True that this chapter does not get much beyond definitions this reviewer for instance misses some words on flows , but anyway it offers a good introductory view of many things. Lie groups for instance. The third chapter would alone justify any previous omisions. This is a neat, motivating and accomplished little first course on de Rham Cohomology. A beauty. The last chapter on Differential Geometry is truly nice. It gives an account on curves form a higher viewpoint that is a bonus in form and content even for those who already know the matter.

Then the author turns to curves in immersed manifolds: first fundamental form and internal geometry.

Next, exterior geometry and the second fundamental form. Here the subject are hypersurfaces, enlarging the usual restriction to surfaces in 3-space. In the end, we get to Gauss Egregium theorem, nothing better in closing. It is clear that the book comes from the experience in the class room, and a good one it seems.

We must thank the author for sharing it with us. Quite efficiently it gets to several deep results avoiding diversions, still giving a sense of pace friendly to the reader. Otherwise it takes matters till the accesible point and proposes literature for further progress. In the last few decades a beautiful new class of non-compact topological groups has been constructed. These are known as Kac-Moody groups and they share most of the structure that compact Lie groups admit. Kac-Moody groups have been shown to be relevant in mathematical physics and further investigation by several mathematicians including the speaker seems to suggest that Kac-Moody groups are surprisingly amenable to homotopical techniques.

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This makes these groups prime candidates for study from the standpoint of homotopy theory. In this talk I'll describe how this perspective on stable homotopy has emerged, and how using algebraic geometry is providing us with a library of new objects in homotopy theory. I will give an introduction to the theory of representation stability, through the lens of its applications in homological stability.

I will first discuss some fairly classical homological stability phenomena: spaces of 0-manifolds e. Riemann's moduli space. I will discuss recent joint work with Oscar Randal-Williams, aimed at calculating the cohomology of BDiff W and related spaces, where W is a smooth 2n-dimensional manifold, Diff W is the topological group of diffeomorphisms of W, and BDiff W is its classifying space.

Surprisingly, the cohomology ring turns out to be partially independent of W through a range of degrees homological stability.

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In this talk, I will discuss how infinite loop spaces can be used to describe the cohomology in this stable range. In this survey talk, I will shall the role that loop groups have played in the construction of topological gauge theories in dimensions 2 and possibly 3.

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## Introduction to Topology

Visa Information. Location MSRI: Simons Auditorium Video Abstract The talk will explain what an operad is, give some examples, and illustrate how operads come up in various mathematical settings. Supplements Notes 5. Supplements Shipley Notes 4. Supplements Location MSRI: Simons Auditorium Video Abstract In this talk, I will given an overview of recent work on formulating the structural properties of algebraic K-theory using the framework for studying homotopical categories provided by the development of higher category theory Supplements Blumberg notes 7.

Location -- Video Abstract -- Supplements Douglas notes 6. Location MSRI: Simons Auditorium Video Abstract I will give an overview of the rich structure in the stable homotopy groups of spheres, both giving a summary of what we do know, and what we don't know. Supplements Isaken notes 5.

## Review: Introduction to Geometry and Topology | EMS

Supplements Westerland notes 3. Supplements Local structure of groups and of their classifying spaces This makes these groups prime candidates for study from the standpoint of homotopy theory Supplements Kitchloo notes 6. Location MSRI: Simons Auditorium Video Abstract : Topological modular forms and its generalizations are objects in stable homotopy that realize a connection to 1-dimensional formal group laws.

In this talk I'll describe how this perspective on stable homotopy has emerged, and how using algebraic geometry is providing us with a library of new objects in homotopy theory Supplements Lawson notes 6. Location MSRI: Simons Auditorium Video Abstract I will give an introduction to the theory of representation stability, through the lens of its applications in homological stability.

Supplements Church notes 7. Supplements R-W Notes 5. Supplements Galatius notes 5. Location MSRI: Simons Auditorium Video Abstract In this survey talk, I will shall the role that loop groups have played in the construction of topological gauge theories in dimensions 2 and possibly 3. Supplements Teleman notes Algebraic Topology. Dwyer notes 5. Goodwillie's calculus of functors Michael Ching Amherst College. Supplements -- Show Detail. Chromatic redshift John Rognes University of Oslo.

## Introductory Topology: Exercises and Solutions

Models for homotopical higher categories Julie Bergner University of Virginia. Computations in the stable homotopy groups of spheres Mark Behrens Massachusetts Institute of Technology. Oliver notes 6. Equivariant homotopy and localization Michael Hopkins Harvard University.