What We Are Reading Today: Introductory Lectures on Equivariant Cohomology

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Updated 31 March 2020

What We Are Reading Today: Introductory Lectures on Equivariant Cohomology

Author: Loring W. Tu

This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology.
Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah-Bott and Berline-Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics.
Assuming readers have taken one semester of manifold theory and a year of algebraic topology, Loring Tu begins with the topological construction of equivariant cohomology.

 


What We Are Reading Today: Introduction to Social Neuroscience

Updated 2 min 56 sec ago

What We Are Reading Today: Introduction to Social Neuroscience

Edited by Stephanie Cacioppo and John T. Cacioppo

Humans, like many other animals, are a highly social species. But how do our biological systems implement social behaviors, and how do these processes shape the brain and biology? Spanning multiple disciplines, Introduction to Social Neuroscience seeks to engage students and scholars alike in exploring the effects of the brain’s perceived connections with others. 

This wide-ranging textbook provides a quintessential foundation for comprehending the psychological, neural, hormonal, cellular, and genomic mechanisms underlying such varied social processes as loneliness, empathy, theory-of-mind, trust, and cooperation.

Stephanie and John Cacioppo posit that our brain is our main social organ. They show how the same objective relationship can be perceived as friendly or threatening depending on the mental states of the individuals involved in that relationship. They present exercises and evidence-based findings readers can put into practice to better understand the neural roots of the social brain and the cognitive and health implications of a dysfunctional social brain. 

This textbook’s distinctive features include the integration of human and animal studies, clinical cases from medicine, multilevel analyses of topics from genes to societies, and a variety of methodologies.