相关论文: Lecture notes on generalized Heegaard splittings

In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of…

This paper is an expanded version of a talk given at the Current Developments in Mathematics Conference last November (2002) on the work of Wilfred Schmid on periods of limits of Hodge structures. The paper begins with an exposition of the…

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.

These are extended notes for my talk at the ICMP 2003 in Lisbon. Our goal here is to demonstrate how natural and fundamental random partitions are from many different points of view. We discuss various natural measures on partitions, their…

We use thin position of Heegaard splittings to give a new proof of Haken's Lemma that a Heegaard surface of a reducible manifold is reducible and of Scharlemann's ``Strong Haken Theorem'': a Heegaard surface for a 3-manifold may be isotoped…

This chapter is based on a series of lectures that I gave at the National University of Singapore in April 2013. The notes survey the representation theory of the cyclotomic Hecke algebras of type A with an emphasis on understanding the KLR…

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…

In 2001, J. Hempel proved the existence of Heegaard splittings of arbitrarily high distance by using a high power of a pseudo-Anosov map as the gluing map between two handlebodies. We show that lower bounds on distance can also be obtained…

This text is the extended version of a talk given at the conference Geometry, Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the Observatoire de Paris. We explore the notion of solder (or soldering form) in differential…

Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly…

These brief lecture notes are intended mainly for undergraduate students in engineering or physics or mathematics who have met or will soon be meeting the Dirac delta function and some other objects related to it. These students might have…

We survey the development and status quo of a subject best described as "generic representation theory of finite dimensional algebras", which started taking shape in the early 1980s. Let $\Lambda$ be a finite dimensional algebra over an…

We generalize the definition of thin position of Scharlemann and Thompson for compact orientable 3-manifolds with torus boundary components and introduce $\alpha$-sloped generalized Heegaard splittings. We examine its relationship to…

These lecture notes are based on an introductory course given by the author at the summer school "Noncommutative Algebraic Geometry" at MSRI in June 2012. The emphasis throughout is on examples to illustrate the many different facets of…

We demonstrate how a 3-manifold, a Heegaard diagram, and a group presentation can each be interpreted as a pair of signed permutations in the symmetric group $S_d.$ We demonstrate the power of permutation data in programming and discuss an…

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…

These lecture notes give a very short introduction to coarsening phenomena and summarize some recent results in the field. They focus on three aspects: the super-universality hypothesis, the geometry of growing structures, and coarsening in…

A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is…

These are the lecture notes for a short course in topological string theory that I gave at Uppsala University in the fall of 2004. The notes are aimed at PhD students who have studied quantum field theory and general relativity, and who…

This paper has been withdrawn by the author, as the proof of Theorem 3.2 contains a flaw; subsequently, both it and Theorem 3.3 are not known to hold. The content of Section 5 has been improved and expanded upon in two separate papers. The…