsible for superfluidity. The Fermi Golden Rule predicts that the lifetime of phonons is proportional to the 5th inverse power of momentum. This was first computed by Beliaev and goes under […]

In 2001 Bolthausen, den Hollander and van den Berg obtained the asymptotics of the probability that the volume of a Wiener sausage at time t is smaller than expected by a fixed muliplicative constant. This asymptotics was given by a variational formula and they conjectured that the best strategy to achieve such a large deviation […]

The Abelian sandpile model on a graph G is a Markov chain whose state space is a subset of the set of functions with integer values defined on the vertices of G. The set of recurrent states of this Markov chain is called the sandpile group and the Abelian sandpile model can be then viewed […]

The KPZ fixed point is conjectured to be the universal space-time scaling limit of the models belonging to the KPZ universality class and it was rigorously constructed by Matetski, Quastel and Remenik (Acta Math., 2021) as a scaling limit of TASEP (Totally Asymmetric Simple Exclusion Process) with arbitrary initial configuration. We set up a new, […]

We introduce a currently hot topic in probability theory, the theory of scaling limits for random fields of gradients in all dimensions. The random fields are a class of model systems arising in the studies of random interfaces, random geometry, Euclidean field theory, the theory of regularity structures, and elasticity theory. After explaining how non-convex […]

We explore a connection between a weak topology on spaces of probability measures, a classical combinatorial problem in matching, and numerical schemes for the solution of PDEs by neural networks.

We prove a fluid limit for the coarsening phase of the condensing zero-range process on a finite number of sites. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is described by a piecewise linear trajectory in the simplex indexed by the sites. The […]

Evolutionary deep neural networks have emerged as a rapidly growing field of research. This talk discusses numerical integrators for such and other classes of nonlinear parametrizations u(t) = Φ(θ(t)) where the evolving parameters θ(t) are to be computed. The primary focus is on tackling the challenges posed by the combination of stiff evolution problems and […]

On Thursday, May 08, 2025, at 12:15 pm, Prof. Dr. Shi Jin (Shanghai Jiao Tong University) will give a talk on “Quantum Computation of partial differential equations and related problems” at TUM, Boltzmannstr. 3, 85748 Garching Forschungszentrum, room 03.06.011. Abstract: Quantum computers have the potential to gain algebraic and even up to exponential speed up compared with its classical counterparts, and can lead to […]

Quantum computing with discrete variable (DV, qubit) hardware is approaching the large scales necessary for computations beyond the reach of classical computers. Separately, hardware containing native continuous-variable (CV, oscillator) systems has received attention as an alternative approach, yet the universal control of such systems is non-trivial. In this talk, I will highlight novel quantum algorithms, […]

In this talk we consider discrete Gaussian free fields with ergodic random conductances on ℤd, d ≥ 2, where the conductances are possibly unbounded but satisfy a moment condition. As our main result, we show that, for almost every realisation of the environment, the rescaled field converges in law towards a continuous Gaussian field. We […]

Let X be a simple random walk in Znd with d≥3 and let tcov be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set Lα of points that have not been visited by time αtcov and prove that it exhibits a phase transition: there exists α∗ so that for all α>α∗ and all ϵ>0 there […]

In this talk, we address the full discretization of Friedrichs’ systems with a two-field structure, such as Maxwell’s equations or the acoustic wave equation in div-grad form. We follow a method of lines approach, where we first discretize in space via the discontinuous Galerkin method. Subsequently, we consider different second-order schemes for time integration, namely […]

In this talk, we address the full discretization of Friedrichs’ systems with a two-field structure, such as Maxwell’s equations or the acoustic wave equation in div-grad form. We follow a method of lines approach, where we first discretize in space via the discontinuous Galerkin method. Subsequently, we consider different second-order schemes for time integration, namely […]

State-sum constructions have numerous applications in both mathematics and physics. In mathematics, they yield invariants for knots and manifolds and serve as a powerful organizing principle in representation theory. To illustrate this principle, we discuss equivariant Frobenius-Schur indicators. In the context of physics, we explain how state-sum models offer a conceptual framework for tensor network […]

In this talk, we consider irreversible translation-invariant interacting particle systems on the d-dimensional hypercubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that […]

We consider a system of closed random walk trajectories interacting by mutual repulsion. This probabilistic model is motivated by its connections to statistical mechanics models, such as the Bose gas, the double dimer model, or the spin O(N) model. We prove the occurrence of macroscopic loops in dimension d>2 (joint with A. Quitmann, 2022) and […]

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter. These estimates are called local laws and they are the cornerstone in most of the recent results […]

Fix some graph or uniform hypergraph F and then consider the following simple process: start with a large complete (hyper)graph Kn on n vertices and iteratively remove (the edge set of) copies of F as long as possible; each selection is made uniformly among all present copies of F at that time. This process is […]

Phase transitions are natural phenomena in which a small change in an external parameter, like temperature or pressure, causes a dramatic change in the qualitative structure of the object. To study this, many scientists (such as Nobel laureates Pauling and Flory) proposed the abstract framework of lattice models. The focus of this talk is on […]

Of great interest in plasma physics is to determine whether excited charged particles in a non-equilibrium state will relax to neutrality or transition to a nontrivial coherent state. Due to the long range interaction between particles, the self-consistent generating electric field oscillates in time and disperses in space like a Klein-Gordon wave, known in the […]

Quantum systems typically reach thermal equilibrium when in weak contact with a large external bath. Understanding the speed of this thermalisation is a challenging problem, especially in the context of quantum many-body systems where direct calculations are intractable. The usual way of bounding the speed of this process is by estimating the spectral gap of […]

Quantum walks (QWs) can be viewed as quantum analogs of classical random walks. Mathematically, a QW is described as a unitary, local operator acting on a grid and can be written as a product of shift and coin operators. We highlight differences to classical random walks and stress their connection to quantum algorithms (see Grover’s […]

We study the generalised Lloyd model, that is, a random Schrödinger operator on the lattice Zd of the form H = −∆ + λV, λ > 0, with Vi = ∑j Tij Wj where Wj ∼ Cauchy(0, 1) are iid random variables. If all coeffi- cients Tij are non-negative one can find an exact formula […]

This talk will provide a partial overview of how to model molecular observables and spectral responses. We shall start with the essential quantum mechanical concepts and formalisms that are used in theoretical chemistry and molecular physics. This will be followed by an illustration of the tools that are involved in numerical simulations in this context.

The dilute Curie-Weiss model is the Ising model on a (dense) Erdös-Rényi graph G(N,p). It was introduced by Bovier and Gayrard in 1990s. There the authors showed that on the level of laws of large numbers the magnetization as well as the free energy behave as they do in the usual Curie-Weiss model (i.e. mean-field […]

Branching Brownian motion, branching random walks, and the F-KPP equation have been the subject of intensive research during the last couple of decades. By means of Feynman-Kac and McKean formulas, the understanding of the maximal particles of the former two Markov processes is related to insights into the position of the front of the solution […]

(Pseudo)spectral methods are popular for solving a wide variety of differential equations and generic optimization problems. Due to favourable approximation properties, such as rapid convergence for smooth functions, they are particularly popular and effective for solving time-independent Schrödinger equations. For example, in the domain of molecular quantum physics, spectral and pseudospectral methods are the building […]

We study the contact process on scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where the neighbourhood of each vertex is updated with a rate depending on its strength. We identify the full phase diagram of metastability exponents in dependence on the tail exponent of the degree distribution and the rate of updating. […]

Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges that will facilitate a transition from NISQ to the fault tolerance era. In this talk, I […]

In Bernoulli percolation, the incipient infinite cluster (IIC) is a version of the “open cluster of the origin at criticality conditioned to be infinite”. Since this event should have probability 0 on Zd, the IIC is constructed via a limiting procedure. For d > 6, several constructions have been given and shown to produce the […]

Regularisation by noise in the context of stochastic differential equations (SDEs) with coefficients of low regularity, known as singular SDEs, refers to the beneficial effect produced by noise so that the singularity from the coefficients is smoothed out yielding well-behaved equations. Kinetic SDEs, also sometimes called second order SDEs, as one typical type of stochastic […]

In this seminar, we will discuss the estimation of group action using the non-commutative Fourier transform. We’ll explore an interesting relationship between the non-commutative Fourier transform and this estimation problem. Next, we’ll optimize our estimation method by leveraging this connection under various conditions, including energy constraints. Finally, we’ll apply the obtained results to uncertainty relations […]

Although many-body quantum simulations have greatly benefited from high-perfor- mance computing facilities, large molecular systems continue to pose formidable challenges. Mixed quantum-classical models, such as Born-Oppenheimer molecular dynamics or Ehren- fest dynamics, have been proposed to overcome the computational costs of fully quantum ap- proaches. However, current mixed quantum-classical models typically suffer from long- standing […]

The Fröhlich polaron models a charged quantum particle interactiong with a polar cystal. Since the moving particle has to drag along a ‘cloud’ of polarization, it appears heavier than it would be without the interaction. An old conjecture of Landau and Pekar states that this so-called effective mass scales as the fourth power of the […]

It is known through classical works of Kac, Salem, Zygmund, Erdös and Gal that lacunary sums behave in several ways like sums of independent random variables, satisfying, for instance, a central limit theorem or a law of the iterated logarithm. We present some recent results on their large deviation behavior, which show that on this […]

We consider a random walk in a truncated cone K_N , which is obtained by slicing cone K by a hyperplane at a growing level of order N. We study the behaviour of the Green function in this truncated cone as N increases. Using these results we also obtain the asymptotic behaviour of the harmonic […]

In this talk, we investigate random convex interfaces which are generated as convex hulls of random point sets. We are interested in their asymptotic behavior when the size of the input goes to infinity. In a first part, we mainly identify average and maximal fluctuations in the radial and longitudinal directions through precise convergence results. […]

Interacting Bose gas at zero temperature is often described by the Bogoliubov approximation. It involves quasiparticles, called phonons, with a rather curious dispersion relation responsible for superfluidity. The Fermi Golden Rule predicts that the lifetime of phonons is proportional to the 5th inverse power of momentum. This was first computed by Beliaev and goes under […]

In 2001 Bolthausen, den Hollander and van den Berg obtained the asymptotics of the probability that the volume of a Wiener sausage at time t is smaller than expected by a fixed muliplicative constant. This asymptotics was given by a variational formula and they conjectured that the best strategy to achieve such a large deviation […]

The Abelian sandpile model on a graph G is a Markov chain whose state space is a subset of the set of functions with integer values defined on the vertices of G. The set of recurrent states of this Markov chain is called the sandpile group and the Abelian sandpile model can be then viewed […]

The KPZ fixed point is conjectured to be the universal space-time scaling limit of the models belonging to the KPZ universality class and it was rigorously constructed by Matetski, Quastel and Remenik (Acta Math., 2021) as a scaling limit of TASEP (Totally Asymmetric Simple Exclusion Process) with arbitrary initial configuration. We set up a new, […]

We introduce a currently hot topic in probability theory, the theory of scaling limits for random fields of gradients in all dimensions. The random fields are a class of model systems arising in the studies of random interfaces, random geometry, Euclidean field theory, the theory of regularity structures, and elasticity theory. After explaining how non-convex […]