Schrödinger's equation is a beautiful piece of mathematics. It fits on just one line and is supposed to accurately describe the behavior of most atoms and molecules of our world. But it is essentially impossible to simulate accurately, due to its very high dimensionality. In this talk I will explain how physicists and chemists have overcome this problem in an impressive way, within a framework called "Density Functional Theory". I will discuss the role that mathematical results have historically played in this revolution and then present more recent results.
In this presentation I give an insight on the connections between finite and infinite Fast-Slow systems by showing that an infinite dimensional Fast-Slow system behaves like a finite dimensional one. To this extend I analyze a specific parameter dependent PDE, which can be considered as a linear Fast-Reaction system. More precisely, I show with the help of Fourier Transforms the convergence of the system to a limit system and proof an infinite dimensional version of the Tikhonov-Fenichel theorem using the possibility to explicitly construct infinite dimensional slow manifolds in the linear case.
Renewable energies, in particular wind and solar power, have become responsible for a large part of the variation in electricity prices in the past years. Moreover, traders are more and more interested in models which can be used to forecast the day-ahead-/intraday-price-spread. In this talk we shed some light on new models for electricity prices using continuous autoregressive processes. In addition, we discuss intraday price modeling and spread forecasting based on Bayesian statistics and artificial intelligence.
The correspondence principle, as stated by Niels Bohr in 1923, is at the root of the traditional results in semi-classical analysis. It offers a natural insight into the world of semi-classical pseudodifferential operators, Egorov Theorem, coherent states, Wigner measures, etc… The aim of this talk will be to present this general setting and explain how recent results of semi-classical analysis express in that frame.
To join via Zoom please request the login details via email to: clayton 'at' cit.tum.de
The nonlinear bending theory for plates describes the mechanics of a thin inextensible and incompressible film. It features a highly non-convex isometry constraint and a quadratic cost on second-order terms. This talk will be about modern developments in deriving an extension of the nonlinear bending theory for new materials, in a mathematically general and rigorous way by means of Gamma-convergence. I will begin by deriving a nonlinear bending theory for prestrained thin films, using convex-integration solutions to the Monge-Ampere equation. I will also discuss nonlinear bending theories for microheterogeneous thin plates and liquid crystal elastomer bilayers.
We study the use of neural networks as nonparametric estimation tools for the hedging of options. To this end, we design a network, named HedgeNet, that directly outputs a hedging strategy given relevant features as input. This network is trained to minimise the hedging error instead of the pricing error. Applied to end-of-day and tick prices of S&P 500 and Euro Stoxx 50 options, the network is able to reduce the mean squared hedging error of the Black-Scholes benchmark significantly. We illustrate, however, that a similar benefit arises by a simple linear regression model that incorporates the leverage effect. (Joint work with Weiguan Wang)
Liquidity risk is typically added exogenously to a market price process. This is conceptually unsatisfying. We build a model, which integrates liquidity risk into the market price process. In particular, we add a liquidity (jump) component to the standard geometric Brownian motion and show that this approach models market prices better than without the liquidity component. Since long positions have to be liquidated at the bid price, we model bid and ask price individually. We verify our model with 50 million bond price data. We suggest that this model should underlie long positions in risk management approaches such as VaR (Value at Risk), ES (Expected Shortfall) and EVT (Extreme Value Theory). The talk is based on a joint work with Robert Engle and Anna van Elst.
For an inverse temperature β>0, we define the β-circular Riesz gas on Rd as any microscopic thermodynamic limit of Gibbs particle systems on the torus interacting via the Riesz potential g(x)=∥x∥^(−s). We focus on the non integrable case d−1<s<d. Our main result ensures, for any dimension d≥1 and inverse temperature β>0, the existence of a β-circular Riesz gas which is not number-rigid. Recall that a point process is said number rigid if the number of points in a bounded Borel set Δ is a function of the point configuration outside Δ. It is the first time that the non number-rigidity is proved for a Gibbs point process interacting via a non integrable potential. We follow a statistical physics approach based on the canonical DLR equations. It is inspired by Dereudre-Hardy-Leblé and Maïda (2021) where the authors prove the number-rigidity of the Sineβ process.
In this talk, we will discuss the extension of the adaptive voter model to hypergraphs. The adaptive voter model is a widely used model in statistical physics which describes the dynamics of opinion formation in societies. The hypergraph extension of this model allows us to capture more complex social effects like peer pressure that cannot be represented by traditional graphs.
We will introduce four different flavors of the hypergraph model and show how subtle differences between the update rules can lead to very different dynamics. Furthermore, we will derive the mean-field moment expansion equations for all four variants of the model. We will also introduce a moment-closure relation for hypergraph motifs as a generalization of the well-known pair approximation on graphs.
The Double Bubble problem is a generalization of the isoperimetric problem asking the following: given two volumes, what are the two shapes admitting these volumes with the smallest perimeter, where the perimeter of the joint boundary is counted once. We study the DB problem over the l_1 norm and show that one can approximate the solutions very well in the discrete lattice. I will also discuss solutions over other norms and their connection to the Euclidean solutions.
Machine learning excels in learning associations and patterns from data and is increasingly adopted in natural-, life- and social sciences, as well as engineering. However, many relevant research questions about such complex systems are inherently causal and machine learning alone is not designed to answer them. At the same time there often exists ample theoretical and empirical knowledge in the application domains. In this talk, I will briefly outline causal inference as a powerful framework providing the theoretical foundations to combine data and machine learning models with qualitative domain assumptions to quantitatively answer causal questions. I will discuss challenges ahead and selected application scenarios to spark interest for integrating causal thinking into data-driven science.
Short bio: Jakob Runge heads the Causal Inference group at the German Aerospace Center’s Institute of Data Science in Jena since 2017 and is guest professor of computer science at TU Berlin since 2021. His group develops theory, methods, and accessible software for causal inference on time series data inspired by challenges in various application domains. Jakob studied physics at Humboldt University Berlin and finished his PhD project at the Potsdam Institute for Climate Impact Research in 2014. For his studies he was funded by the German National Foundation (Studienstiftung) and his thesis was awarded the Carl-Ramsauer prize by the Berlin Physical Society. In 2014 he won a $200.000 Fellowship Award in Studying Complex Systems by the James S. McDonnell Foundation and joined the Grantham Institute, Imperial College London, from 2016 to 2017. In 2020 he won an ERC Starting Grant with his interdisciplinary project CausalEarth. On https://github.com/jakobrunge/tigramite.git he provides Tigramite, a time series analysis python module for causal inference. For more details, see: www.climateinformaticslab.com.
The Kronecker covariance structure for array data posits that the covariances along comparable modes, such as rows and columns, of an array are similar. For example, when modelling a multivariate time series, it might be assumed that each individual series follows the same AR process, up to changes in scale, while at each particular timepoint the observations across series have the same correlation structure. Over and above being a plausible model for many types of data, the Kronecker covariance assumption is especially useful in high-dimensional settings, where unconstrained covariance matrix estimates are typically unstable. In this talk we explore the information geometric aspects of the estimation of Kronecker covariance matrices. The asymptotic properties of two estimators, the maximum likelihood estimator and an estimator based on partial traces, are contrasted. It is shown that the partial trace estimator is inefficient, where the relative performance of this estimator can be quantified in terms of a principle angle between tangent spaces. This principle angle can be related to the eigenvalues of the underlying Kronecker covariance matrix. By defining a rescaled version of the partial trace operator, an asymptotically efficient correction to the partial trace estimator is proposed. This estimator has a closed-form expression and also has a useful equivariance property. An orthogonal parameterization of the collection of Kronecker covariances is subsequently motivated by the rescaled partial trace estimator. Orthogonal parameterizations imply that the components of the parameterization are asymptotically independent, which in the Kronecker case has implications for tests concerning row and column covariances.
Recently, Panchenko identified the limit of the free energy of a vector spin glass model as a variant of the Parisi formula. However, due to the non-constant spin self-overlap, the proof is more technical than that of the classical Sherrington-Kirkpatrick (SK) model. In this talk, I will present a proof that is simpler on both technical and conceptual levels. This is achieved by introducing a self-overlap correction term to the Hamiltonian, which cancels the self-overlap terms in Guerra's interpolation. As a result, we can identify the limit with the classical form of the Parisi formula, similar to the proof for the SK model. Furthermore, the correction term can be removed by solving a simple Hamilton-Jacobi equation, yielding a formula resembling Panchenko’s.
The main concern of this talk is the presentation of a multi-scale numerical scheme for an efficient resolution of the electron-ion dynamics in thermonuclear fusion plasmas. The starting mathematical model is based on a multi-species Fokker-Planck kinetic model, conserving mass, total momentum and energy, as well as satisfying Boltzmann’s H-theorem. Firstly, an asymptotic limit is performed, letting the electron/ion mass ratio going towards zero, ε->0, in order to obtain a reduced model, consisting of a thermodynamic equilibrium state for the rapid electrons (adiabatic regime), whereas the slow ions remain kinetic. Secondly, we develop a first numerical scheme, based on a Hermite spectral method, and perform numerical simulations to investigate in more details this (kinetic towards macroscopic) asymptotic electron/ion limit. The specificity of this method is the fact that it permits considerable improvements in simulation time for small ε-values, as in such regimes very few Hermite modes have to be taken into account.
We will begin with a quick reminder of algebraic K-theory, and a few classical, vanishing results for negative K-theory. The talk will then focus on a striking 2019 article by Antieau, Gepner and Heller - it turns out that there are K-theoretic obstructions to the existence of bounded t-structures.
The result suggests many questions. A few have already been answered, but many remain open. We will concentrate on the many possible directions for future research.
I will present a model for vapor-liquid-solid growth of nanowires where liquid drops are described as local or global volume-constrained minimizers of the capillarity energy outside a semi-infinite convex obstacle modeling the nanowire. I will first discuss global existence of minimizers and then, in the case of rotationally symmetric nanowires, I will explain how the presence of a sharp edge affects the shape of local minimizers and the validity of Young's law. Finally, I will present some recent regularity results for local minimizers and the connections of this problem with an isoperimetric inequality outside convex sets.
A spanning tree of a finite connected graph G is a connected subgraph of G that touches every vertex and contains no cycles. In this talk we will consider uniformly drawn spanning trees of ``high-dimensional’‘ graphs, and show that, under appropriate rescaling, they converge in distribution as metric-measure spaces to Aldous’ Brownian CRT. This extends an earlier result of Peres and Revelle (2004) who previously showed a form of finite-dimensional convergence. Based on joint works with Asaf Nachmias and Matan Shalev.
The talk reviews some growth processes describing the evolution of clusters consisting of atomic parts called monomers. The growth and shrinkage can only occur by adding and removing single monomers, specified through a rate kernel depending solely on the involved clusters' size.
First, I discuss the exchange-driven growth model, obtained as the mean-field limit of stochastic particle systems (zero-range process). Under a detailed balance condition on the kernel, the model's longtime behavior can be described entirely. Here, the total mass density, determined by the initial data, acts as an order parameter in which the system shows a phase separation.
Next, we consider the model for a family of product kernels, which do not satisfy a detailed balance condition. After a suitable rescaling to self-similar variables, the equation becomes a discrete Laplace with a power-law as diffusion coefficient, which in particular degenerates at the origin and grows at infinity. We will see that the solution converges to a stretched exponential self-similar profile.
Lastly, we consider the now-classic Becker-Döring system to which an injection of monomers and a depletion of large clusters is added. These equations have been extensively used to model chemical-physical systems, especially bubbleator dynamics. The model approximates a transport equation with a conservation law entering the boundary condition by formal asymptotics. For the limit model, a Hopf bifurcation is shown, indicating temporal oscillations in the model.
joint works with Constantin Eichenberg, Barbara Niethammer, Robert Pego, and Juan Velazquez.