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…
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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.
Introduces statistical optimal transport for probabilistic lectures.
problem No specific problem stated; focuses on introduction.
method Lecture-based introduction to statistical optimal transport.
result Provides an introduction to statistical optimal transport.
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 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…
The existence theorem for mapping cylinder neighborhoods is discussed as a prototypical example of controlled topology and its applications. The first of a projected series developed from lectures at the Summer School on High-Dimensional Topology, Trieste Italy 2001
Lectures on linearized Kapustin-Witten equations on half-line.
problem Analyzing differential operator from Kapustin-Witten equations.
method General theorems of R. Mazzeo and E. Witten applied.
result Instances of asymptotic solutions found.
The present article surveys some mathematical aspects of the BCOV holomorphic anomaly equations introduced by Bershadsky, Cecotti, Ooguri and Vafa. It grew from a series of lectures the authors gave at the Fields Institute in the Thematic Program of Calabi-Yau Varieties in the fall of 2013.
These notes are based on a series of five lectures given at the 2009 Villa de Leyva Summer School on Geometric and Topological Methods for Quantum Field Theory. The purpose of the lectures was to give an introduction to differential-geometric methods in the study of holomorphic vector bundles on a compact connected Rie…
This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants and Floer homol…
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…
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.
The source of these notes is a series of lectures given at the CIMPA's summer school "Recent Topics in Geometric Analysis".
This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with deg…
Discussing curvature flows and their applications.
problem Analyzing expanding curvature flows.
method Classical aspects of expanding curvature flows.
result First applications of curvature flows.
Lectures on surface evolution through singularities.
problem Analyzing the mean curvature flow of surfaces and their singularities.
method Analysis of neck and conical singularities, using monotonicity formulas, epsilon-regularity, weak solutions, and blowup techniques.
result Unique evolution through neck singularities, nonuniqueness through conical singularities.
This is a survey of recent results on manifolds with positive curvature from a series of lecture given in Guanajuato, Mexico in 2010. It also contains some hitsorical comments.
This lecture notes are an expanded version of the course given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher …
This preprint was split in two and became the first two parts of a four-part series (arXiv:1405.1956, arXiv:1405:1955, and two in preparation). The remaining relevance of this preprint is due to the series of videotaped lectures (wClips) that are linked within.
Topics in polar actionsmath.DG
Lectures on polar actions and their properties in Riemannian geometry.
problem Characterizing polar actions and understanding their properties.
method Analyzing isometric actions on Riemannian manifolds, using normal slice theorem and principal orbit type theorem.
result Characterization of polar actions in terms of integrability of the distribution of normal spaces to the principal orbits.
The aim of this short lecture series is to expose the students to the beautiful theory of lattices by, on one hand, demonstrating various basic ideas that appear in this theory and, on the other hand, formulating some of the celebrated results which, in particular, shows some connections to other fields of mathematics.…
Explains how knots relate to 4D shapes.
problem Understanding 4D shapes through knot theory.
method Combines knot theory with 4D manifold topology.
result Connects 4D shapes to knot theory and other geometries.
This article arose from a series of three lectures given at the Banach Center, Warsaw, during period of 24 March to 13 April, 2003. Morse functions are useful tool in revealing the geometric formation of its domain manifolds . They define the handle decompositions of from which the additive homologies $H_{\ast}(…
Lectures on symplectic aspects of surface degenerations at KIAS.
problem Exploring symplectic structures in surface degenerations.
method Expository account of symplectic aspects of cyclic quotient surface singularities.
result Discussion of symplectic structures in surface degenerations.
Lecture notes on singular foliations, smooth and holomorphic.
problem Understanding singular foliations in geometry.
method Review of foundations, recent tools from non-commutative geometry, and homotopic notions.
result Introduction of various homotopic notions and open questions.
Overview of high-dimensional time series regression methods.
problem Estimation and inference with high-dimensional time series data.
method Limit theory for high-dimensional dependent data, asymptotic theory for time series regression, statistical learning methods.
result Main limit theory results and asymptotic theory for high-dimensional time series regression.
In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recen…
These notes are based on a series of lectures given by the first author at the school of `Poisson 2010', held at IMPA, Rio de Janeiro. They contain an exposition of the theory of super- and graded manifolds, cohomological vector fields, graded symplectic structures, reduction and the AKSZ-formalism.
These notes are based on a lecture series given at the Park City Math Institute in the summer of 2013. The notes are intended as a leisurely introduction to the Kähler-Ricci flow on compact Kähler manifolds, aimed at graduate students with some background in differential geometry.
In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of …
Lecture notes on Finslerian geometry.
problem No specific problem stated; covers Finslerian geometry.
method Lecture notes.
result No specific key result mentioned.
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.
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 …
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.
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.
Explains Bernstein theorems for various geometric PDEs.
problem Bernstein problem for minimal surface, Monge-Ampère, and special Lagrangian equations.
method Expository review of existing theorems and systems.
result Discussion of Bernstein theorems for different geometric PDEs.
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.
Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.
problem Optimal functional inequalities and their stability.
method Compactness theorems, characterization of optimizers, and quantitative stability analysis.
result Characterization and stability of optimizers for Sobolev inequalities and their fractional generalizations.
Lectures on mean curvature flow and its related equations.
problem Singularity formation, nonuniqueness, and topological change in motion by mean curvature.
method Analyzes motion by mean curvature flow and related equations.
result Exploration of singularity formation, nonuniqueness, and topological change.
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.
Notes on Khovanov and knot Floer theories' stable homotopy types.
problem Understanding stable homotopy types in Khovanov and knot Floer theories.
method Introduction to Khovanov and knot Floer theories' stable homotopy types.
result Introduction of stable homotopy types in Khovanov and knot Floer theories.
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…
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.
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.
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.
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…
The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Ja…