[Mini-course]: Lectures on Capacities | Professor Daniel Spector, National Taiwan Normal University

 

Title:  Riemann and Lebesgue Integration

Abstract:  The Riemann integral is perfectly suited for consideration of volume, surface area, arc length, and integration of functions in classical analysis - when the sets in question are smooth and the functions in question continuous.  In this talk, we introduce these ideas and explain the progression from Riemann integration to Lebesgue integration, emphasizing in particular the powerful tools one obtains from this construction.

 

 

Title:  Capacitary Integration

Abstract:  The Lebesgue integral provides one with a satisfactory tool for many purposes in mathematical analysis.  Yet in the modeling of natural phenomena, with the introduction of partial differential equations, integrals which are not Lebesgue integral makes a prominent appearance - capacitary integrals.  In this talk we discuss this motivation for capacitary integration, with examples, explain the differences with Lebesgue integration, and show the usefulness of these non-standard objects.

 

 

Title: Capacitary Sobolev Inequalities and Applications

Abstract:  The study of capacities and Capacitary Sobolev Inequalities is now more than half a century old, and yet there are still a number of open research questions to investigate concerning them.  In this talk we discuss in more detail Capacitary Sobolev inequalities with an emphasis on a subject with the most recent activity - Capacitary Sobolev Inequalities around L1.  Open problems will be mentioned.