Academic year 2023/204:

Semester 1: MM course on  Statistical Mechanics (together with C. Spitoni), Tuesdays 11-13 BBG 005

Content:

Statistical mechanics is a branch of physics aiming at understanding the laws of the macroscopic behaviour of systems that are composed of many microscopic components (e.g. solids, gases or liquids). Critical phenomena such as phase transitions involve a drastic change in the macroscopic state by tuning some model parameters.

Those principles are extremely universal and are used to study many component systems and their behaviour beyond physics, namely in chemistry, biology or complex systems.

 In this course we aim at giving the mathematical foundation for studying many component systems on the lattice and in the continuum space. Moreover, we would like to  motivate the theory of Gibbs measures starting from basic principles in classical mechanics.

In particular we will treat:

• Gibbs ensembles and thermodynamic limits.
• Infinite volume Gibbs measures and DLR formalism in the lattice and in the continuum.
• Variational characterisation.
• Peierls contour method.
• Cluster expansion and polymer models.
  Piragov Sinai theory and its application to the Blume-Capel model.
• Particle Systems in Continuum: Gibbsian formalism for superstable interactions.
  Example: Lebowitz-Mazel-Presutti model

For more information please contact me or C. Spitoni.
 

Academic year 2022/2023

1) Semester 1: Measure theoretic probability (MM), together with Dalia Terhesiu, Wednesdays 2-4 p.m. in UU campus

Content:

In this course we discuss the measure- and Lebesgue integration theory that is relevant in probability theory.
We introduce some vital concepts in probability theory, such as conditional expectations, the Radon-Nikodym theorem, martingale convergence theorems, characteristic functions and why they are characteristic, the Brownian motion.
Furthermore we provide rigorous proofs for two central convergence theorems in probability: the Strong Law of Large Numbers and the Central Limit Theorem.

Academic year 2021/2022

1) Semester 1: Measure theoretic probability (MM), together with Dalia Terhesiu, Wednesdays 2-4 p.m. on Zoom

Content:

In this course we discuss the measure- and Lebesgue integration theory that is relevant in probability theory.
We introduce some vital concepts in probability theory, such as conditional expectations, the Radon-Nikodym theorem, martingale convergence theorems, characteristic functions and why they are characteristic, the Brownian motion.
Furthermore we provide rigorous proofs for two central convergence theorems in probability: the Strong Law of Large Numbers and the Central Limit Theorem.

2) Semester 2: USCIMAT22 Probability and Networks at UCU, Tuesdays 9-11 and Thursdays 13.30-15.30

Content:

Probability theory is fundamental in all mathematical and data-driven sciences. This course
introduces the core concepts of the field that are useful for students across the natural and social
sciences. The course balances interesting applications and examples with a depth of conceptual and
theoretical understanding that goes beyond merely pragmatic methods and skills. Topics include
discrete and continuous probability distributions, combinatorics, conditional probability,
computations with random variables, expectation and variance, the law of large numbers, and the
central limit theorem.
The interdisciplinary study of networks is recently receiving much attention. It reveals unexpected
connections between otherwise separate fields such as sociology, ecology, economics, cognitive
neuroscience and computer science. Network thinking provides new ways to understand our strongly
connected world. This approach has generated new tools for the analysis and understanding of
complex systems in both the social and natural world. This part of the course will discuss how to
describe and quantify networks, provide means to analyse network data (using among others graph
theory) and explain how to build and analyse concrete mathematical models, e.g. of the spread of
diseases or of financial crises.

Academic year 2020/2021

1) Semester 2: (WISM569) Master seminar on Mathematical Statistical Mechanics (together with C. Spitoni), Mondays 9-11 on Teams

Content:

Statistical mechanics is a branch of physics aiming at understanding the laws of the macroscopic behaviour of systems that are composed of many microscopic components (e.g. solids, gases or liquids). Critical phenomena such as phase transitions involve a drastic change in the macroscopic state by tuning some model parameters.

Those principles are extremely universal and are used to study many component systems and their behaviour beyond physics, namely in chemistry, biology or complex systems.

 In this course we aim at giving the mathematical foundation for studying many component systems on the lattice and in the continuum space. Moreover, we would like to  motivate the theory of Gibbs measures starting from basic principles in classical mechanics.

In particular we will treat:

• Gibbs ensembles and thermodynamic limits.
• Infinite volume Gibbs measures and DLR formalism in the lattice and in the continuum. Variational characterisation.
• Peierls contour method.Cluster expansion and polymer models.
• Piragov Sinai theory and its application to the Blume-Capel model.Particle Systems in Continuum: Gibbsian formalism for superstable interactions.
• Example: Lebowitz-Mazel-Presutti model.
• For more information please contact me or C. Spitoni.

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2) Semester 2: (USCIMAT22) Mathematical modelling of networks, Tuesdays 9-11 and Thursdays 1.30-3.30

Content:

The interdisciplinary study of networks is recently receiving much attention. It reveals unexpected connections between otherwise separate fields such as sociology, ecology, economics, cognitive neuroscience and computer science. Network thinking provides new ways to understand our strongly connected world. This approach has generated new tools for the analysis and understanding of complex systems in both the social and natural world.

The course will discuss how to describe and quantify networks, provide means to analyse network data (using among others graph theory) and explain how to build and analyse concrete mathematical models, e.g. of the spread of diseases or of financial crises.

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3) Semester 2: (WISB263) Mathematical statistics

Content:

This course gives an introduction to the mathematical theory of statistics. In statistics, data are analysed by assuming they are the outcome of a (partly) unknown stochastic model. Mathematical statistics gives methods to extract information about this stochastic model in the most efficient way. There will be emphasis on fundamental notions and methods of statistics, theory of parametric and non-parametric estimators and limit theorems.

Subjects in particular are:

• statistical models
• multivariate normal distribution
• concepts of convergence
• sufficient and complete statistics
• confidence-areas
• Fisher information
• Cramer-Rao lower bound
• statistical tests
• likelihood ratio test