Related papers: IMPANGA lecture notes on log canonical thresholds
The paper describes behavior of log-inflection points of curves in $(\mathbb{C}^*)^2$ under passing to the tropical limit. We show that such points accumulate by pairs at the midpoints of bounded edges in the limiting tropical curve.…
This represents a talk given at the International Conference for Basic Science, July 2025. We review the theory of canonical bases of quantum groups and its relation with the theory of total positivity.
These informal notes were prepared in connection with a lecture at a high school mathematics tournament, and provide an overview of some examples of metric spaces and a few of their basic properties.
These notes grew out of two lectures I have given on CAT(0) cube complexes. I've tried to keep the material elementary and self-contained in order to keep the material easily accessible and to provide an elementary introduction on the topic…
The questions I have been asked during the 5th International School on Field Theory and Gravitation, have compelled me to give an account of the premises that I consider important for a beginner's approach to Loop Quantum Gravity. After a…
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is…
We study a pair consisting of a smooth variety over a field of positive characteristic and a multi-ideal with a real exponent. We prove the finiteness of the set of minimal log discrepancies for a fixed exponent if the dimension is less…
These notes are based on lectures given at the Les Houches Summer School in 2011, which was centered on the general topic "Theoretical Physics to face the challenge of LHC". In these lectures I reviewed a number of topics in the field of…
These notes are very informal notes on the Langlands program. I had some pleasure in daring to ask colleagues to explain to me the importance of some of the recent results on Langlands program, so I thought I will record (to the best of my…
These lecture notes provide a relatively self-contained introduction to field theoretic methods employed in the study of classical and quantum phase transitions.
These third-year lecture notes are designed for a 1-semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second-year subjects: multivariable calculus, introduction to…
Gaussian latent tree models, or more generally, Gaussian latent forest models have Fisher-information matrices that become singular along interesting submodels, namely, models that correspond to subforests. For these singularities, we…
Scaling limits of critical percolation models show major differences between low and high dimensional models. The article discusses the formulation of the continuum limit for the former case. A mathematical framework is proposed for the…
Introductory lectures notes on cosmology, aimed at master students in physics and based on a course held at Milano University (in Italian).
These are notes based on a course that I gave at the University of Chicago in Fall 2016 on "Loop measures and the loop-erased random walk." This is not intended to be a comprehensive view but rather a personal selection of some key ideas…
These are lecture notes on scale calculus and M-polyfolds written for a graduate course at UNICAMP March-June 2018 and an advanced mini-course given during the biannual meeting of Brazilian mathematicians, CBM-32, at IMPA in August 2019.
The present technical notes offer a brief summary of the essential points of electromagnetism at the undergraduate physics level. Some problems are presented at the end of each section; those with solutions are marked with an asterisk.
We present a conceptually clear introduction to quantum theory, deriving the theory from scratch from the point of view of quantum information. Different subsets of these lectures were taught to a wide variety of audiences, including…
An introductory lectures on cosmology at ITEP Winter School for students specializing in particle physics are presented. Many important subjects are not covered because of lack of time and space but hopefully the lectures may serve as a…
These are lecture notes compiled for a short lecture series at the 2023 Condensed Matter Summer School at the University of Minnesota. They are designed to be conversational and fun, and not to take the place of review articles that do a…