Filter off: No filter for categories. Modify filter
In complex systems, external parameters often determine the phase in which the system operates, i.e., its macroscopic behavior. For nearly a century, statistical physics has extensively studied systems' transitions across phases, (universal) critical exponents, and related dynamical properties. In this talk I will consider the functionality of systems, notably operations in socio-technical ones, production in economic ones and possibly information-processing in biological ones, where timing is of crucial importance. I will introduce a stylised model on temporal networks with the magnitude of delay-mitigating buffers as the control parameter. I will show that the model exhibits temporal criticality, a novel form of critical behavior in time. I will characterize fluctuations near criticality, commonly referred to as "avalanches'', and identify the corresponding critical exponents. I will also show that real-world temporal networks, too, exhibit temporal criticality.
Least-squares assignments are a common way to partition a data set in clustering applications. While the combined choice of best clusters and representative sites is a known hard problem, linear programming methods and polyhedral theory provide insight into closely related tasks. We show that it is efficient to decide whether a given clustering can be represented as a least-squares assignment, and extend the related algorithm to arrive at soft-margin multiclass support vector machines. Further, we connect the search for optimal sites for a given clustering to volume computations for normal cones of an associated vertex in a certain polyhedron. This leads to new measures for the robustness of clusterings and explains why popular algorithms like k-means work well in practice.
In light of the log-Brunn-Minkowski conjecture, various attempts have been made to define the geometric means of convex bodies. Many of these constructions are fairly complex and/or fail to satisfy some natural properties one would expect of such a mean. To improve our understanding of potential geometric mean definitions, we study the closely related p-means of convex bodies, with the usual definition extended to two series ranging over all p in [-∞,∞]. We characterize their equality cases and obtain (in almost all instances tight) inequalities that quantify how well these means approximate each other. Based on our findings, we propose a fairly simple definition of the geometric mean that satisfies the properties considered in recent literature, and discuss potential axiomatic characterizations. Finally, we conclude that some of these properties are incompatible with approaches to proof the log-Brunn-Minkowski conjecture via geometric means.
Tate-Shafarevich groups measure the obstruction to the solvability of certain types of (systems of) polynomial equations in several variables over the field Q of rational numbers. They are also connected to some deep conjectures in arithmetic geometry. Recently, analogues have been studied over other fields such as Q(t), the field of rational functions with rational coefficients. In my lecture I shall present classical and recent aspects of this fascinating subject.
The aim of the present talk is to discuss HJM cross currency models that can serve as the basis for the simulation of exposure profiles in the xVA context. Such models need to take into account the asymmetries that arise in the different currency denominations in view of the benchmark reform: for example, while in the EUR area Euribor is still the dominant interest rate benchmark, the situation in the US is much more complex due to the introduction of SOFR and alternative forward looking unsecured rates such as the Bloomberg BSBY or the Ameribor 90T. The impact of the Libor transition on the structure of cross currency swap is also an aspect we would like to address. In summary we would like to: - Provide, in a HJM setting, a unified treatment of forward looking and backward looking rates with and without a credit/liquidity component, i.e. consider a HJM setting for a general underlying index in each currency area. - Properly link such general single currency HJM models by means of cross currency processes that capture the cross currency basis. - Analyze cross currency swaps with arbitrary combinations of interest rate indexes and collateral rates in the different currency areas i.e. with and without Libor discontinuation. This is a joint work with Silvia Lavagnini (BI Oslo)
The goal of this work is to disentangle the roles of volume and participation rate in the price response of the market to a sequence of orders. To do so, we use an approach where price dynamics are derived from the order flow via no arbitrage constraints. We also introduce in the model sophisticated market participants having superior abilities to analyse market dynamics. Our results lead to two square root laws of market impact, with respect to executed volume and with respect to participation rate. This is joint work with Bruno Durin and Grégoire Szymanski.
The Amazon rainforest, as one of the world’s largest ecosystems vulnerable to climate change, presents a crucial topic of study regarding the potential transition from forest to savanna states in the region. The presence of multiple modes in the tree cover distribution has been attributed to the existence of multistable states in the system. However, theoretical findings suggest that multimodality can also arise within a monostable system. This study further explores the feasibility of generating multimodality within the dynamical system. It introduces two monostable models: the first involves two variables following either a step or sigmoid function of tree cover, while the second includes two variables (temperature and precipitation) with unimodal input, along with an additional unknown variable featuring bimodal input. The second model incorporates productivity and mortality dynamics dependent on temperature and precipitation characteristics. The interaction between the lower and higher levels of the step or sigmoid function of tree cover could lead to the emergence of trimodality in the first model, while the presence of an additional variable with bimodal input played a pivotal role in generating trimodality in the second model. Both models are capable of generating multimodal frequency distributions of tree cover, highlighting that the presence of multistability alone is not a prerequisite for multimodality. However, the identification of pertinent empirical variables for these models remains indispensable.
Motivated by the tradeoff between exploitation and exploration in reinforcement learning, we study a continuous-time entropy-regularized mean variance portfolio selection problem in the presence of jumps. A first key step is to derive a suitable formulation of the continuous-time problem. In the existing literature for the diffusion case (e.g., Wang, Zariphopoulou and Zhou, Mach. Learn. Res. 2020), the conditional mean and the conditional covariance of the controlled dynamics are heuristically derived by a law of large numbers argument. In order to capture the influence of jumps, we first explicitly model distributional controls on discrete-time partitions and identify a family of discrete-time integrators which incorporate the additional exploration noise. Refining the time grid, we prove convergence in distribution of the discrete-time integrators to a multi-dimensional Levy process. This limit theorem gives rise to a natural continuous-time formulation of the exploratory control problem with entropy regularization. We solve this problem by adapting the classical Hamilton-Jacobi-Bellman approach. It turns out that the optimal feedback control distribution is Gaussian and that the optimal portfolio wealth process follows a linear stochastic differential equation, whose coefficients can be explicitly expressed in terms of the solution of a nonlinear partial integro-differential equation. We also provide a detailed comparison to the results derived by Wang and Zhou (Math. Finance, 2020) for the exploratory portfolio selection problem in the Black-Scholes model. The talk is based on joint work with Thuan Nguyen (Saarbrücken).
In this talk, I will present a general framework which can be used to analyze the scaling limits of various stochastic spatial "population" models. Such models include ternary Branching Brownian motion subject to majority voting and several examples of interacting particle systems motivated by biology. The approach is based on moment duality and a PDE methodology introduced by Barles and Souganidis which can be used to study the asymptotic behaviour of rescaled reaction-diffusion equations. In the limit, the models exhibit phase separation which is governed by a global-in-time, generalized notion of mean-curvature flow. This talk is based on joint work in progress with Thomas Hughes (Bath).
Identifying similarities and constructing joint latent spaces between heterogeneous, multivariate observation data sets is a challenge. Even in case classical canonical correlation analysis (CCA) is not applicable, nonlinear techniques can often successfully uncover the relationship between the observations. I will discuss our research in this direction: a concept that we call "jointly smooth functions". The construction of these functions is fully data-driven, and relies on spectral methods from manifold learning and a Dirichlet energy-based definition of smoothness. I will illustrate the theoretical results on simple examples, discuss our efficient implementation, and show improvements over existing nonparametric and kernel CCA techniques for real physiological signals in sleep stage identification, the construction of effective parameters in dynamical systems, and positional alignment from different video camera feeds of a race track. Related paper (SIAM Journal on Mathematics of Data Science, Vol. 4, Iss. 1 (2022)): https://doi.org/10.1137/21M141590X
In a popular Kuramoto-Sakaguchi model of globally coupled phase oscillators, the phase shift in coupling is a constant. We extend this model to a situation where mutual phase shifts are i.i.d. random numbers. In the first part of the talk, a nontrivial distribution of the phase shifts is assumed. We show that after the averaging over the phase shifts, a system without disorder appears with a new effective coupling function, which is the convolution of the original coupling function with the distribution of the phase shifts. In the second part of the talk, a situation with maximally frustrating disorder is considered, where the distribution of the phase shifts is uniform. In this case, the averaged coupling vanishes, and a standard synchronization transition is not observed. Nevertheless, some order in the phase dynamics appears as the coupling strength grows. We characterize it via a novel correlation order parameter, and via the properties of the frequency entrainment. The talk is based on the preprints https://arxiv.org/abs/2307.12563 and https://arxiv.org/abs/2401.00281.
Ginzburg-Landau fields are a class of models from statistical mechanics that describe the behavior of interfaces. The so-called Helffer-Sjöstrand representation relates them to a random walk in a time-dependent random environment. In the talk I will introduce these objects and survey some of the known results. I will then describe joint work with Wei Wu and Ofer Zeitouni on the asymptotics of the maximum of the Ginzburg-Landau fields in two dimensions.
Any collection of n compact convex planar sets K_1,…,K_n defines a vector of n over 2 mixed areas V(K_i, K_j) for 1≤i<j≤n. We show that for n≥4 these numbers satisfy certain Plücker-type inequalities. Moreover, we prove that for n=4 these inequalities completely describe the space of all mixed area vectors (V(K_i,K_j):1≤i<j≤4). For arbitrary n≥4 we show that this space has a semialgebraic closure of full dimension. As an application, we obtain an inequality description for the smallest positive homogeneous set containing the configuration space of intersection numbers of quadruples of tropical curves.
Joint work with Ivan Soprunov.
We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multi-scale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology, can be derived from an interacting particle model. As sub-Riemannian geometries play an important role in the Citti-Sarti-Petitot model of the visual cortex, this paper provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field stage to curvature flows which are used in image processing.
In the first part of the talk, we condition a Brownian motion on spending a total of at most $s > 0$ time units outside a bounded interval and discuss the behavior of the resulting process in the context of entropic repulsion. Moreover, we explicitly determine the exact asymptotic behavior of the probability that a Brownian motion on $[0,T]$ spends limited time outside a bounded interval, as $T \to \infty$. This is joint work with Frank Aurzada (Darmstadt) and Martin Kolb (Paderborn). In the second part, we condition a Brownian motion on having an atypically small $L_2$-norm on a long time interval and identify the resulting process as a well-known one. This is joint work with Frank Aurzada (Darmstadt) and Mikhail Lifshits (St. Petersburg).