"Multi-type logistic branching processes with selection: frequency process and genealogy for large carrying capacities"

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage. We first study the frequency of the disadvantageous type and show that, once the population approaches the carrying capacity, its evolution converges to a Gillespie-Wright-Fisher diffusion process. We then study the dynamics backward in time: we fix a time horizon at which the population is at carrying capacity and we study the ancestral relations of a sample of individuals. We prove that, provided that the advantageous and disadvantageous branching measures are ordered, this ancestral line process converges to the moment dual of the limiting diffusion. This talk is based on joint work with Julian Kern.

"Map enumeration and random noncommutative geometries"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: In this talk I will discuss an approach to quantum gravity that constructs path integrals over spaces of finite dimensional spectral triples, called Dirac ensembles. Such integrals can be realized as bi-tracial multi-matrix integrals and can be studied via connections to map enumeration. Recent work has added matter fields to these models, and it has been shown that certain models can be rephrased as 3-colour problem and loop gas models.

"A stochastic Gordon-Loeb model for optimal security investment under clustered cyber-attacks"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 17:15

Abstract:  We develop a continuous-time stochastic model for optimal cybersecurity investment under the threat of cyberattacks. The arrival of attacks is modeled using a Hawkes process, capturing the empirically relevant feature of clustering in cyberattacks. Extending the Gordon-Loeb model, each attack may result in a breach, with breach probability depending on the system’s vulnerability. We aim at determining the optimal cybersecurity investment to reduce vulnerability. The problem is cast as a two-dimensional Markovian stochastic optimal control problem and solved using dynamic programming methods. We perform a numerical study of the value function and the associated optimal investment strategy in cyber-security, highlighting the impact of randomly arriving clustered cyber-attacks. Based on a joint work with G. Callegaro, C. Fontana and C. Hillairet.

"The FBSDE approach to sine-Gordon"

Research Seminar Stochastics at FU Berlin, SR 009, Arnimallee 6, 10 AM

"Stochastic Modified Flows, Mean-Field Limits and Dynamics of Stochastic Gradient Descent"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room, 17:15

Abstract: We introduce a new class of limiting dynamics for stochastic gradient descent in the small learning rate regime, called stochastic modified flows. These dynamics are described by stochastic differential equations driven by infinite-dimensional noise, offering two key improvements over the so-called stochastic modified equa tions: they feature regular diffusion coefficients and capture multi-point statistics. Additionally, we present distribution-dependent stochastic modified flows, which we prove describe the fluctuating limiting behaviour of stochastic gradient descent in the combined small learning rate and infinite-width scaling regime. This is joint work with Benjamin Gess, Rishabh Gvalani, and Sebastian Kassing. 

"Aspects of metastability in molecular simulation"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room, 16:15

Abstract: The objective of this talk will be to present an approach we developed over the past 10 years to analyze metastable stochastic processes using quasi-stationary distributions. The interest of this approach is that it can be used to make a rigorous link between Markov processes with values in a continuous state space, and Markov processes with values in a discrete state space (i.e. the indices of the metastable states). This is motivated by questions from molecular simulation, in particular to justify models based on jump Markov processes between states (kinetic Monte Carlo models) and to analyze related sampling algorithms (accelerated dynamics). 

"A Navier-Stokes/Mullins-Sekerka system with different densities: weak solutions"

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: In this talk, I would like to introduce a model for the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier–Stokes and Mullins–Sekerka type parts, and can be obtained from the asymptotic limit of the diffuse interface model introduced by Abels, Garcke, and Grün in 2012. I will introduce a new notion of weak solutions and prove its global in time existence, together with a consistency result. This new notion of solution allows to include the case of different densities of the fluids, a sharp energy dissipation principle à la De Giorgi, as well as a weak formulation of the constant contact angle condition at the boundary, which were left open in the previous notion of solution proposed by H. Abels and M. Röger in 2009. This is a joint project with H. Abels.

"Quantitative mixing properties for Gibbs fields on bounded degree graphs"

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: Mixing properties are an important object of study within the theory of Gibbs measures. In this talk I will focus on two quantitative mixing properties: exponential ϕ-mixing (also called weak mixing) and exponential ψ-mixing (also called ratio weak mixing). These mixing conditions are well studied for Gibbs fields on square lattices. For instance, the unique Gibbs state of the Ising model above the critical temperature exhibits both types of mixing. In this talk we shall consider random fields on arbitrary graphs with bounded degree. While exponential ϕ-mixing is known to hold whenever Dobrushin's uniqueness condition is satisfied, exponential ψ-mixing has been studied very little in this general setting. The main result I'm going to present shows that for Markovian random fields ϕ-mixing with sufficiently high exponential decay rate implies ψ-mixing with slower, but still exponential decay rate. This generalizes a result of Alexander for square lattices and allows to formulate a general criterion for exponential ψ-mixing of Gibbs measures in terms of the corresponding Dobrushin constant. I will illustrate how this criterion applies to Ising and Potts models on regular trees. 

"Linear Monge is All You Need"

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: In this talk, we explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space, based on our recent work with Yoav Zemel. A key feature of our approach is its simplicity: relying only on elementary arguments from linear operator theory, we are able to derive explicit results without resorting to Kantorovich duality or Otto's Calculus. We provide a complete characterisation of both the Monge and Kantorovich problems in this context, regardless of the degeneracy of their measures. Furthermore, we show a simple way to construct all possible Wasserstein geodesics connecting two Gaussian measures. Finally, we generalise our results to characterise Wasserstein barycenters of Gaussian measures, borrowing the idea of Procrustes distance from statistical shape analysis.

"Kernel ridge regression for spherical responses"

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: The aim is to propose a novel nonlinear regression framework for responses taking values on a hypersphere. Rather than performing tangent space regression, where all the sphere responses are lifted to a single tangent space on which the regression is performed, we estimate conditional Frechet means by minimizing squared distances on the nonlinear manifold. Yet, the tangent space serves as a linear predictor space where the regression function takes values. The framework integrates Riemannian geometry techniques with functional data analysis by modelling the regression function using methods from vector-valued reproducing kernel Hilbert space theory. This formulation enables the reduction of the infinite-dimensional estimation problem to a finite-dimensional one via a representer theorem and leads to an estimation algorithm by means of Riemannian gradient descent. Explicit checkable conditions on the data that ensure the existence and uniqueness of the minimizing estimator are given.

"An integrable bound for rough stochastic partial differential equations"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: Random dynamical systems theory is often challenging to apply to stochastic partial differential equations (SPDEs), especially in the presence of multiplicative noise. This difficulty primarily arises from the inapplicability of the Kolmogorov continuity theorem in infinite-dimensional settings. 

Rough path theory provides a framework for constructing solutions in a path-wise manner, thereby enabling the definition of random dynamical systems (RDS) for such SPDEs. While this theory has been successful in this direction, it also introduces certain challenges: the solution framework often entails a loss of spatial and temporal regularity, rendering standard methods for deriving a priori estimates inapplicable. 

Moreover, purely algebraic arguments are, in some sense, insufficient for establishing integrable bounds when Gaussian rough paths are involved. In addition, obtaining an integrable bound for the solution is a crucial step once the framework of random dynamical systems is adopted. In this talk, we show that when the Cameron–Martin space associated with the noise exhibits suitable regularity properties, these difficulties can be overcome. In particular, this includes the case of fractional Brownian motion in the regime where the Hurst parameter exceeds 1/4. This, in turn, opens several avenues for analyzing the dynamical behavior of the solutions, including the study of invariant manifolds and stability. The results presented are based on joint work with Alexandra Blessing.

"Finite volume methods for regularized Dean-Kawasaki equations"

Research Seminar Stochastics at FU Berlin, SR 009, Arnimallee 6, 10 AM

"Advancing Decision-Efficiency in (Pre)-clinical Research via Novel Sequential Frameworks"

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: The escalating costs and ethical imperatives of drug development necessitate statistical frameworks that prioritize both efficiency and rigorous evidence generation. Conventional fixed-sample designs often lack the flexibility to adapt to emerging data, leading to potentially redundant animal testing or delayed decision-making. This talk introduces a suite of novel sequential testing procedures designed to optimize the transition from pre-clinical discovery to clinical validation.

We explore three primary pillars of application:

• Pre-clinical Statistics: We demonstrate how sequential probability ratio tests (SPRT) and group sequential designs can be adapted for small-sample laboratory studies, ensuring ethical "stop-for-efficacy" or "stop-for-futility" boundaries are met without compromising the Type I error rate.

• Generalized Pairwise Comparisons (GPC): We extend the sequential framework to GPC, a versatile method for analyzing multiple prioritized endpoints. This allows researchers to assess the net clinical benefit of a treatment sequentially, integrating diverse outcomes (e.g., survival, toxicity, and biomarkers) into a single, unified decision metric.

• Digital Biomarkers: Addressing the high-frequency, longitudinal nature of data from wearable devices, we propose sequential monitoring techniques that detect treatment signals in real-time. These methods account for the inherent "noise" and autocorrelation in digital health data, facilitating faster "go/no-go" decisions in early-phase trials.

By bridging these methodologies, this talk provides a roadmap for implementing adaptive evidence synthesis that reduces sample size requirements and accelerates the identification of promising therapeutic candidates.

"An integrable bound for rough stochastic partial differential equations"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: Random dynamical systems theory is often challenging to apply to stochastic partial differential equations (SPDEs), especially in the presence of multiplicative noise. This difficulty primarily arises from the inapplicability of the Kolmogorov continuity theorem in infinite-dimensional settings. 

Rough path theory provides a framework for constructing solutions in a path-wise manner, thereby enabling the definition of random dynamical systems (RDS) for such SPDEs. While this theory has been successful in this direction, it also introduces certain challenges: the solution framework often entails a loss of spatial and temporal regularity, rendering standard methods for deriving a priori estimates inapplicable. 

Moreover, purely algebraic arguments are, in some sense, insufficient for establishing integrable bounds when Gaussian rough paths are involved. In addition, obtaining an integrable bound for the solution is a crucial step once the framework of random dynamical systems is adopted. In this talk, we show that when the Cameron–Martin space associated with the noise exhibits suitable regularity properties, these difficulties can be overcome. In particular, this includes the case of fractional Brownian motion in the regime where the Hurst parameter exceeds 1/4. This, in turn, opens several avenues for analyzing the dynamical behavior of the solutions, including the study of invariant manifolds and stability. The results presented are based on joint work with Alexandra Blessing.

"A microlocal approach to the stochastic nonlinear Dirac equation"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: We present a novel framework for the study of a wide class of nonlinear Fermionic stochastic partial differential equations of Dirac type, which is inspired by the functional approach to the λ Φ^3 model. The main merit is that, by realizing random spinor fields within a suitable algebra of functional-valued Dirac distributions, we are able to use specific techniques proper of microlocal analysis. These allow us to deal with renormalization using an Epstein-Glaser perspective, hence without resorting to any specific regularization scheme. As a concrete example we shall use this method to discuss the stochastic Thirring model in two Euclidean dimensions and we shall comment on its applicability to a larger class of Fermionic SPDEs.

Based on joint work with A. Bonicelli, C. Dappiaggi and P. Rinaldi -- Math.Phys.Anal.Geom. 27 (2024) 3, 16.

"Boundary attainment conditions for stochastic Volterra equations"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 17:15

Abstract: In this presentation, I will discuss boundary attainment conditions for one-dimensional stochastic Volterra equations (SVEs) of convolution type.

In the first part of the talk, I will present an Osgood-type test for explosion to infinity of SVEs driven by additive noise, featuring kernels from a family that includes the fractional kernel. I will also investigate stability results for explosion times with respect to the kernels, including the case of an Euler-Maruyama approximation scheme. In the second part, I will present a Feller-type test that establishes, on a general open interval of the real line, necessary and sufficient conditions for boundary attainment of solutions to SVEs with possibly multiplicative noise. Here, I will consider dynamics governed by nonsingular kernels, which preserve the semimartingale property of the processes while introducing memory effects through a path-dependent drift. I will also show an application of these results to the Volterra square-root diffusion. 

"Dynamic Portfolio Choice with Intertemporal Hedging and Transaction Costs"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 16:15

Abstract: When returns are partially predictable and trading is costly, utility maximizing investors track a target portfolio at a constant trading speed. The target portfolio is optimal for a frictionless market, where asset returns are scaled back to account for trading costs and volatilities are adjusted to proxy the “execution risk” of holding assets that are costly to trade and exposed to volatile states. The trading speed solves an optimal execution problem, which describes how the legacy portfolio inherited from the past is traded towards the target portfolio in an optimal manner. Unlike for period-by-period mean-variance preferences as in Garleanu and Pedersen (2013), the target portfolio hedges changes in investment opportunities, and both it and the trading speed are linked and depend on execution risk. We set the problem out first in an “absolute” framework – price shocks independent of the price level and investors have CARA preferences – and then in a “relative” framework, with price shocks scaled by price levels and CRRA preferences.

"Equivalence between local and global Hadamard States with Robin boundary conditions on half-Minkowski spacetime"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 16:15, Joint session with the Geometry Seminar

Abstract: We construct the fundamental solutions and Hadamard states for a Klein-Gordon field in half-Minkowski spacetime with Robin boundary conditions in arbitrary dimensions using a generalisation of the Robin-to-Dirichlet map. On the one hand this allows us to prove the uniqueness and support properties of the Green operators. On the other hand, we obtain a local representation for the Hadamard parametrix that provides the correct local definition of Hadamard states, capturing `reflected' singularities from the spacetime timelike boundary. This allows us to prove the equivalence of our local Hadamard condition and the global Hadamard condition with a wave-front set described in terms of generalized broken bicharacteristics, obtaining a Radzikowski-like theorem in half-Minkowski spacetime. 

Joint work with B. Costeri, R. D. Singh and B. Juárez-Aubry -- ArXiv: 2509.26035 [math-ph]

"Duality in Non-Markovian Stochastic Control problems using Rough Stochastic Differential Equations"

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA 748, 11 AM

Abstract: The classical stochastic optimal control framework is heavily based on the Markovianity of the underlying dynamics e.g. through the use of the dynamic programming principle and the subsequent derivation of the Hamilton-Jacobi-Bellman equation. In this talk we will focus on a certain class of non-Markovian stochastic control problems arising e.g.from optimal control under stochastic volatility. In particular we will consider controlled doubly-stochastic differential equations driven by two independent noises B and W, where the coefficients depend progressively on the noise W. Extending previous work of [Diehl, Friz,Gassiat 2017] by methods from BS(P)DE-theory, we are then able to relate this stochastic control problem to a penalized version of the original control problem, where W can now be treated as a ‚frozen‘ i.e. deterministic (but irregular) path. Most importantly this ‚dual problem‘ is now Markovian and may thus be treated by classical methodology. The main technical tool allowing us to handle the dual problem will be the recently introduced theory of Rough Stochastic Differential Equations (RSDEs) [Friz, Hoquet, Lê „21], which provides a generalized framework of classical SDE Theory and Lyons’ Rough Differential Equations. This is joint work with Peter Bank, Peter K. Friz and Fillipo de Feo.

"An SPDE-based proof on jointly invariant measures for KPZ/Burger"

Research Seminar Stochastics at FU Berlin, SR 009, Arnimallee 6, 10 AM

"A Mean-Field Game Analysis of Systemic Risk under Capital Constraints"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room room 405/406, 17:15

Abstract: We analyze the effect of regulatory capital constraints on financial stability in a large homogeneous banking system using a mean-field game (MFG) model. Each bank holds cash and a tradable risky asset. Banks choose absolutely continuous trading rates in order to maximize expected terminal equity, with trades subject to transaction costs. Capital regulation requires equity to exceed a fixed multiple of the position in the tradable asset; breaches trigger forced liquidation. The asset drift depends on changes in average asset holdings across banks, so aggregate deleveraging creates contagion effects, leading to an MFG.

We discuss the coupled forward-backward PDE system characterizing equilibria of the MFG, and we solve the constrained MFG numerically. Experiments demonstrate that capital constraints accelerate deleveraging and limit risk-bearing capacity. In some regimes, simultaneous breaches trigger liquidation cascades.

The last part of the presentation is devoted to the mathematical analysis of a related model with time-smoothed contagion as in, e.g., Hambly, Ledger and Sojmark (2019) or Campi and Burzoni (2024). We characterize optimal strategies for a given evolution of the system, establish the existence of an MFG equilibrium and discuss limit results for a finite but large homogeneous banking system

"Rough SDEs and Robust Filtering for Jump-Diffusions"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room room 405/406, 16:15

Abstract: Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma has paved the way for a unified theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise. As an application, we will then investigate the existence of a robust representation of the conditional distribution in a stochastic filtering model for multidimensional correlated jump-diffusions. 

"Curvature-driven pattern formation in biomembranes: A gradient flow approach"

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

"Causal Inference (on time series data)"

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: This talk is intended as a broad overview of graphical causal inference, including causal discovery, where the task is to learn a causal graph, and causal effect estimation, where a causal graph is already assumed and the task is to quantify causal effects. In both cases a focus will be on statistically robust estimation strategies, including a theoretical optimality result for adjustment-based causal effect estimation. I will also illustrate practical workflows in my software package Tigramite.

"Yamaguti algebras and noncrossing partitions"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: Among various types of algebras originating in differential geometry as "tensorial algebras" of an affine connection, those known as the Lie-Yamaguti algebras still remain an algebraic mystery. In September 2025, A.Das introduced an "associative" version of Yamaguti algebras which are supposed to serve as envelopes of Lie-Yamaguti algebras. Together with Frédéric Chapoton, we found a combinatorial description of free Yamaguti algebras and of the Yamaguti operad via noncrossing partitions. I'll report on that work.

"Multisymplectic structures in Lagrangian Field Theory"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:15

Abstract: According to Cariñena, Crampin and Ibort, multisymplectic structures are “the field theoretical analogues of the symplectic structures used in geometrical mechanics”. Multisymplectic geometry can therefore be used to study classical field theories from a geometric point of view but it also took on a mathematical life on its own being a generalization of symplectic geometry. Recent developments in the field of multisymplectic geometry like the definition of the $L_\infty$-algebra of Hamiltonian forms introduced by Rogers in 2012 and the introduction of homotopy momentum maps by Callies, Frégier, Rogers, and Zambon in 2016 provide a new starting point to reconsider long-standing conceptual problems in the understanding of symmetries of classical field theories.

In this talk I will provide a brief introduction to multisymplectic geometry as well as a crash course on Lagrangian field theory in the variational bicomplex on the infinite jet bundle. We will use this set-up to investigate homotopy momentum maps in Lagrangian field theory and the problem of reduction of multisymplectic structures with Hamiltonian symmetries.

This is joint work with Christian Blohmann and part of the talk is based on arXiv:2505.09492.

"Spectral truncations and operator systems in noncommutative geometry"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 10:15

Abstract: Spectral truncations are compressions of spectral triples by spectral projections for the Dirac operator. This formalism was introduced by Connes--van Suijlekom to reflect constraints on the availability of spectral data, and they advocate for considering operator systems rather than C*-algebras in noncommutative geometry. Connecting to the setting of Rieffel's compact quantum metric spaces and Kerr--Li's operator Gromov--Hausdorff distance, it makes sense to ask about convergence of spectral truncations. I will report on recent progress on this question for tori and compact quantum groups.

"Talagrand-Type Transport Inequalities for Path Spaces over Carnot Groups"

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA 748, 11 AM

"Uncertainty quantification for a model for a magnetostrictive material involving a hysteresis operator"

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: A model for a magnetostrictive material involving a generalized Prandtl-Ishlinskii-operator was derived in [1]. In this presentation some parameters in this model are supposed to have a fixed value while other parameters are supposed to be uncertain.

Using results of measurements, values for the parameters of the first kind are determined and for those parameters with uncertain values Uncertainty Quantification (UQ) is used to determine random densities to describe these parameters and their uncertainties. These results are used do perform forward UQ and the results of forward UQ are compared with measured data. This extends some of the results in [2,3].

Joint work with Carmine Stefano Clemente, Daniele Davino, Ciro Visone. 

References

[1] D. Davino, P. Krejčí, and C. Visone, Fully coupled modeling of magneto-mechanical hysteresis through ‘thermodynamic’ compatibility. Smart Mater. Struct., 22(9), 0950099 (2013).

[2] O. Klein, D. Davino, and C. Visone, On forward and inverse uncertainty quantification for models involving hysteresis operators. Math. Model. Nat. Phenom. 15, 53 (2020).

[3] O. Klein, On forward and inverse uncertainty quantification for a model for a magneto mechanical device involving a hysteresis operator. Applications of Mathematics, 68, 6 (2023)

"Reversible birth-and-death dynamics in continuum: Free-energy dissipation and attractor properties"

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: The theory of Gibbs point processes, describing interacting systems of particles in continuous space, is by now a classical topic in probability and statistical mechanics and for various questions, somewhat satisfactory results have been obtained. On the lattice, these results on the equilibrium behavior of large systems of interacting particles are complemented by a detailed study of associated dynamics. In contrast, very little is known about dynamical aspects in continuum, in particular out-of-equilibrium dynamics and convergence to equilibrium. In this talk, I will review basic terms of the probabilistic theory of statistical mechanics and present our first investigation of the long-time behavior of birth-and-death dynamics in continuum space. Our main results there confirm the physical intuition that the free energy is non-increasing in time and moreover governs the long-term behavior of large systems of interacting particles.

"A Unified Framework for Pattern Recovery in Penalized Estimation"

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: We consider the framework of least-squares penalized estimation where the penalty term is given by a polyhedral norm, or more generally, a polyhedral gauge, which encompasses methods such as LASSO and. generalized LASSO, SLOPE, OSCAR, PACS and others. Each of these estimators can uncover a different structure or “pattern”of the unknown parameter vector. We define a novel and general notion of patterns based on subdifferentials and formalize an approach to measure pattern complexity. For pattern recovery, we provide a minimal condition for a particular pattern to be detected with positive probability, the so-called accessibility condition. We make the connection to estimation uniqueness by showing that uniqueness holds if and only if no pattern with complexity exceeding the rank of the X-matrix is accessible. Subsequently, we introduce the noiseless recovery condition which is a stronger requirement than accessibility and which can be shown to play exactly the same role as the well-known irrepresentability condition for the LASSO – in that the probability of pattern recovery is bounded by 1/2 if the condition is not satisfied. Through this, we unify and extend the irrepresentability condition to a broad class of penalized estimators using an interpretable criterion. We also look at the “gap” between accessibility and the noiseless recovery condition and discuss that our criteria show that it is more pronounced for simple patterns. Finally, we prove that the noiseless recovery condition can indeed be relaxed when turning to so-called thresholded penalized estimation: in this setting, the accessibility condition is already sucient (and necessary) for sure pattern recovery provided that the signal of the pattern is large enough. We demonstrate how our findings can be interpreted through a geometrical lens throughout the talk and illustrate our results for the Lasso as well as other estimation procedures.

Related references:

P. Graczyk, U. Schneider, T. Skalski and P. Tardivel, A Unified Framework for Pattern Recovery in Penalized and Thresholded Estimation and its Geometry; https://doi.org/10.1007/s10957-025-02863-6

U. Schneider and P. Tardivel, The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation; https://jmlr.org/papers/v23/21-0420.html

K. Ewald and U. Schneider, On the Distribution, Model Selection Properties and Uniqueness of the Lasso Estimator in Low and High Dimensions; https://doi.org/10.1214/20-EJS1687

"Boundaries, corners and edge modes: semi-local observables for electromagnetism"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: For gauge field theories on spacetimes with (asymptotic) boundaries, the gauge transformations acting non-trivially at boundaries and/or corners have a distinguished role compared to those acting purely in the bulk, where the latter are viewed as redundancies, the former play an important role in boundary/infrared features of these theories. We investigate the quantum electromagnetic field on spacetimes with boundaries and corners, and construct algebras of semi-local observables sensitive to the boundary gauge group. We discuss the role of these observables in gluing procedures and superselection theory for electromagnetism, as well as the existence of physically well behaved (Hadamard) states on these algebras.

"The spatial Muller’s ratchet"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room, 17:15

Abstract: In this talk we will consider the spatial Muller’s ratchet introduced by Foutel Rodier and Etheridge (2020). This particle system represents a spatial model of an asexual population with birth and death rates that depend on the local population density. For each particle, we keep track of the number of mutations of its genotype. Moreover, at each birth event with positive probability the offspring receives an additional mutation resulting in a lower birth rate.

We show that under appropriate re-scaling, the process converges weakly to an infinite system of PDEs, confirming non-rigorous computations of Foutel-Rodier and Etheridge. Under certain conditions, we can analyse these PDEs and consider the behaviour of travelling waves exploring an empty habitat. Finally, we will also answer the question whether deleterious mutations can surf population waves. Throughout we will discuss some of the technical difficulties that arise when dealing with a non-monotone particle system with infinitely many types and long-range interactions.

This is joint work with Joao de Oliveira Madeira and Sarah Penington.

"Scaling Limits of Line Models in Degenerate Environment" 

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room, 16:15

Abstract: We consider a 2-dimensional model of random walk in random environment known as line model. The environment is described by two independent families of i.i.d. random variables dictating rates of jumps in vertical, respectively horizontal directions, and whose values are constant along vertical, respect. horizontal lines.

When jump rates are heavy- tailed in one of the directions, and given by a stable distribution, we prove that the random walk becomes superdiffusive in that direction, with an explicit scaling limit written as a two-dimensional Brownian motion time-changed (in one of the components) by a Kesten- Spitzer process. In the case of a fully degenerate environment, a sufficient condition for non explosion is provided, and conjectures on the associated scaling limit are formulated. 

This is joint work with Henri Elad Altman.

"Dynamic accessibility percolation"

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: Accessibility percolation is a simple model in evolutionary biology describing how a population driven by the evolutionary forces of selection and mutation explores a fitness landscape. Mathematically, the fitness landscape is modeled by attaching random weights to the vertices of a graph. Then, accessible percolation asks whether there are paths of increasing fitness of a certain length. I will review some of the progress in this area and then consider the question what happens if the fitness landscape changes over time. In particular, I will focus on the case when the underlying graph is a regular tree. Depending on the ratio of depth to width of the tree, we will see different scaling regimes for the time it takes to see an increasing path. Some of the proofs rely on adapting techniques from the area of noise sensitivity for Boolean functions. 

"Discrete signature tensors for persistence landscapes"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: Signature tensors of paths are a versatile tool for data analysis and machine learning. Recently, they have been applied to persistent homology, by embedding barcodes into spaces of paths. Among the different path embeddings, the persistence landscape embedding is injective and stable, however it loses injectivity when composed with the signature map. Here we address this by proposing a discrete alternative. The critical points of a persistence landscape form a time-series, of which we compute the discrete signature. We call this association the discrete landscape feature map (DLFM). We give results on the injectivity, stability and computability of the DLFM. We apply it to a knotted protein dataset, capturing sequence similarity and knot depth with statistical significance.

"Scaling Deep Residual Networks: an Optimization perspective"

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA 748, 11 AM

Abstract: We study the gradient-based training dynamics of deep residual networks (ResNets) initialized at random. Focusing first on architectures with linear dependence on the weights, we show that, in the regime of diverging depth L, fixed embedding dimension D, and appropriate scaling, the discrete training dynamics converges almost surely, pathwise, to a gradient flow on the Cameron–Martin space H^1. This result provides a precise infinite-depth limit for ResNet training. We also discuss the key mathematical and engineering challenges that arise when extending this analysis to the fully non-linear setting.

"Nonlinear dynamics of complex biophysical processes"

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: Mathematical and computational approaches have become essential tools in modern healthcare and quantitative biomedicine, providing a framework to understand, predict, and control complex biological processes. Many biophysical systems – ranging from cellular signaling and electrophysiology to tissue dynamics – are inherently nonlinear and involve multiple interacting time and spatial scales.

In this talk, I will discuss how nonlinear dynamical systems theory can be used to model and analyze complex biophysical processes, with a particular emphasis on electrophysiological phenomena. Starting from mathematical models of excitable cells described by systems of nonlinear differential equations, I will illustrate how multiscale interactions can give rise to rich dynamical behavior. Tools from bifurcation theory will be used to uncover mechanisms underlying transitions between physiological and pathological states.

As a representative example, I will highlight abnormal oscillatory dynamics arising from ion current interactions, such as early afterdepolarizations, which are linked to disease, pharmacological effects, and cellular stress and are believed to play a key role in the initiation of arrhythmias. Moving from the cellular to the tissue level, I will also discuss how collective dynamics and synchronization phenomena can emerge, including wave propagation and pattern formation, as observed in excitable media. Numerical simulations using modern open-source computational frameworks will be presented to illustrate these concepts.

Overall, the talk aims to demonstrate how nonlinear dynamics provides unifying principles for understanding complex behavior in biophysical systems across scales.

"Averaging principle for semilinear slow–fast rough partial differential equations"

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA 748, 11 AM

"Iterative Schemes for Markov perfect Equilibria"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 4 PM c.t.

Abstract: We study Markov perfect equilibria in continuous-time dynamic games with finitely many symmetric players. The corresponding Nash system reduces to the Nash-Lasry-Lions equation for the commonvalue function, also known as the master equation in the mean-field setting. In the finite-state space problems we consider, this equation becomes a nonlinear ordinary differential equation admitting a unique classical solution. Leveraging this uniqueness, we prove the convergence of both Picard and weighted Picard iterations, yielding efficient computational methods. Numerical experiments confirm the effectiveness of algorithms based on this approach. This is joint work with Mathieu Laurière (NYU Shanghai), Mete Soner (Princeton), and Qinxin Yan (Princeton).

"Shape and dynamics of the term structure in multi-factor interest rate models"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 5 PM c.t.

Abstract: We examine the shapes that are attained by the forward- and yield-curve in several multi-factor interest rate models, in particular in the two-factor Vasicek model and the Svensson family of models. We provide a complete classification of all attainable shapes and partition the parameter space of each family according to these shapes. Building upon these results, we then examine the consistent dynamic evolution of the Svensson family under absence of arbitrage. Our analysis shows that consistent dynamics restrict the set of attainable shapes, and we demonstrate that certain complex shapes can no longer appear after a deterministic time horizon. As mathematical tools, the theory of total positivity and envelopes of plane curves are employed. The talk is based on joint work with Felix Sachse.

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: We establish a weak-strong uniqueness principle for the two-phase Mullins-Sekerka equation in ambient dimension d = 2 and 3: As long as a classical solution to the evolution problem exists, any weak De Giorgi type varifold solution (see for this notion the recent work with Stinson, Arch. Ration. Mech. Anal. 248, 8, 2024) must coincide with it. In particular, in the absence of topology changes such weak solutions do not introduce a mechanism for (unphysical) non-uniqueness. We also derive a stability estimate with respect to changes in the data. I will explain our method which is based on the notion of relative entropies for interface evolution problems, a reduction argument to a perturbative setting, and a stability analysis in this perturbative regime relying crucially on the gradient flow structure of the Mullins–Sekerka equation.

This is joint work with Julian Fischer, Tim Laux and Theresa M. Simon.

"The quenched Edwards—Wilkinson equation with Gaussian disorder"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room, 17:15

Abstract: In this talk I will introduce the quenched Edwards–Wilkinson equation, which models the growth of an interface among an obstacle field. Due to the elasticity effect of the laplacian, obstacle may slow down or stop the growth of the interface. When the driving force is low and there is enough disorder of the obstacle field, the interface may get pinned. But for a large enough driving force, there is a positive speed of propagation of the interface.

I will give the intuition for this phenomenon, mention what is done in the literature and then will turn to this equation with a Gaussian disorder, which is white in the spatial component. Due to the irregularity we need tools from Rough Paths, like the (stochastic) sewing lemma and regularisation by noise in order to show well-posedness. I will explain the idea behind these tools and how we apply them.

This is joint work with Toyota Matsuda and Jaeyun Ji.

"Markov Perfect Equilibria in Discrete Finite-Player and Mean-Field Games"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room, 16:15

Abstract: We study dynamic finite-player and mean-field stochastic games within the framework of Markov perfect equilibria (MPE). Through the Kuramoto mean field game, we investigate the emergence of multiple self-organizing equilibria with complex time-dependent dynamics. This is in contrast to continuous-time finite player games, which admit unique MPE in the absence of monotonicity conditions. While discrete-time problems generally do not admit unique MPE, we show that uniqueness is remarkably recovered when the time steps are small. This result, established without relying on any monotonicity conditions, underscores the importance of inertia in dynamic games.

This is joint work with Mete Soner (Princeton) and Atilla Yilmaz (Temple)

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: U-statistics are a fundamental class of estimators that generalize the sample mean and underpin much of nonparametric statistics. Although extensively studied in both statistics and probability, key challenges remain. These include their inherently high computational cost—addressed partly through incomplete U-statistics—and their non-standard asymptotic behavior in the degenerate case, which typically requires resampling methods for hypothesis testing. This talk presents a novel perspective on U-statistics, grounded in hypergraph theory and combinatorial designs. Our approach bypasses the traditional Hoeffding decomposition, which is the the main analytical tool in this literature but is highly sensitive to degeneracy. By fully characterizing the dependence structure of a U-statistic, we derive a new Berry–Esseen bound that applies to all incomplete U-statistics based on deterministic designs, yielding conditions under which Gaussian limiting distributions can be established even in the degenerate case and when the order diverges. Moreover, we introduce efficient algorithms to construct incomplete U-statistics of equireplicate designs, a subclass of deterministic designs that, in certain cases, enable to achieve minimum variance. To illustrate the power of this novel framework, we apply it to kernel-based testing, focusing on the widely used two-sample Maximum Mean Discrepancy (MMD) test. Our approach leads to a permutation-free variant of the MMD test that delivers substantial computational gains while retaining statistical validity. 

“A Pretty Pre-amble to Pre-Lie Algebras from Combinatorial Species”Algebraic analysis of integrable hierarchies”

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: We present an introduction to pre-Lie algebras, an algebraic structure closely related to Lie algebras. From a combinatorial perspective, pre-Lie algebras are connected to the notion of insertion of objects of a given nature into one another. In recent years, pre-Lie algebras have found numerous applications in algebra, combinatorics, quantum field theory, and numerical analysis.

To better understand the nature of pre-Lie algebras, we employ the framework of species, a categorification of the concept of generating functions. This perspective allows us to describe, in a combinatorial way, several algebraic properties of pre-Lie algebras. In particular, we present a “pre-Lie-like” notion of symmetric operads, called Nested Pre-Lie operads (NPL for short). After giving several examples of NPL operads, we explain how to construct NPL-algebras, in the same way algebraic structures emerge from operads by considering algebras over operads.

To do so, we use a new variant of species based on polynomial functions. This is joint work with Dominique Manchon, Hedi Regeiba, and Imen Rjaiba.

"Graph Theory Produces Optimal Fault-Tolerant Cat States"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: One classic quantum state is known as the cat state. It is, in a sense, the simplest possible highly entangled state, and is very useful for a variety of quantum protocols, including syndrome extraction. It is therefore of great interest to be able to prepare cat states fault-tolerantly, meaning that small numbers of errors do not propagate to more qubits. Recent attempts to construct fault-tolerant circuits for cat state preparation have used brute force techniques to find sets of flags that will detect any badly-propagating faults. In this talk, I will showcase recent results that allow us to quickly and easily reproduce and often exceed these brute force techniques by studying 3-regular graphs and applying some techniques from elementary (and time-permitting, advanced) graph theory. Furthermore, I will show how we can use graph theory to prove that our constructions are optimal and discuss how we can improve the quantum gate count for fault-tolerant cat state preparation from O(n log n) to O(n).

“Fixed Points of Autoencoders: Existence and Stability”

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA 748, 11 AM

Abstract: The talk will consist of two parts. In the first part, we introduce Artificial Neural Networks (ANNs) in the context of classical approximation theory with additional insight from neural networks in the brain. We next explain how ANNs work using a simple example of classification of handwritten digits. Historically the development of ANNs was done using empirical methods.  In the second part of the talk, we demonstrate how well-developed mathematical theories and classical theorems can be used to understand and improve the performance of ANNs. Specifically, for several types of ANNs, we use spectral results from Random Matrix Theory and Fixed Point Theorems for stochastic operators. Here we focus on the existence and stability results for fixed points of autoencoder ANNs which have the same dimension of input and output layers. In the case of light-tailed initialization of random i.i.d. weight matrices we prove the existence of the unique stable fixed point and establish a connection between scale of the variance of weights and contraction properties of autoencoders. We conclude by presenting numerical results for both light- and heavy-tailed initialization.

“Anomalous diffusions in inhomogeneous media”

Research Seminar Stochastics at FU Berlin, SR 009, Arnimallee 6, 10 AM

“From regularity theory for elliptic equations to invariance principle for random walks”

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: This talk addresses regularity for elliptic equations under merely averaged ellipticity conditions and establishes sharp criteria ensuring local boundedness and Harnack inequalities. These results extend the classical De Giorgi–Nash–Moser theory and yield new applications to variational integrals with (p, q)-growth. In the second part, the same regularity ideas are applied to the random conductance model, leading – under optimal moment assumptions – to a quenched invariance principle and a local limit theorem. The results highlight a close connection between regularity theory and stochastic homogenization.

"Penalized MLE in high-dimensional logistic regression"

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: This talk focuses on a popular logistic regression model in the case of a large or even infinite parameter dimension, limited sample size, and possible model misspecification. Recent studies highlighted new effects and phenomena appearing in highly or overparametrized regimes; see e.g. Bartlett et al. (2020), Sur and Cand`es (2019); Cand`es and Sur (2020), Bach (2024), Montanari et al. (2025), among others. The properties of the random design in high-dimensional regimes give rise to significant challenges for statistical analysis and inference. Existing strategies to mitigate these issues include tailored gradient-based methods with appropriate step-size choices and stopping criteria, as well as alternative regularization techniques for the logistic loss. In line with Cheng and Montanari (2022), this work examines a simple ridge-penalized formulation in possibly infinite-dimensional settings. The central question addressed is how to select the ridge tuning parameter while preserving standard accuracy guarantees and enabling valid inference.

“Algebraic analysis of integrable hierarchies”

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

"Portfolio Selection in Contests"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 5 PM c.t.

Abstract:  In an investment contest with incomplete information, a finite number of agents dynamically trade assets with idiosyncratic risk and are rewarded based on the relative ranking of their terminal portfolio values. We explicitly characterize a symmetric Nash equilibrium of the contest and rigorously verify its uniqueness. The connection between the reward structure and the agents' portfolio strategies is examined. A top-heavy payout rule results in an equilibrium portfolio return distribution with high positive skewness, which suffers from a large likelihood of poor performance. Risky asset holding increases when competition intensifies in a winner-takes all contest. This is joint work with Yumin Lu.

“Mean Field Games, Mean Field Type Control Problems, and their Associated Equations”

Research Seminar Rough Analysis and Stochastic Dynamics TU Berlin online at 11 AM

"Exponential convergence of fictitious-play FBSDEs in finite player stochastic differential games"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 4 PM c.t.

Abstract: We study finite player stochastic differential games on possibly bounded spatial domains. The equilibrium problem is formulated through the dynamic programming principle, leading to a coupled Nash system of HJB equations and, in probabilistic form, to a corresponding Nash FBSDE with stopping at the first exit from the parabolic domain (covering both boundary and terminal conditions). The main focus of the talk is the analysis of a fictitious-play procedure applied at the level of FBSDEs. At each iteration, a player solves a best-response FBSDE against fixed opponent strategies, giving rise to a sequence of fictitious-play FBSDEs. We show that this sequence converges exponentially fast to the Nash FBSDE. In unbounded domains, this holds under a small-time assumption; in bounded domains, exponential convergence is obtained for arbitrary horizons under additional regularity conditions. For completeness, we also discuss how the fictitious-play FBSDE is approximated by a numerically tractable surrogate FBSDE, which itself converges exponentially to the fictitious-play equation. Since the surrogate FBSDE admits a standard time-discrete approximation of order 1/2, this provides a transparent overall error structure for the numerical approximation of the Nash FBSDE. We conclude with representative numerical illustrations of the full approximation scheme.

“Robust Filtering: Correlated Noise and Multidimensional Observation”

Research Seminar Stochastics at FU Berlin, SR 009, Arnimallee 6, 10 AM

“A simple approach to the chain rule for gradient systems”

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: For constructing solutions for gradient systems, one usually approximates the problem, e.g. via the minimizing movement scheme, and then passes to the limit. By this way one obtains an energy-dissipation inequality and has to prove that this inequality is indeed an equality. For this one has to establish a chain rule for the energy functional within the class of curves satisfying only the natural dissipation and slope bounds.

We present a simple approach that is based on the classical duality pairing in Lp spaces instead of the often used metric approach in measure spaces. We show that this approach allows the treatment of a reasonably large class of nonlinear diffusion equations. Moreover, it is flexible enough to be generalized to systems of diffusion equations.

"Numerical approximation of stochastic differential games by deep FBSDE fictitious play"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room, 17:15

Abstract: We study the numerical approximation of multi-player stochastic differential games by combining fictitious play with a deep learning method for coupled FBSDEs. Starting from the game formulation, we outline how the equilibrium conditions lead to a coupled system of HJB equations and, equivalently, to a coupled FBSDE system. Within fictitious play, the full system is decomposed into a sequence of single-player coupled FBSDEs, where at each iteration the opponents feedback maps are fixed. This best-response problem is then replaced by a surrogate FBSDE that is suitable for numerical approximation.
The main focus of the talk is the numerical solution of this single surrogate FBSDE. We illustrate that a direct extension of the deep BSDE method encounters difficulties for strongly coupled systems, and we introduce an alternative algorithm designed to address these issues. An a posteriori error estimate is provided, together with numerical results for individual coupled FBSDEs and for the iterative fictitious-play procedure applied to the underlying stochastic differential game.

"Stochastic Volterra Equations on Convex Domains"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room

Abstract: We will present in this talk sufficient conditions on the kernel and on the coefficients to get the existence of a solution that stays in a convex domain. The underlying tool is an approximation scheme that also stays in this domain. We will first consider the case of convolution kernels and then more general double kernels.

Applications include: a comparison result for scalar SVEs, existence of solutions possibly with a jump component, weak second-order approximation schemes for SVEs with multi-factor kernels such as the multi-factor approximation of the rough Heston model.

"Inferring diffusivity from killed diffusions"

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: We consider diffusion of independent molecules in an insulated Euclidean domain with unknown diffusivity parameter. At a random time and position, the molecules may bind and stop diffusing in dependence of a given `binding potential'. The binding process can be modeled by an additive random functional corresponding to the canonical construction of a `killed' diffusion Markov process. We study the problem of conducting inference on the infinite-dimensional diffusion parameter from a histogram plot of the `killing' positions of the process. We show first that these positions follow a Poisson point process whose intensity measure is determined by the solution of a certain Schrödinger equation. The inference problem can then be re-cast as a non-linear inverse problem for this PDE, which we show to be consistently solvable in a Bayesian way under natural conditions on the initial state of the diffusion, provided the binding potential is not too `aggressive'. In the course of our proofs we obtain novel posterior contraction rate results for high-dimensional Poisson count data that are of independent interest. These theoretical results are accompanied by a numerical illustration of the algorithm via standard MCMC methods. (Joint work with Richard Nickl.)

"KP Solitons: Tropical Curves meet Grassmannians"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: The KP equation, a partial differential equation describing nonlinear wave motion, has solutions linked to algebraic curves. Solitons, a special class of solutions, arise from rational nodal curves. Kodama and Williams explored real regular solitons and their connection to totally positive Grassmannians. Building on Abenda and Grinevich’s work, I will discuss the interplay between real regular solitons, dual graphs of singular curves, matroids, and cells in the totally positive Grassmannian. I will focus on the case of degenerations to banana (or dipole) graphs. This is based on forthcoming work with Simonetta Abenda, Türkü Özülüm Çelik, and Yelena Mandelshtam.

"How to measure quantum fields? Implementing a causal measurement scheme"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: While measurement processes in standard quantum mechanics are well understood, the extension of these ideas to quantum field theory (QFT) remains a key challenge. In particular, ensuring that measurements respect fundamental principles such as relativistic causality is crucial. A persistent issue concerning measurements in QFT is, though, that the usual axioms for QFT alone are insufficient to prevent superluminal signaling. In this talk, I will discuss a recent proposal by Fewster and Verch for a local, covariant and causal measurement framework in algebraic QFT. In particular, I will discuss completeness of the framework and motivate its underlying assumptions focussing on the concrete setting of a free scalar field and Gaussian measurements. We conclude that the Fewster-Verch approach is suitable to model typical measurements in QFT. 

The talk is based on joint work with Miguel Navascués (Lett Math Phys 115, 115 (2025), https://doi.org/10.1007/s11005-025-02001-3).

"Ergodicity of φ^4_3"

Research Seminar Stochastics at FU Berlin, SR 009, Arnimallee 6, 10 AM

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: Physics-informed statistical learning is an emerging area of spatial and functional data analysis that integrates observational data with prior physical knowledge encoded by partial differential equations (PDEs). We propose an iterative Majorization–Minimization scheme for functional Principal Component Analysis of random fields in a general Hilbert space, formulated under the practically relevant assumption of partial observability of the data. By combining differential penalties with finite element discretizations, our approach recovers smooth principal component functions while preserving the geometric features of the spatial domain. The resulting estimation procedure involves solving a smoothing problem, which may become computationally demanding for large-scale datasets. After establishing the well-posedness of this smoothing problem under mild assumptions on the PDE parameters, we develop an efficient iterative algorithm for its solution. This framework enables the practical analysis of massive functional datasets at the population level, ranging from physics-informed fPCA to functional clustering, with applications to environmental and neuroimaging data. 

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: Motivated by engineering and Photonics research on open resonators in structured deterministic or stochastic environments, the talk introduces rigorous randomizations of absorbing and conservative boundary conditions on Lipschitz boundaries. As underlying PDEs, we choose div-grad acoustic systems, which can be also considered as dimensionally reduced Maxwell equations. We give a description of random m-dissipative boundary conditions that produce acoustic operators with almost surely (a.s.) compact resolvents, and so, also with a.s. discrete spectra, which may be interpreted as stochastic point processes. Based on these results, examples of mathematically convenient randomizations are constructed in terms of eigenfunctions of Laplace–Beltrami operators. It will be shown that, for these special randomizations, the resolvent compactness is connected with the Weyl law on the boundary. If time allows us, the asymptotics of the Laplace–Beltrami eigenvalues on non-smooth boundaries will be also discussed. The talk is based on the paper https://doi.org/10.1016/j.jmaa.2025.129985.

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-ESH

Abstract: Consider a dynamical network model featuring mobile stations on the Euclidean plane. The initial locations of the stations are given by a homogeneous Poisson point process. The stations are all moving at a constant speed and in a random direction. Consider fixed observers located in the Euclidean plane, which are served by the mobile stations. Each observer stays connected to the nearest station at any given point of time. Since the stations are moving, an observer disconnects and connects with different stations over time, by always selecting whichever station is the closest. This gives rise to a dynamical version of the Poisson–Voronoi tessellation. The focus of this talk is on the sequence of "handover" events of a typical observer, which are the epochs when its association changes. This defines a point process on the time axis, the "handover point process". We show that this point process is stationary and we determine its main properties, in particular its intensity and the joint distribution of its inter-event times. We also analyze the handover Palm distributions of several variables of practical interest. This includes the distance to the closest mobile station and the point process of all other mobile stations at handover epochs. The analysis is conducted both in the single-speed and in the multi-speed scenarios. It leads to the identification of the three dimensional state variables that "Markovize" the association dynamics. The analysis is based on a specific system of non-compact particles. The motivations are in the modeling of low or medium orbit satellite wireless communication networks. The model studied here is a planar "caricature" of this problem, which is initially defined on the sphere.

Joint work with Sanjoy Kumar Jhawar.

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 5 PM c.t.

Abstract: We frame dynamic persuasion in a partial observation stochastic control game with an ergodic criterion. The receiver controls the dynamics of a multidimensional unobserved state process. Information is provided to the receiver through a device designed by the sender that generates the observation process. 

The commitment of the sender is enforced and an exogenous information process outside the control of the sender is allowed. We develop this approach in the case where all dynamics are linear and the preferences of the receiver are linear-quadratic.

We prove a verification theorem for the existence and uniqueness of the solution of the HJB equation satisfied by the receiver’s value function. An extension to the case of persuasion of a mean field of interacting receivers is also provided. We illustrate this approach in two applications: the provision of information to electricity consumers with a smart meter designed by an electricity producer; the information provided by carbon footprint accounting rules to companies engaged in a best-in-class emissions reduction effort. In the first application, we link the benefits of information provision to the mispricing of electricity production. In the latter, we show that when firms declare a high level of best-in-class target, the information provided by stringent accounting rules offsets the Nash equilibrium effect that leads firms to increase pollution to make their target easier to achieve.

This is a joint work with Prof. René Aïd, Prof. Giorgia Callegaro and Prof. Luciano Campi.

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 4 PM c.t.

Abstract: A rough volatility model is a stochastic volatility model for an asset price process with rough volatility, meaning that the Hölder regularity of the volatility path is less than one half. In this talk, we will focus on the asymptotic behavior of implied volatility for short maturities under such models, and show that the large deviation principle for rough volatility models provides the short-time asymptotic behavior of implied volatility. Rough path theory sheds light on the calculus of these asymptotics. 

"Quenched invariance principle for random conductance models on Delaunay triangulations"

Berlin Probability Colloquium at WIAS, Erhard-Schmidt lecture room, 17:15-18:15

Abstract: The random conductance model is one of the most prominent models for a random walk in random environment. One of the main questions of interest in this context is if the random walk satisfies a quenched functional central limit theorem a.k.a. quenched invariance principle.

In this talk we will first review some techniques to show a quenched invariance principle for random walks on the integer lattice. Then we consider a class of random simple point processes and the random walk under random conductances on the associated Delaunay triangulation. We present comparison results between balls in the graph distance on the Delaunay triangulation (a.k.a. the chemical distance) and the Euclidean distance. Using this result we derive a quenched invariance principle for the random walk under suitable ergodicity and moment assumptions on both, the point process and the conductances.

This task is based on a joint work with Alessandra Faggionato, Martin Slowik and Yuki Tokushige.

"Early stopping in fixed-design regression with projection estimators"

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: The present work analyzes the behavior of early stopping rules in the context of fixed-design regression. The setting is the one of high-dimension with a limited amount of available data. In this setting reducing the dimension is a natural purpose and choosing how many features should be considered at the (non-linear) feature space level is a classical question. Projection estimators of the regression function are sequentially considered, each of them being associated with a nested collection of feature spaces. With no assumptions on the smoothness of the regression function, a first statement is given which proves that the (data-driven) early stopping rule based on the discrepancy principle a non-improvable rate O(1/p(n)). By contrast if one makes stronger smoothness assumptions, considering a smoothed version of the empirical risk for choosing the dimension of the feature space allows for improving the rate of convergence up to being minimax optimal. These rates are made explicit in particular for Random Features when the regression function does belong to an RKHS. Along the talk, we will discuss technical tools involving concentration inequalities for matrices under sub-Gaussian assumptions as well as the notion of critical equation combined with Random Linear Algebra results, which help deciding the level of smoothing to use.

"An H-convergence-based implicit function theorem and homogenization of nonlinear non-smooth elliptic systems"

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: see here

"Higher Gauge Flow Models"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: Generative Flow Models (GFMs) learn time-dependent vector fields that transport simple base distributions into complex data, with Flow Matching providing a unifying training principle that connects diffusion, score-based, and optimal transport approaches. In this talk, I will first introduce GFMs through the lens of Flow Matching, clarifying how trajectories, objectives, and sampling schemes are interconnected. Building on this foundation, I will present Gauge Flow Models, where the dynamics are equipped with a gauge connection so that flows evolve partially equivariantly under a specified symmetry group—injecting symmetry-aware inductive biases that improve both efficiency and generalization. I will then extend this framework to Higher Gauge Flow Models, which endow the dynamics with an L-\infty (higher-Lie) algebra structure, allowing the representation of higher symmetries and multi-way interactions directly within the flow equations. I will conclude with an outlook on open challenges and potential future directions.

"Absolutely convergent cyclotomic conical zeta values"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: Zeta values which are discrete sums on the one dimensional cone ]0, +\infty[ generalise to multizeta values which are discrete sums on k-dimensional Chen cones 0<x_k<...<x_1 with k in N. Going from Chen cones to general convex polyhedral cones leads to conical zeta values which in turn generalise to cyclotomic conical zeta functions when inserting a  U(1)-valued character in the sum. In this talk, we show that absolutely convergent cyclotomic conical zeta values span the same space as absolutely convergent cyclotomic multiple zeta values.

For this purpose, we regularise cyclotomic conical zeta functions by means of regularised conical zeta values. We then implement subdivisions of cones combined with a rescaling resulting from symmetry properties of the roots of the unity, to reduce them to regularised cyclotomic multiple zeta values. To do so, we first reinterpret regularised conical zeta values as regularised cyclotomic matrix zeta values built from matrices. This way, we can describe transformations on the matrices involved in the cyclotomic matrix zeta values induced by subdivisions of cones applied to regularised conical zeta values. These are some of the operations on matrices we use to write regularised cyclotomic multiple zeta values as rational linear combinations of regularised cyclotomic multiple zeta values. We give a necessary and sufficient criterion for the absolute convergence of cyclotomic matrix zeta values and view absolute convergent cyclotomic multiple zeta values as limits of regularised cyclotomic matrix zeta values. In the limit we obtain that absolutely convergent cyclotomic conical zeta values can be written as  rational linear combinations of regularized cyclotomic multiple zeta values.

This is joint work with Li Guo, Sylvie Paycha and Zhiyao Zhang.

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, 11 AM

Abstract: In this third and final talk, we apply the theory developed in the previous parts to study the robustness of a multidimensional stochastic filtering model with correlated continuous and jump noise. As a key ingredient for such robustness results, we establish exponential moment bounds for rough stochastic integrals. Specifically, we show that rough stochastic integrals are BMO^{p-var} processes, a notion introduced in [Lê2022] for continuous paths, and establish their exponential integrability by extending a version of the John—Nirenberg inequality from [Lê2022] to our setting. These results finally allow us to derive robustness and model misspecification estimates for the filtering problem, completing the programme developed throughout this mini-course.

"Small-scale and large-scale problems in function spaces"

Research Seminar Stochastics at FU Berlin, SR 009, Arnimallee 6, 10 AM

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: This talk focuses on the interplay between type and ancestry in two different multitype population models. In the first part, we briefly discuss the long-term behavior of critical multitype branching processes conditioned on survival, both with respect to the forward and the ancestral processes. Despite substantial differences in forward-time behavior and required techniques, their ancestral processes retain key structural similarities to the supercritical case. The main part of the talk then focuses on structured populations divided into d colonies, where individuals migrate at rates proportional to a global scaling parameter K. We sample N(K) individuals evenly across colonies and trace their ancestral lineages backward in time. Within each colony, coalescence occurs at a constant rate as in the Kingman coalescent. We encode the system's state as a d-dimensional vector of empirical measures, recording both current lineage locations and the colonies of their sampled descendants. Our focus is on how the sample size affects the asymptotic behavior of this process as K→∞ (representing fast migration), distinguishing two regimes: the critical-sampling regime (N(K)∼K) and the large-sampling regime (N(K)≫K). After suitable time-space rescaling, we prove convergence to d-dimensional coagulation equations in both sampling regimes. In the critical regime, the solution admits a representation via a multitype birth-death process; in the large-sample regime, via the entrance law of a multitype Feller diffusion. 

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: In this talk, we present the second-order asymptotic development of the Cahn-Hilliard functional under Dirichlet boundary conditions via Γ-convergence. We begin by reviewing results from the literature on the asymptotic expansion of the Cahn-Hilliard functional. Subsequently, we discuss our ongoing research focused on the Cahn-Hilliard functional with Dirichlet boundary  conditions. In particular, we examine the case where no interior interfaces are present and highlight several open questions for future investigation. This seminar is based on work in collaboration with Irene Fonseca and Giovanni Leoni.

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: Spatial confounding is a fundamental issue in spatial regression models which arises because spatial random effects, included to approximate unmeasured spatial variation, are typically not independent of covariates in the model. This can lead to significant bias in covariate effect estimates. The problem is complex and has been the topic of extensive research with sometimes puzzling and seemingly contradictory results. We will give an introduction to spatial confounding and discuss some suggested solutions for dealing with it, including the formalisation as a structural equation model and spatial+, where spatial variation in the covariate of interest is regresssed away first and remaining residuals are then used to identify the relevant effect.

In the second part of the presentation, we develop a broad theoretical framework that brings mathematical clarity to the mechanisms of spatial confounding, relying on an explicit analytical expression for the resulting bias. We see that the problem is directly linked to spatial smoothing and identify exactly how the size and occurrence of bias relate to the features of the spatial model as well as the underlying confounding scenario. Using our results, we can explain subtle and counter-intuitive behaviours. Finally, we propose a general approach for dealing with spatial confounding bias in practice, applicable for any spatial model specification. When a covariate has non-spatial information, we show that a general form of the so-called spatial+ method can be used to eliminate bias. When no such information is present, the situation is more challenging but, under the assumption of unconfounded high frequencies, we develop a procedure in which multiple capped versions of spatial+ are applied to assess the bias in this case. We illustrate our approach with an application to air temperature in Germany.

"Distributions of positive type and entanglement quantum field theory"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: Distributions of positive type arise naturally in quantum field theory as two-point distributions. Motivated by this application, one would like to perform various constructions with such distributions, but a key obstruction is that cut-off functions f(x,y) that equal 1 in a neighbourhood of the diagonal x=y cannot be of positive type. After reviewing basic definitions and results on distributions of positive type, I will show how this obstruction can be overcome in Euclidean space by using test-functions of positive type and finding lower bounds on their Fourier transforms. If time permits I will show how this workaround allows us to construct quantum states with interesting entanglement properties.

"The small mass limit of SDEs with arbitrary state-dependent friction driven by Lévy noise"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 4 PM c.t.

Abstract: The small mass limit is also known as the Smoluchowski-Kramers approximation. It was first proposed by  Smoluchowski (1916) and Kramers (1940) and is used in many mathematical and physical  studies to describe the motion approximation problem of small mass particles. The small mass limit is the justification for using the first order equation to describe the motion of a small particle disturbed by a Wiener process instead of using the Newton second-order equation.

 We develop an approach to derive the small mass limit for stochastic differential equations with state dependent friction driven by non-Gaussian Levy noise. For the case where the Levy noise has a finite second moment, we identify the limiting equation in probability, with respect to Skorokhod topology as the mass tends to zero. For the case where the Levy noise is-stable, we use interlacing method, which provides effective estimate results for a system under an -stable Levy noise, with rigorous error estimates. Then, we obtain the same limiting equation as the case that Levy noise has a nite second moment. In particular, compared to Gaussian noise, there are two more terms in the limit equation that are related to jumps, one is expressed entirely in terms of the solution itself and its jumps, and the other is expressed entirely by the integral of the state-dependent friction matrix with respect to jump increments. Finally, we give numerical simulation results to illustrate the validity of our theory.

"Neural networks, PDEs and control"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 5 PM c.t.

Abstract: Optimal control problems often involve the solution of high dimensional nonlinear PDEs, which is a key computational bottleneck. In this talk we will consider how neural networks can be used as a computational tool for these problems, how simple test cases can work deceptively well, and how fine details of the approach can lead to different results.

 Based on joint work with Justin Sirignano, Deqing Jiang and Jackson Hebner.

"On an a priori bound for the Kuramoto–Sivashinsky equation"

Research Seminar Stochastics at FU Berlin, SR 009, Arnimallee 6, 10 AM

"Random Tensors, an introduction"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: Random tensor models were introduced in the early '90s as generating functions for random geometries. Over the years they fond application in probability theory, data science and (conformal) field theory. In this talk I will give a brief introduction to the field and present some more classical and some more recent results.

"Strong generalized holomorphic principal bundles"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: We explore a classical problem within the framework of generalized complex (GC) geometry, a structure introduced by Hitchin and further developed by Gualtieri and Cavalcanti - namely, the development of an appropriate bundle theory in this setting. In this talk, we present the theory of strong generalized holomorphic (SGH) principal bundles, along with their connections and curvatures. These bundles interpolate between holomorphic and flat symplectic bundles. This is a joint work with Mainak Poddar.

"Kolmogorov equations for stochastic Volterra processes with singular kernels"

Research Seminar “Rough Analysis and Stochastic Dynamics” at TU Berlin, MA 748, 11 AM

Abstract: We associate backward and forward Kolmogorov equations to a class of fully nonlinear Stochastic Volterra Equations (SVEs) with convolution kernels K that are singular at the origin. Working on a carefully chosen Hilbert space H1, we rigorously establish a link between solutions of SVEs and Markovian mild solutions of a Stochastic Partial Differential Equation (SPDE) of transport-type. Using this Markovian lift, we obtain novel Itˆo formulae for functionals of mild solutions and, as a byproduct, show that their laws solve corresponding Fokker–Planck equations. Finally, we introduce a natural notion of “singular” directional derivatives along K and prove that (conditional) expectations of SVE solutions can be expressed in terms of the unique solution to a backward Kolmogorov equation on H1. Our analysis relies on stochastic calculus in Hilbert spaces, the reproducing kernel property of the state space H1, as well as crucial invariance and smoothing properties that are specific to the SPDEs of interest. In the special case of singular power-law kernels, our conditions guarantee well-posedness of the backward equation either for all values of the Hurst parameter H, when the noise is additive, or for all H > 1/4 when the noise is multiplicative. Time permitting, we shall discuss applications to mathematical finance as well as a few open problems. Based on joint work with Alexandre Pannier (Université Paris Cité).

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: We introduce a non-local transport distance on the space of point processes and analyse the induced geometry. We show — among other things — that the Ornstein–Uhlenbeck semigroup is the gradient flow of the specific relative entropy, functional inequalities like a Talagrand inequality, and exponential convergence of the Ornstein–Uhlenbeck flow to the Poisson point process. Towards the end, we discuss ongoing work on the extension of the framework adapted to Papangelou point processes.

Based on joint work with Martin Huesmann and Matthias Erbar.

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: Markov Modulated Stochastic Processes (MMSP) are a more flexible class of Stochastic Processes which capture phase changes arising in economies by allowing jumps in drift and volatility, linked to hidden states of a Markov chain. These models have been used to model option prices, renewable energy markets as well as for risk quantification. While Bayesian inference methods exists for simpler regime-switching models, we aim to extend it to more complex MMSPs. Our approach involves applying Bayesian estimation techniques to recover the hidden states and the parameters associated with each state of the Markov Chain. We propose Markov Chain Monte Carlo algorithms to perform Bayesian inference for MM-SPs. This will allow for a more data-driven analysis of asset returns with regime shifts and jumps. In the first part of the talk we review well-known estimation techniques for the CIR-process which is the solution of

dXt = (a+ bXt)dt+ σ X+t dWt

based on the paper Estimation in the CIR-Process (O.& Ryd´en. Scand, Econometric Theory. 1997). Finallywe want to extend this to a model with hidden variables, namely to

dXt = a(Zt) + b(Zt)Xtdt+ σ(Zt) X+s dWs

where Z is a continuous-time Markov chain. This is aspecial case of a general regime switching stochastic process.

dXt = β(Zt,Xt)dt+ σ(Zt,Xt)dWζ(Zt)

We consider two special cases

dXt = a(Zt) + b(Zt)Xtdt+ σ(Zt) X+s dWs.

For two special cases We formulate For parameter estimation we present a new version of the EM (Expectation Maximizer)-approach which was in a similar way used in the paper On the estimation of regime-switching L´evy models (Chevalier & Goutte, Stud. Nonlinear Dyn. E. 2017). Finally, we discuss the potential modification of the EM-algorithm, if we consider conditional least square minimization instead of likelihood maximisation in the ”M”-step of the EM-algorithm.

"Strong generalized holomorphic principal bundles"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:30

Abstract: We explore a classical problem within the framework of generalized complex (GC) geometry, a structure introduced by Hitchin and further developed by Gualtieri and Cavalcanti - namely, the development of an appropriate bundle theory in this setting. In this talk, we present the theory of strong generalized holomorphic (SGH) principal bundles, along with their connections and curvatures. These bundles interpolate between holomorphic and flat symplectic bundles. This is a joint work with Mainak Poddar.

"The Geometry of Rough Space"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: How can we differentiate functionals on rough path space? In this talk, I’ll be discussing how we can  approach this question, as well as some of my results.

"Semi-static variance-optimal hedging with self-exciting jumps"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 5 PM c.t.

Abstract: The aim of this talk is to investigate a quadratic, i.e., variance-optimal, semi-static hedging problem in an incomplete market model where the underlying log-asset price is driven by a diffusion process with stochastic volatility and a self-exciting jump process of Hawkes type. More precisely, we aim at hedging a claim at time T 0 by using a portfolio of available contingent claims, so to minimize the variance of the residual hedging error at time T. In order to improve the replication of the claim, we look for a hybrid hedging strategy of semi-static type, in which some assets are continuously rebalanced (the dynamic hedging component) and for some other assets a buy-and-hold strategy (the static component) is performed. We discuss in detail a specifc example in which the approach proposed is applied, i.e., a variance swap hedged by means of European options, and we provide a numerical illustration of the results obtained. (In cooperation with G. Callegaro, P. Di Tella and B. Ongarato, to appear on Mathematics of Operations Reasearch)

"Mean-Variance Portfolio Selection by Continuous-Time Reinforcement Learning: Algorithms, Regret Analysis, and Empirical Study"

Research Seminar Stochastic Analysis and Finance at HU Berlin, 1.115 (Johann-von-Neumann-Haus, Rudower Chaussee 25), 4 PM c.t.

Abstract: We study continuous-time mean-variance portfolio selection in markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes yet the coefficients of these processes are unknown. Based on the recently developed reinforcement learning (RL) theory for diffusion processes, we present a general data-driven RL algorithm that learns the pre-committed investment strategy directly without attempting to learn or estimate the market coefficients. For multi-stock Black-Scholes markets without factors, we further devise a baseline algorithm and prove its performance guarantee by deriving a sublinear regret bound in terms of Sharpe ratio. For performance enhancement and practical implementation, we modify the baseline algorithm and carry out an extensive empirical study to compare their performance, in terms of a host of common metrics, with a large number of widely used portfolio allocation strategies on SP 500 constituents. The results demonstrate that the proposed continuous-time RL strategy is consistently among the best especially in a volatile bear market, and decisively outperforms the model-based continuous-time counterparts by signifficant margins.

"The Geometry of Rough Path Space"

Research Seminar “Rough Analysis and Stochastic Dynamics” at TU Berlin, MA 748, 11 AM 

Abstract: How can we differentiate functionals on rough path space? In this talk, I’ll be discussing how we can approach this question, as well as some of my results.

"Generalized gauge theories on Lie algebroids"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: In this presentation we first introduce the formalism of Cartan geometry and present specific applications to describe gravitational theories by building action functionals from Characteristic classes and invariant polynomials.

In the second part we introduce the notion of generalized connections on Lie algebroids. We show they can be used to describe ordinary gauge theories (Ehresmann connections for particle physics and Cartan connections for gravitational theories) with a Higgs sector and the BRST formalism (used as a renormalization tool for the gauge theories) from a single unified mathematical structure corresponding to the Atiyah Lie algebroid of a principal bundle.

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: Under certain conditions, the largest eigenvalue of a sample covariance matrix undergoes a well-known phase transition when the sample size f and data dimension p diverge proportionally. 

In the subcritical regime, this eigenvalue has fluctuations of order $n^{-2/3}$ that can be approximated by a Tracy-Widom distribution, while in the supercritical regime, it has fluctuations of order $n^{-1/2}$ that can be approximated with a Gaussian distribution. However, the statistical problem of determining which regime underlies a given dataset is far from resolved. We develop a new testing framework and procedure to address this problem. In particular, we demonstrate that the procedure has an asymptotically controlled level, and that it is power consistent for certain alternatives. Also, this testing procedure enables the design of a new bootstrap method for approximating the distributions of functionals of the leading sample eigenvalues within the subcritical regime---which is the first such method that is supported by theoretical guarantees.

This talk is based on a joint work with Miles E. Lopes (UC Davis).

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: In this talk, we study a spatial model for dormancy in a random environment via a two-type branching random walk in continuous-time, where individuals switch between dormant and active states depending on the current state of a fluctuating environment (responsive switching). The branching mechanism is governed by the same random environment, which is here taken to be a simple symmetric random walk. We will interpret the presence of this random walk as a trap which attempts to kill the individuals whenever it meets them. The responsive switching between the active and dormant state is defined so that active individuals become dormant only when a trap is present at their location and remain active otherwise. Conversely, dormant individuals can only wake up once the environment becomes trap-free again. We quantify the influence of dormancy on population survival by analyzing the long-time asymptotics of the expected population size. The starting point for our mathematical considerations and proofs is the Parabolic Anderson Model via the Feynman-Kac formula. In particular, we investigate the quantitative role of dormancy by extending the Parabolic Anderson Model to a two-type random walk framework.

Joint work with Leo Tyrpak.

Langenbach-Seminar at WIAS, Library, room 411, 14:15

Abstract: Modeling concentrated ion mixtures in solvents like water is a complex research area with key applications in biology (e.g., ion transport through protein channels) and electrochemistry (e.g., batteries). 

In this talk, I will present a finite volume scheme for modeling the diffusion of ions in constrained geometries using a degenerate Poisson-Nernst-Planck system with size exclusion yielding cross-diffusion. The proposed method utilizes a two-point flux approximation and is part of the exponentially fitted scheme framework. The scheme is shown to be thermodynamically consistent, as it ensures the decay of some discrete version of the free energy. Classical numerical analysis results - existence of discrete solution, convergence of the scheme as the grid size and the time step go to 0 - follow. The long-time behavior of the scheme is also investigated, both from a theoretical and numerical point of view. Numerical simulations confirm our findings, but also point out some possibly very slow convergence towards equilibrium of the system under consideration.

This is a joint work with Clément Cànces and Maxime Herda.

"New directions for signature varieties"

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00

Abstract: Varieties of signature tensors were introduced by Améndola-Friz-Sturmfels in 2019. In this informal talk, I will give an overview of the works that followed this groundbreaking article, introduce the halfshuffle map M_p adjoint to a polynomial map p and hint current WIP.

Based on joint work with Annika Burmester, Steven Charlton, Laura Colmenarejo, Abhiram Kidambi, Felix Lotter. Special thanks to Francesco Galuppi.

"Finite Gröbner basis for quantum permutation groups"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: Quantum symmetries of discrete structures such as graphs have been studied extensively in the last two decades. Algorithmic approaches via a non commutative gröbner bases computation are a new development, with the issue, that the problems is in general undecidable and in consequence making this appraoch theoretically infeasible. However, experiments and recent theoretical results suggest that this approach is promising in practice.

First, I will outline the fundamental principles of quantum automorphism groups, with a focus on graphs and matroids. Next, I will discuss recent developments in the quantum analogue of symmetric groups, specifically the discovery of a finite Gröbner basis. This renders the word problem solvable for this class and makes symmetry detection simpler for quantum automorphism groups.

Furthermore, I will provide a brief overview of the generalisation of the quantum symmetries of matroids to quantum isomorphisms using non-local games.

This is based on a series of articles resulting from collaboration with Daniel Corey, Michael Joswig, Julien Schanz, Simon Schmidt, Leonard Schmitz and Moritz Weber.

"On the density of singular SDEs with fractional noise and applications to McKean-Vlasov equations"

Research Seminar “Rough Analysis and Stochastic Dynamics” at TU Berlin, MA 748, 11 AM 

Abstract: We investigate the properties of solutions to SDEs with distributional drift and fractional Brownian noise. Well-posedness of such equations has been widely studied in recent years. However, the standard tools to estimate the law of the solution are not available in this setting as the noise is not Markovian. The results presented include that the (conditional) law of the unique solution enjoys a certain regularity in space and integrability in time and its density has upper and lower Gaussian bounds. As a consequence, novel results on existence and uniqueness of solutions to a related McKean-Vlasov equation are presented. Joint work with Lucio Galeati, Alexandre Richard and Etienne Tanré.

Langenbach-Seminar at WIAS, rooms 405/406, 14:15

Abstract: We present new results that extend the De Giorgi-Nash-Moser theory to a class of hypoelliptic equations naturally arising in kinetic theory. These operators are mathematically characterized by two parts: a diffusion part governed by the (fractional) Laplace operator in some set of variables and a first order operator that contains the directions of missing ellipticity. A key ingredient to prove our results is a Poincare inequality, which we derive from the construction of suitable trajectories. The trajectories we rely on are quite exible and allow us to consider equations with an arbitrary number of Hormander commutators and whose diffusive part is either local (second order) or nonlocal (fractional order). We later combine the Poincare inequality with a L-2-L-infinity estimate, a Log-transformation and a classical covering argument (Ink-Spots Theorem) to deduce Harnack inequalities and Hölder regularity along the line of De Giorgi method. This talk is based on a series of papers in collaboration with F. Anceschi, H. Dietert, J. Guerand, A. Loher and C. Mouhot. 

Forschungsseminar Mathematische Statistik at WIAS, Erhard-Schmidt lecture room, 10 AM

Abstract: In this work we consider the problem of releasing a differentially private statistical summary that resides on a Riemannian manifold. We present extensions of the Laplace and K-norm mechanisms that utilizes intrinsic distances and volumes on the manifold. We also consider in detail the specific case where the summary is the Fréchet mean of data residing on a manifold. We demonstrate that our mechanism is rate optimal and depends only on the dimension of the manifold, not on the dimension of any ambient space, while also showing how ignoring the manifold structure can decrease the utility of the sanitized summary. Lastly, we illustrate our framework in three examples of particular interest in statistics: the space of symmetric positive definite matrices, which is used for covariance matrices, the sphere, which can be used as a space for modeling discrete distributions, and Kendall's 2D planar shape space.

"When a branching process grows like its mean"

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: A classical question in the study of discrete-time branching processes is determining when the normalized population size admits a non-degenerate limit. In the Bienaymé–Galton–Watson setting, where each individual produces a random number X of offspring and  reproduction is identically distributed across generations, the well-known Kesten–Stigum theorem states that such a limit exists if and only if
E[X logX]<∞. In this talk, we extend this result to branching processes in varying environments, where the offspring distribution may change from generation to generation, and discuss some ongoing work.

"Free cumulants and freeness for unitarily invariant random tensors"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: I will present our recent work with Benoît Collins and Razvan Gurau concerning the construction of free cumulants for unitarily invariant random tensors, the resulting formulation of tensorial freeness in terms of asymptotic moments, and the resulting generalization of non-commutative probability spaces.

"Algorithmic analysis of control systems with affine input and polynomial state dynamics"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Abstract: We provide simple algorithms for the formal analysis of deterministic continuous-time control systems whose dynamics is affine in the input and polynomial in the state (in short, polynomial systems). We consider the following semantic properties: input-output equivalence, input independence, linearity, and analyticity. Our approach is based on Chen-Fliess series, which provide a unique representation of the dynamics of such systems via their generating series (in noncommuting indeterminates). Our starting point is Fliess' seminal work showing how the semantic properties above are mirrored by corresponding combinatorial properties on generating series. Next, we observe that the generating series of polynomial systems coincide with the class of shuffle-finite series, a nonlinear generalisation of Schützenberger's rational series which we have recently studied in the context of automata theory and enumerative combinatorics. We exploit and extend recent results in the algorithmic analysis of shufflef-finite series to show that the semantic properties above are decidable for polynomial systems.

"Energy increment on abstract measure spaces"

Seminar "Interacting Random Systems" at WIAS Berlin, 11:30, WIAS-405-406

Abstract: A relevant concept in the proof of the weak regularity lemma (as in the original version) is that of the energy of a sigma algebra. We shall first introduce the notion of a semiring together with some examples. We then revisit the classical argument in the proof of the weak regularity lemma in its most abstract sense and from a probabilistic point of view. We shall finally mention a noncommutative martingale convexity inequality from Ricard and Xu (2014) to argue a variant of it on Lp spaces. Applications concern, for instance, large dense graphs, arithmetic progressions, and hypergraphs.

"Hibler's time-periodic sea ice model on ℝ2"

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 14:15

Abstract: In 1979, Hibler introduced a model for the large-scale dynamics of sea ice, modeling it as a fluid with viscous-plastic rheology. The resulting momentum balance equation contains a stress tensor, which depends nonlinearly on the strain rate and on the ice strength. Here, the ice strength is given by the ice thickness characteristics- the mean ice thickness and the ice compactness-, which are in turn coupled to the system via continuity equations. In this talk, we consider a regularized version of Hibler’s model on the whole space R2 in a time-periodic framework. Our approach is based on interpreting the system as an abstract quasi-linear evolution equation and decomposing the linearized equation into a stationary and a purely oscillatory part. We will focus in particular on the challenges arising from the nonlinearities and the unboundedness of the domain and discuss possible strategies to overcome these difficulties. 

"Quantum path signatures"

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 15:00

Paper Abstract: We elucidate physical aspects of path signatures by formulating randomised path developments within the framework of matrix models in quantum field theory. Using tools from physics, we introduce a new family of randomised path developments and derive corresponding loop equations. We then interpret unitary randomised path developments as time evolution operators on a Hilbert space of qubits. This leads to a definition of a quantum path signature feature map and associated quantum signature kernel through a quantum circuit construction. In the case of the Gaussian matrix model, we study a random ensemble of Pauli strings and formulate a quantum algorithm to compute such kernel.