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MI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

Although many-body quantum simulations have greatly benefited from high-performance computing facilities, large molecular systems continue to pose formidable challenges. Mixed quantum-classical models, such as Born-Oppenheimer molecular dynamics or Ehrenfest dynamics, have been proposed to overcome the computational costs of fully quantum approaches. However, current mixed quantum-classical models typically suffer from long-standing consistency issues. In this talk, we present a fully Hamiltonian theory of quantum-classical dynamics based on a geometric approach and Koopman wave functions. The resulting model appears to be the first to ensure a series of consistency properties, beyond the positivity of quantum and classical densities. We also exploit Lagrangian trajectories to formulate a finite-dimensional closure scheme for numerical implementations, the "Koopmon method". Numerical experiments demonstrate that the Koopmon method is able to capture effects beyond Ehrenfest dynamics in both the classical and the quantum sectors.

8101.02.201 (BC1 2.02.01) (Parkring 11, 85748 Garching)

The OSCAR project is a collaborative effort to shape a new computer algebra system, written in Julia. OSCAR is built on top of the four "cornerstone systems" ANTIC (for number theory), GAP (for group and representation theory), polymake (for polyhedral and tropical geometry) and Singular (for commutative algebra and algebraic geometry). We present examples to showcase the current version 0.14.0. This is joint work with The OSCAR Development Team, currently lead by Wolfram Decker, Claus Fieker, Max Horn and Michael Joswig.

\[ \] Interested participants can also install OCSCAR before the workshop. More information about the installation can be found here: https://www.oscar-system.org/install/

Online: attend (945856)

Networks of coupled dynamical systems are successful models in diverse fields of science ranging from physics to neuroscience. The network interaction structure impacts the dynamics, in fact, many malfunctions are associated with disorders in the network structure. Yet, typically we cannot measure the interaction structure, we only have access to multivariate time series of nodes’ states. This led to much effort to reconstruct the network from multivariate data. This reconstruction problem is ill-posed for large networks leading to the reconstruction of false network structures. In this talk, I put forward an approach that uses the network dynamics’ statistical properties to ensure the exact reconstruction of weakly coupled sparse networks. Moreover, this approach exhibits robustness against noise. To demonstrate its efficacy, I illustrate its reconstruction power using experimental multivariate time series data obtained from optoelectronic networks.

8101.02.201 (BC1 2.02.01) (Parkring 11, 85748 Garching)

After a demo of the OSCAR system, we introduce the mrdi file format and discuss the advantages of using serialization for collaborative work and scientific research. We demonstrate how users can benefit from OSCAR's built-in serialization mechanism, which employs that file format. Key applications include the reproduction of mathematical results computed with OSCAR and the interoperability between OSCAR and other software applications.

2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

For a time dependent family of probability measure $(\mu_t)_{t\he 0}$ we consider a kinetic-type evolution equation $\partial \mu_t/\partial t + \mu_t = Q \mu_t$, where $Q$ is the smoothing transformation. During the talk we will present probabilistic representation of a solution of this equation in terms of continuous time branching random walks. Moreover, assuming that $\mu_0$ belongs to the domain of attraction of a stable law, we describe asymptotic behaviour of $\mu_t$. Literature: [1] Bogus, B., Marynych, SPA 2020 [2] B., Kolesko, Meiners, EJP 2021 [3] B., Dyszewski, Marynych, SPA 2023

MI 00.06.11 HS3 (Boltzmannstr. 3, 85748 Garching)

Partial differential equations are the backbone of many results in applied Mathematical Finance. For example, prices of derivative contracts can be characterized in terms of linear, parabolic, second-order PDEs, and solutions of optimal investment problems are intimately linked to fully nonlinear, parabolic or elliptic second-order equations of Hamilton-Jacobi-Bellman type. In this talk, we discuss to which extend this relationship can also be reversed in that existence, uniqueness, and regulartiy results on certain PDEs can be given a financial interpretation. In particular, we consider several examples including optimal investment problems for institutional or private investors and optimal execution problems for large agents such as hedge funds. We conclude with an outlook on work in progress on a new notion of differentiability designed to construct regular solutions of degenerate second-order PDEs.

MI 00.06.11 HS3 (Boltzmannstr. 3, 85748 Garching)

The rise of quantum computing threatens the security of currently deployed cryptographic systems that are used, for instance, to secure the internet, as well as pretty much anything else that has a computer chip in it. In this presentation I intend to give a broad overview over the impact of these developments on our cryptographic world and discuss possible solutions to the impending catastrophe, with a particular focus on the mathematical aspects of post-quantum cryptography.

MI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

There currently is a significant interest in understanding the Edge of Stability (EoS) phenomenon, which has been observed in neural networks training, characterized by a non-monotonic decrease of the loss function over epochs, while the sharpness of the loss (spectral norm of the Hessian) progressively approaches and stabilizes around 2/(learning rate). Reasons for the existence of EoS when training using gradient descent have recently been proposed—a lack of flat minima near the gradient descent trajectory together with the presence of compact forward-invariant sets. In this paper, we show that linear neural networks with a quadratic loss function satisfy the first assumption and also a necessary condition for the second assumption. More precisely, we prove that the gradient descent map is non-singular, the set of global minimizers of the loss function forms a smooth manifold, and the stable minima form a bounded subset in parameter space. Additionally, we prove that if the step-size is too big, then the set of initializations from which gradient descent converges to a critical point has measure zero.

MI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We analyze the asymptotic behavior of a multiscale cholera model where the population is structured by the pathogen load inside the individuals. A threshold for the pathogen load divides the population in susceptible and infected people. The pathogen load is increased by booster events (representing the uptake of pathogens from the environment) which describe a stochastic process. The immune response of the body degrades the pathogens time-continuously. The booster events depend on the pathogen in the environment which we assume to be constant due to time-scale arguments. On the population level, this results in a transport equation with non-local terms. We derive a non-negative stationary solution for the pathogen load of infected people. Furthermore, we prove the existence of a stable pathogen distribution for the susceptible people following the standard approach for fragmentation-aggregation equations from Gyllenberg and Heijmans. Combining the stationary solution for infected people and the pseudo-stationary solution for susceptible people weighted by the amount of infected respectively susceptible people depending on time, we obtain an ODE system (SI-model), which we hope will approximate the original dynamics well.

MI 03.04.011 (Boltzmannstr. 3, 85748 Garching)

Motivated from statistical physics, the Dean-Kawasaki Equation aims to describe the density of a system of fluctuating particles. However, it was shown by Konarovskyi, Lehmann and von Renesse that in the space of probability measures the equation only admits solutions given by empirical measures. But what happens if we allow solutions and initial conditions with infinite mass, e.g. starting from the Lebesgue measure? In this talk we will show that even allowing infinite mass, the Dean-Kawasaki equation only admits solutions if its initial condition is a suitable multiple of an empirical measure. Joint work with Vitalii Konarovskyi, arXiv 2311.10006.

MI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

The Modified Massive Arratia Flow is a model of infinitely many sticky Brownian particles where the diffusion scaled proportionally to the aggregate mass of the particles. The model was introduced by Konarvovskyi and later studied by Konarovskyi and Renesse who showed that the diffusive behaviour of the model is governed locally by the quadratic Wasserstein distance. In this talk we present a central limit theorem for the occupation measure of the process in the case of countably many starting points. A central ingredient of the proof is quantitative decorrelation estimates in terms of the alpha-mixing coefficient for which we present explicit non-standard coupling constructions.

Joint work with Vitalii Konaroskyi (Hamburg) and Andrey Dorogovtsev (Kyiv).

MI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

The Banach-Mazur distance is a well-established notion of convex geometry with numerous important applications in the fields like discrete geometry or local theory of Banach spaces. This notion has already been extensively studied by many different authors, but the vast majority of established results are of the asymptotic nature. The non-asymptotic properties of Banach-Mazur distance seem to be quite elusive and even in very small dimensions they are surprisingly difficult to establish. Actually there are rather few situations in which the Banach-Mazur distance between a pair of convex bodies was determined precisely. One example illustrating this difficulty is the case of the cube and the cross-polytope, as their Banach-Mazur distance was known only in the planar case. In this talk we prove that the distance between the cube and the cross-polytope is equal to 9/5 in the dimension three and it is equal to 2 in dimension four.

MI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

The n-dimensional simplex is a convex body well-known for its numerous remarkable features and it was studied extensively also from the point of view of the Banach-Mazur distance. Our starting point is the following well-known and interesting property: the n-dimensional simplex is equidistant (in the Banach-Mazur distance) to all symmetric convex bodies with the distance equal to n. Moreover, it is known the simplex is the unique convex body with this property. It is therefore natural to ask if the simplex is the unique convex body that is equidistant to all symmetric convex bodies, but not necessarily with the distance equal to n. We answer this question negatively in the planar case. For all 7/4 < r < 2 we provide a general construction of a family of convex bodies K, which are with the distance r to every symmetric convex body. It should be noted that this distance r coincides with the asymmetry constant of K and the construction is based on some basic properties of the asymmetry constant.

8101.02.110 (Parkring 13, 85748 Garching-Hochbrück)

Graphs are often used as representations of conditional independence structures of random vectors. In stochastic processes, one may use graphs to represent so-called local independence. Local independence is an asymmetric notion of independence which describes how a system of stochastic processes (e.g., point processes or diffusions) evolves over time. Let A, B, and C be three subsets of the coordinate processes of the stochastic system. Intuitively speaking, B is locally independent of A given C if at every point in time knowing the past of both A and C is not more informative about the present of B than knowing the past of C only. Directed graphs can be used to describe the local independence structure of the stochastic processes using a separation criterion which is analogous to d-separation. In such a local independence graph, each node represents an entire coordinate process rather than a single random variable.

\[ \] In this talk, we will describe various properties of graphical models of local independence and then turn our attention to the case where the system is only partially observed, i.e., some coordinate processes are unobserved. In this case, one can use so-called directed mixed graphs to describe the local independence structure of the observed coordinate processes. Several directed mixed graphs may describe the same local independence model, and therefore it is of interest to characterize such equivalence classes of directed mixed graphs. It turns out that directed mixed graphs satisfy a certain maximality property which allows one to construct a simple graphical representation of an entire Markov equivalence class of marginalized local independence graphs. This is convenient as the equivalence class can be learned from data and its graphical representation concisely describes what underlying structure could have generated the observed local independencies.

\[ \] Deciding Markov equivalence of two directed mixed graphs is computationally hard, and we introduce a class of equivalence relations that are weaker than Markov equivalence, i.e., lead to larger equivalence classes. The weak equivalence classes enjoy many of the same properties as the Markov equivalence classes, and they provide a computationally feasible framework while retaining a clear interpretation. We discuss how this can be used for graphical modeling and causal structure learning based on local independence.