Yesterday I made a preliminary program for the first two months

of the masterclass *non-commutative geometry*. It is likely that

the program will still undergo changes as at the moment I included only

the mini-courses given by Bernhard

Keller and Markus Reineke but several other people have

already agreed to come and give a talk. For example, Jacques Alev (Reims),

Tom Lenagan (Edinburgh),

Shahn Majid (London),

Giovanna Carnovale (Padua) among others. And in

may, Fred assures me, Maxim Kontsevich will give a couple of talks.

As for the contents of the two courses I will be

teaching I changed my mind slightly. The course *non-commutative
geometry* I teach jointly with Markus Reineke and making the program

I realized that I have to teach the full 22 hours before he will start

his mini-course in the week of March 15-19 to explain the

*few*

thingshe needs, like :

things

To derive all the

counting of points formulas, I only need from your course:

–

the definition of formally smooth algebras basic properties, like

being

hereditary

– the definition of the component

semigroup

– the fact that dim Hom-dim Ext is constant along

components. This I need

even over finite fields $F_q$, but I

went through your proof in “One quiver”,

and it works. The

key fact is that even over $F_q$, the infinitesimal lifting

property implies smoothness in the sense Dimension of variety =

dimension of

(schematic) tangent space in any $F_q$-valued

point. But I think it’s fine for

the students if you do all

this over C, and I’ll only sketch the (few)

modifications for

algebras over $F_q$.

So my plan is to do all of

this first and leave the (to me) interesting problem of trying to

classify *formally smooth algebras* birationally to the second

course *projects in non-commutative geometry* which fits the title

as a lot of things still need to be done. The previous idea to give in

that course applications of non-commutative orders to the resolution of

singularities (in particular of quotient singularities) as very roughly

explained in my three talks on non-commutative geometry@n I now

propose to relegate to the *friday afternoon seminar*. I’ll be

happy to give more explanations on all this (in particular more

background on central simple algebras and the theory of (maximal)

orders) if other people work through the main part of the paper in the

seminar. In fact, all (other) suggestions for seminar-talks are welcome

: just tell me in person or post a comment to this post.

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