This live online course gives an introduction to the theory of concentration inequalities with some basic applications to matrix computations. The course assumes only a basic level of experience with probability, linear algebra, and matrix computations. It does not assume exposure to high-dimensional probability or randomized matrix computations. The course will consist of four 90-minute lectures on scalar concentration, matrix concentration, subspace embeddings and randomized SVD. Each lecture takes place online from 18:30-20:00 from Modal 3rd to Thursday 6th May 2021.
Further Information and registration under: https://igdk1754.ma.tum.de/IGDK1754/CC_Tropp
Large human crowds constitute a fascinating research area for mathematical modelling. Even though the individuals forming the crowd are extremely complex, dominant emergent behaviors like density-velocity profiles and lane formation can be captured with relatively simple models. In this talk, we discuss recent "first principle", agent-based approaches, coarser cellular automatons, as well as modelling ideas from fluid dynamics. We also show how machine learning methods can provide a separate path to understanding crowds, with both strengths and shortcomings compared to the traditional approach.
Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816
We will introduce recent non-reversible generalizations of the Edge-Reinforced Random Walk and its motivation in Bayesian statistics for variable order Markov Chains. The process is again partially exchangeable in the sense of Diaconis and Freedman (1982), and its mixing measure can be explicitly computed. It can also be associated to a continuous process called the *-Vertex Reinforced Random Walk, which itself is in general not exchangeable. If time allows, we will also discuss some properties of that process.Based on joint work with S. Bacallado and C. Sabo.
The field of evolutionary genetics is profoundly rooted in stochastic mathematical theory and since several years the theory has been extended to model the evolution of full genomes. Indeed, large amount of full genome data are becoming available for human but also non-model organisms. Several methods based on the Sequential Markovian coalescent (SMC) have been developed to use sequence data to uncover population demographic history. While these methods can be applied in principle to all possible species, they have been developed based on the human biological characteristics and have main limitations such as assuming sexual reproduction and no overlap of generations. However, in many plants, invertebrates, fungi and other taxa, these assumptions are often violated due to different ecological and life history traits, such as self-fertilization, long term dormant structures (seed or egg-banking) or large variance in offspring production. I will first describe a novel SMC-based method which we developed to infer 1) the rates of seed/egg-bank and of self-fertilization, and 2) the populations' past demographic history. We also apply our method to Arabidopsis thaliana, Daphnia pulex and to detect seed banking in different populations of the wild tomato species Solanum chilense. Finally, I will show that we can even extend this method to detect and to date the changes of selfing / seed banking in time. I will conclude by discussing more general class of mathematical stochastic models and methods which should be developed for applicability to all species.
Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816
Link: https://lmu-munich.zoom.us/j/581970248?pwd=SkdUZS8yWE53Ri9MVnBBMzMrb3RrQT09
Meeting-ID: 581 970 248, Kenncode: 062401
We prove that the speed of a $\lambda$-biased random walk on a supercritical Galton-Watson tree (with/without leaves) is differentiable for $\lambda$ such that the walk is ballistic and obeys a central limit theorem. We also give an expression of the derivative using a certain 2-dimensional Gaussian random variable, which naturally arise as limits of functionals of a biased random walk. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres. In particular, an important role is played by moment estimates of regeneration times, which are locally uniform in $\lambda$. This talk is based on a joint work with Adam Bowditch (University College Dublin).
The deterministic and fractional approaches can be used for in silico research to formalize the bacterial communications in terms of reaction-diffusion mathematical models. The current study continues more in depth the development of reaction-diffusion models of bacterial quorum sensing with a focus on the following directions. The use of external enzymes underlies an alternative way of reducing communication in pathogenic bacteria that may leads to a loss of pathogeneity. We propose an optimal control problem for bacterial quorum sensing degrading under the impact of external enzymes. The problem is to find the minimum of the objective functional that permits one to reduce the signaling molecules and simultaneously to limit the total amount of enzymes. This approach allows the valid strategy of enzyme impact to be specified. The second direction is presented by the time-fractional diffusion-wave modification of the quorum sensing model as a generalization of the classical model in order to analyze different dynamical regimes of the biological system. The study is aimed at developing numerical techniques to solve the time-fractional diffusion-wave problem with application to bacterial communication processes.
Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816
Roundtable jointly organized with the Institute of Mathematics and Applications (IUMA), University of Zaragoza, Spain To register for the next event, please fill this form: https://forms.gle/LMzC9QGkkNmCT4nE6
Networks everywhere Prof. Ernesto Estrada
Abstract The use of networks to represent the "skeleton" of complex systems is ubiquitous nowadays across the sciences. A network represents the entities of the system as nodes and the interrelation among them are represented as edges interconnecting the nodes. These networks account for a variety of systems at different size-scales ranging from social, infrastructural and ecological to cellular and molecular systems in biology.
In this talk I will introduce the topic by illustrating a collection of problems and their solutions from a mathematical perspective. They include, for instance, the problem of representation of different kinds of systems, particularly ecological ones, and the use of different kinds of networks. Another structural problem is related to the characterization of the "importance" of nodes in a network, know as node centrality.
I will illustrate the problem on the basis of detecting essential proteins in a proteome. I will mention the problem of network robustness to random failures and intentional attacks, particularly in infrastructural systems, and will finish this section with the analysis of communication in networks illustrated by the study of brain connectome. Other problems are related to the dynamics on networks. I will give some examples of synchronization in biological systems, diffusion on social networks and epidemics propagation at different scales, all illustrated by real-world examples.
The Mathematics of the Heart Prof. Esther Pueyo
Abstract Cardiovascular diseases represent the first cause of death in Europe, accounting for 45% of all deaths. The way the cardiovascular system, and particularly the heart, have been investigated so far is remarkably changing. Mathematics, together with other disciplines like physics and engineering, are ever more being used to help in the prevention, diagnosis and treatment of cardiovascular diseases.
In this talk, I will describe how mathematics can be key to understand the function of the heart. I will show examples where mathematical modeling and numerical simulation have been combined with clinical and experimental research to derive novel tools that can be used by cardiologists to improve medical decision-making. Additionally, I will show how mathematical methods can be integrated into the construction of personalized digital hearts to have virtual replicas where to test novel therapies or to be used for the prediction of abnormal cardiac behavior.
Traffic Guard: two different mathematical views of a public / private successful collaboration project Prof. Ruben Vigara Prof. Ruben Vigara
Abstract Román Guerra is Artificial Vision Manager in Lector Vision (LV), a Spanish company that applies Computer Vision (CV) algorithms worldwide to Intelligent Transport Systems (ITS). Rubén Vigara, from Universidad de Zaragoza (UNIZAR), is an expert in low-dimensional topology, a research field far away from daily life problems. Since 2015 both mathematicians have collaborated in a successful research project in which UNIZAR team developed and coded CV algorithms for new LV products. We present this project from their two different viewpoints.
Mathematics of Porous Media Prof. Carmen Rodrigo
Abstract We are interested in problems related to fluid flow through deformable and/or fractured porous media. These problems appear in many areas of application such as geothermal energy extraction, petroleum engineering, CO2 storage, hydraulic fracturing or cancer research, among many others.
The numerical simulation of this type of problems has become a topic of increasing importance, and with this purpose the study of appropriate discretization techniques and the design of efficient solution methods have to be investigated. Robust discretizations with respect to the physical parameters are needed for this type of problems to obtain reliable numerical solutions. Another important aspect in the numerical simulation of these problems deals with the efficient solution of the large systems of algebraic equations obtained after discretization, since this is the most consuming part when real simulations are performed.
‘Maths in Society’ is a lecture series started by the graduate students of the Mathematics Department at the Technical University of Munich. We are conducting our first event on 12th May 2021 (Wednesday) at 18:00 CET. A lecture on ‘The Future of Mathematics in the Age of Artificial Intelligence’ will be delivered by Prof Gitta Kutyniok (from LMU Munich).
Abstract: We currently witness the impressive success of artificial intelligence (AI) in real-world applications, ranging from science to public life. Such type of approaches often outperforms classical model-based and mathematically founded approaches. At the same time, AI methods such as deep learning still lack a profound theoretical understanding, sometimes even being referred to as "alchemy". This poses an enormous challenge to, but also a tremendous chance for mathematics as a field.
The goal of this lecture is to first provide an introduction into this research area. We will then focus on two aspects. We first ask: What is the future of areas such as inverse problems or partial differential equations, whose change can already by now be observed, and why are hybrid methods such important? We will discuss this also based on several examples. Second, we will show to which extent mathematics can contribute to this new world of artificial intelligence, and how in turn this might also change mathematics.
To register for the event, please fill this form: https://forms.gle/LMzC9QGkkNmCT4nE6
For more information, please check out this website: https://www.ma.tum.de/en/news-events/studies-information/maths-in-society.html
Link: https://lmu-munich.zoom.us/j/581970248?pwd=SkdUZS8yWE53Ri9MVnBBMzMrb3RrQT09
Meeting-ID: 581 970 248, Kenncode: 062401
A linear competition process is a continuous time Markov chain defined as follows. The process has N (N\ge 1) non-negative integer components. Each component increases with the linear birth rate or decreases with a rate given by some linear function of other components. A zero value is an absorbing state for each component: should a component become zero ("extinct"), it would stay zero for good. For all possible interactions we show that a.s. eventually only a (possibly, random) subset of non-interacting components can survive. A similar result holds for the relevant generalized Pólya urn model with removals.(Based on a joint work with Serguei Popov and Vadim Shcherbakov.)