Nigel Higson: A rapid tour through noncommutative geometry

Nigel Higson: A rapid tour through noncommutative geometry - from integral equations and the spectral theorem to the index theorem and beyond
Nigel Higson is professor for mathematics at Penn State University in University Park (Pennsylvania, USA).
Hilbert’s theory of integral equations, now more than one hundred years old, is a famously successful amalgam of matrix algebra and analysis. One way to look at Alain Connes’ much newer noncommutative geometry is to see it as an evolution of the theory of integral equations that incorporates geometric ideas into Hilbert’s work, and I shall try to develop this perspective in my lecture. I will focus on just one construction, that of Connes’ tangent groupoid, but even so there will be opportunities to glimpse at Weyl’s asymptotic law, the Atiyah-Singer index theorem, and recent work of Bismut on orbital integrals and the hypoelliptic Laplacian with v=0.
This talk was part of the conference on »Bivariant K-theory in Geometry and Physics«
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1. Lead 00:00:00
2. Hilbert's spectral theorem 00:00:08
3. Integral operators as an algebra 00:09:06
4. Principal symbol 00:19:53
5. Two examples 00:29:29
6. The continuous field of C*-algebras 00:37:48
7. Elliptic operators 00:46:28
8. Bismut's theory of the hypoelliptic Laplacian 00:53:24
9. Questions from the audience 00:58:32

[@giuliocasa1304]

This professor is really a great mathematician! And it's a pleasure to listen while he's explaining the most advanced math.

[@Christopher-Lloyd_SIMON]

Hello, does someone have an idea of where to find the so refred "Emil's video's" at about 30min of Nigel Higson's very good talk ?

[@herodeonhopeful]

Maybe something Emil Prodan had shown during the conference?

[@NoNTr1v1aL]

Amazing video!

[@user-ub1qi3ye5k]

Are you Arab

[@SystemsMedicine]

It’s interesting to hear the language of algebraic topology/homotopy theory diffusing into grad student physics lectures over the years. [I know it’s a theory institute… but still…]

[@UnforsakenXII]

: D

[@slbumkim2925]

Being has a tendency to 'return' to clusters
='the nature of solidarity' -a desire for empathy -(Wave)-(yin)
and also,
Being has a tendency to 'exist' as individuals
='the nature of self-expension' -a desire for breed-(Particle)-(Yang)
Likewise, humans have two elements.
We must realize that we all have both left and right elements
-Solidarity and Self Reliance-
No one has only one element.
so 'Sum' derived from 'two poles' , (thesis, antithesis, synthesis)
To develop intellect and ethics by harmonizing the two,
It is good to realize it and balance it properly
But A few people polarized the crowd(political partisanship)
without balancing themselves.
And They stole only the sum, only the synthesis from the triangle composition.
Now We all have to get out of this deceptive situation.
This is not the time for us to hate each other.
We have to track down those who have been manipulating us.