Spring 2023

Conley’s topological approach to dynamics provides a framework for deducing qualitative and global information about dynamics that is robust with respect to perturbations of the underlying system. Current methods for applications begin with a uniform discretization of the phase space into cubes, followed by problem-specific subdivisions if needed. At fine enough levels of resolution, the computational cost of these methods can become prohibitively expensive as dimension increases. We will introduce a machine learning approach for discretizing the phase space that aims to overcome the dimension problem. Preliminary experimental results using this method will be discussed.

Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki, which is a Cahn-Hilliard-like equation together with a nonlocal term. Unlike the Cahn-Hilliard model, even on one-dimensional spatial domains the steady state bifurcation diagram of the Ohta-Kawasaki model is still not fully understood. We therefore present computer-assisted proof techniques which can be used to validate and continue its bifurcation points. This includes not only fold points, but also pitchfork bifurcations which are the result of a cyclic group action beyond forcing through $\Z_2$ symmetries.

Mathematical modeling is widely used to study within-host viral dynamics. However, to the best of our knowledge, for the case of SARS-CoV-2 such analyses were mainly conducted with the use of viral load data and for the wild type (WT) variant of the virus. In addition, only few studies analyzed models for in vitro data, which are less noisy and more reproducible. In this work we collected multiple data types for SARS-CoV-2-infected Caco-2 cell lines, including infectious virus titers, measurements of intracellular viral RNA, cell viability data and percentage of infected cells for the WT and Delta variants. We showed that standard models cannot explain some key observations given the absence of cytopathic effect in human cell lines. We propose a novel mathematical model for in vitro SARS-CoV-2 dynamics, which included explicit modeling of intracellular events such as exhaustion of cellular resources required for virus production. The model also explicitly considers innate immune response. The proposed model accurately explained experimental data. Attenuated replication of the Delta variant in Caco-2 cells could be explained by our model on the basis of just two parameters: decreased cell entry rate and increased cytokine production rate.

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In this talk, we explore the challenges faced in analyzing time-varying systems with multi-scale dynamics. While traditional methods model these systems using ordinary differential equations (ODE), the direct analysis of such models is often difficult due to poorly measured parameters and numerous variables. To overcome these challenges, we propose a novel approach based on combinatorics and algebraic topology. We move away from classical ODE analysis and towards a more robust, scalable, and computable description of global dynamics in terms of annotated graphs.

We describe some new ideas in inverse scattering problem for screens in euclidian spaces. Especially we study the inverse problem including one passive measurement. By this we mean the physical problem where the transmitter is keeping fixed as well as the energy but the the measurement angle is allowed to vary. In mathematical terms the far field is known only for one fixed incoming wave and fixed wave number. Compared to standard fixed energy inverse scattering problem the dimension of the data for us is exactly half of the latter oner. At the end of the talk we give one simple result of this kind but by using a method that is promising to solve many more of such problems. On our way we let R.H. Mellin to meet J-B. J. Fourier and D. Hilbert on a half-line.

This is join study with Emilia Blåsten from the Lappeenranta University of Technology, Petri Ola from the University of Helsinki and Sadia Sadique from the Tallinn University of Technology.

I will discuss the inverse conductivity problem on recovering interior information about the electrical conductivity of an object/body from exterior electrical measurements. I will show how one can reconstruct the exact shape and position (called inclusions) of very general inhomogeneities from a known conductivity coefficient, using the monotonicity in terms of the conductivity coefficient of the associated Neumann-to-Dirichlet map. The inhomogeneities can be of several different types, with parts that are finite positive and negative perturbations, parts that are superconducting or insulating, and parts originating from a Muckenhoupt weight (e.g. leading to degenerate or singular problems). If time permits it, I will also discuss how to handle more practical electrode models and noisy measurements in a rigorous way.

Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe recent work on a real-space formulation of quantum electrodynamics for single photons interacting with two-level atoms. It is shown that the probability amplitude of a photon obeys a nonlocal PDE. Applications to quantum optics in random media will be described, where there is a close relation to kinetic equations for PDEs with random coefficients.

We will discuss interaction networks that spontaneously form in particulate-based systems. These networks, most commonly known as `force chains’ in granular systems, are dynamic structures which are by now known to be of fundamental importance for the purpose of revealing the underlying physical causes of a number of physical phenomena involved in statics and dynamics of particulate-based systems. The presentation will focus on applications of algebraic topology, and in particular of persistent homology (PH) to analysis of such networks found in both simulations and experiments. PH allows for a simplified representation of complex interaction field in both two and three spatial dimensions in terms of persistent diagrams (PDs)that are essentially point clouds. These point clouds could be compared in a meaningful manner, meaning that they allow for the analysis of both static and dynamic properties of the underlying systems. It is important to point out that such representation is robust with respect to small perturbations, which is crucial in particular when applying the method to the analysis of experimental data. In the second part of the talk, we will focus on interaction networks in both simulated and experimental systems experiencing stick-slip, intermittent type of dynamics, with focus on exploring predictability potential of the considered topological measures.

We present a combinatorial topological method to compute the dynamics of a parameterized family of ODEs. A discretization of the state space of the systems is used to construct a combinatorial representation from which recurrent versus non-recurrent dynamics is extracted. Algebraic topology is then used to validate and characterize the dynamics of the system. We will discuss the combinatorial description and the algebraic topological computations and will present applications to the dynamics of gene regulatory networks.

We consider an approximate, transformation optics-based cloaking scheme that incorporates a Drude-Lorentz model to account for the dispersive properties of the cloak. We show that on one hand, perfect (far field) cloaking is impossible at any frequency for any incident field, but on the other hand, one can achieve approximate cloaking for any finite band of frequencies, as the resonant frequency of the Drude-Lorentz term approaches infinity.

Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description of global dynamics. In this paper, a sheaf-theoretic approach to their continuation is developed. The algebraic structures are cast into a categorical framework to study their continuation systematically and simultaneously. Sheaves are built from this abstract formulation, which track the algebraic data as systems vary. Sheaf cohomology is computed for several classical bifurcations, demonstrating its ability to detect and classify bifurcations.

We consider the problem of understanding a dynamical system via finite samples of that system. The most straightforward approach to addressing this problem is to use the data to generate a model and then analyze the dynamics of that model using standard techniques. However, because long term dynamics may qualitatively change under arbitrarily small perturbations of a system it is unclear how reliable the conclusions that we arrive at in this manner will be; that is, it is difficult to quantify the probability that the predictions we make are correct. In recent work, Batko et al. address this problem by combining Gaussian processes with multivalued dynamics. This talk will discuss this general framework as well as considering the special case of a Weiner process that is conditioned to pass through a finite set of points and the dynamics generated by iterating a sample path from this process. In both the general and special cases, topological techniques (Conley theory) are used to characterize the global dynamics and deduce the existence, structure and approximate location of invariant sets. Most importantly, these techniques determine the probability (or confidence) that this characterization is correct.