Lecture notes on Finslerian geometry.
problem No specific problem stated; covers Finslerian geometry.
method Lecture notes.
result No specific key result mentioned.
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Explains the Schwarz lemma in lecture notes.
problem None explicitly stated; focuses on explanation.
method Expository notes on the Schwarz lemma.
result Explains the Schwarz lemma.
Lecture notes on advanced geometry equations and subvarieties.
problem Understanding generalized Monge-Ampère equations and their subvarieties.
method Sketches results by Yau, Demailly-Paun, the author, and Datar-Pingali.
result Use these results to study the Hodge conjecture.
Lecture notes introduce differential geometry using sheaves and differential operators.
problem Exploring differential geometry concepts.
method Using sheaves, differential operators, and horizontal subbundles.
result Presented an approach to fundamental differential geometry structures.
These are lecture notes that are based on the lectures from a class I taught on the topic of Spectral Graph Methods at UC Berkeley during the Spring 2015 semester.
These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.
Lecture note for data science students on machine learning basics and advanced topics.
problem No specific problem stated; preparing students for advanced machine learning.
method Introduction of basic machine learning concepts and advanced topics.
result Students are prepared to study advanced machine learning topics.
These are lecture notes from a series of lectures at the SMF summer school on "Geometric and Quantum Topology in Dimension 3", June 2014. The focus is on Heegaard Floer homology from the perspective of sutured Floer homology.
Notes on Harmonic Mapsmath.DG
Notes on harmonic maps between manifolds, existence and regularity covered.
problem Existence and regularity of harmonic maps between Riemannian manifolds.
method Lecture-based approach covering harmonic maps, pluriharmonic maps, and related theorems.
result Coverage of existence and regularity of harmonic maps, including Siu-Sampson formula and Donaldson-Corlette theorem.
Lecture notes on Heegaard Floer homology for 3-manifolds and knots.
problem Understanding Heegaard Floer homology for 3-manifolds and knots.
method Explains Heegaard diagrams, Heegaard Floer homology, and knot Floer homology.
result Relationship between knot and 3-manifold invariants.
These lecture notes give an introduction to the Kahler-Ricci flow. They are based on lectures given by the authors at the conference "Analytic Aspects of Complex Algebraic Geometry", held at the Centre International de Rencontres Mathematiques in Luminy, in February 2011.
Lecture notes on optimization for machine learning, derived from a course at Princeton University and tutorials given in MLSS, Buenos Aires, as well as Simons Foundation, Berkeley.
Lecture notes on crystallography and discrete surfaces.
problem Mathematical modeling of crystal structures.
method Variational principle and discrete surface theory.
result Most symmetric crystal structures identified.
Lecture notes on mean curvature flow for beginners.
problem Existence and uniqueness of mean curvature flow solutions.
method Conditional existence and weak-strong uniqueness theory.
result Accessible introduction for graduate students.
Lecture notes on curves in complex projective plane from a topological viewpoint.
problem Understanding curves in complex projective plane from a topological perspective.
method Topological analysis of curves in complex projective plane.
result Curves in complex projective plane have unique topological properties.
These notes gather recent results on robust statistical learning theory. The goal is to stress the main principles underlying the construction and theoretical analysis of these estimators rather than provide an exhaustive account on this rapidly growing field. The notes are the basis of lectures given at the conference…
Lecture notes on advanced linear regression methods.
problem Understanding the properties of linear regression estimators in high dimensions.
method Proposition-proof exploration of least squares, ridgeless, ridge, and lasso estimators.
result Detailed analysis of the existence, uniqueness, relations, computation, and non-asymptotic properties of these estimators.
These notes grew out of a lecture series given at RIMS in the summer of 2001. The lecture series was aimed at a broad audience that included many graduate students. Its purpose lay in familiarizing the audience with the basics of 3-manifold theory and introducing some topics of current research. The first portion of th…
Lecture notes on kernel functions and Random Fourier Features.
problem Understanding and approximating kernel functions in machine learning.
method Mathematical background and proofs of concentration results.
result Estimation of error in Random Fourier Features approximation.
Advanced mathematical relativity course for math and physics students.
problem Mathematical foundations of general relativity.
method Comprehensive lecture notes covering advanced topics.
result Comprehensive coverage of mathematical relativity for advanced students.
These are lecture notes mainly aimed at graduate students on selected aspects of generalized geometry: in particular generalized complex and Kaehler structures and generalized holomorphic bundles. They are based on lectures given in March 2010 at the Chinese University of Hong Kong.
These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The…
These notes briefly summarize the lectures for the Summer School "Optimal transportation: Theory and applications" held by the second author in Grenoble during the week of June 22-26, 2009. Their goal is to describe some recent results on Brenier's variational models for incompressible Euler equation.
Lecture notes on Heegaard Floer homology for beginners.
problem Introducing Heegaard Floer homology to newcomers.
method Illustrates algebraic structures via grid homology, then defines Heegaard Floer homology geometrically.
result Defines and describes key properties of Heegaard Floer homology.
This is a set of lecture notes for a course given at the 2005 Summer School in Poisson Geometry held at ICTP-Trieste.
Notes for a short lecture series, covering exploded manifolds, the moduli stack of curves in exploded manifolds, and a tropical gluing formula for Gromov-Witten invariants: a gluing formula providing a degeneration formula for Gromov-Witten invariants in normal-crossing degenerations. I gave the original lecture series…
This is an expanded version of the lecture course the second author gave at Winterbraids VI in Lille in February 2016. Version 2: revision incorporating referee remarks.
Lecture Notes in Lie Groupsmath.DG
These lecture notes in Lie Groups are designed for a 1--semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This landmark theory of the 20th Century mathematics and physics gives a rigorous foundation to modern dynamics, as well as field and gauge theories in physics, engi…
These lecture notes are based on a mini-course given by the author at the sixth KAWA Winter School on March 23-26, 2015 at the Centro De Giorgi of Scuola Normale Superiore in Pisa. They provide an introduction to the study of the Kahler-Ricci flow on compact Kahler manifolds, and a detailed exposition of some recent de…
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
These are lecture notes from the Clay Mathematics Institute summer school ``Floer Homology, Gauge Theory, and Low Dimensional Topology'' Alfred Renyi Institute; www.claymath.org/programs/summer_school/2004/. The main goal of these notes is to sketch a proof of Giroux correspondence between open book decompositions of t…
This is an informal set of lecture notes on moduli spaces of curves based on a set of lectures given at the ICTP last summer. It begins at an elementary level and discusses the genus 1 case in detail. The notes then give an informal treatement of Teichmuller space and higher genus moduli spaces. The point of view is ge…
The materials accompany a lecture short course presented at the 2011 Park City Mathematics Institute, Graduate Summer School on Moduli Spaces of Riemann Surfaces. The lectures were part of/coordinated with an overall program, including lectures by Ursula Hamenstadt on Teichmueller Theory, Andy Putman on Mapping Class a…
This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at ETH Zurich and the Humboldt University Berlin in 2009/2010. The notes are still incomplete, but due to recent requests from readers, I've decided to make a presentable half-finished version available here.…
These are largely expanded notes from lectures on Higgs moduli and abelianisation given in Angers, France (2014) and Guaruja, Brazil (2015). Dedicated to Ugo Bruzzo on his 60-th birthday. Version 2: minor corrections.
These notes are an expanded version of an introductory lecture on contact geometry given at the 2001 Georgia Topology Conference. They are intended to present some of the "topological" aspects of three dimensional contact geometry.
Geometry of geodesicsmath.DG
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
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 for advanced bachelor students or early master students with little to no previ…
We present a short overview on the strongest variational formulation for gradient flows of geodesically -convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer…
These lecture notes, which were designed for the Summer School "Heegaard-Floer Homology and Khovanov Homology" in Marseilles, 29th May - 2nd June, 2006, provide an elementary introduction to Khovanov homology. The intended audience is graduate students with some minimal background in low-dimensional and algebraic topol…
Notes on symmetric spacesmath.DG
These are notes from a lecture course on symmetric spaces by the second author given at the University of Pittsburgh in the fall of 2010.
These are notes for four lectures on higher structures in M-theory as presented at workshops at the Erwin Schroedinger Institute and Tohoku University. The first lecture gives an overview of systems of multiple M5-branes and introduces the relevant mathematical structures underlying a local description of higher gauge …
These lecture notes review the topological string theory and its applications to mathematics and physics. They expand on material presented at the Takagi Lectures of the Mathematical Society of Japan on 21 June 2008 at Department of Mathematics, Kyoto University.
This survey is based on a series of five lectures, given May 3--7, 2010, at the Centre de Recerca Matematica, Barcelona. The goal of the lectures was to present aspects of the theory of foliation dynamical systems which have particular importance for the classification of foliations of compact manifolds. The lectures e…
Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.
problem Understanding non-Kähler complex threefolds and conifold transitions.
method Review of basics, description of topological features, survey of recent developments.
result Survey of recent developments on the geometrization of conifold transitions.
Discussing curvature flows and their applications.
problem Analyzing expanding curvature flows.
method Classical aspects of expanding curvature flows.
result First applications of curvature flows.
Lecture notes on link homologies and knotted surfaces, focusing on 4D obstructions.
problem Understanding knotted surfaces using link homology theories.
method Overview of link homology theories, introduction to Khovanov homology, and computational techniques.
result New insights into the homomorphisms assigned to link cobordisms for knotted surfaces.
Lecture Notes on Linear Neural Networks: A Tale of Optimization and Generalization in Deep Learningcs.LG
Lecture notes on linear neural networks for deep learning optimization and generalization.
problem Understanding optimization and generalization in deep learning models.
method Mathematical tools and dynamical systems theory.
result Potential of mathematical tools to enhance understanding of deep learning.