Advisement on 4-manifolds: please suggest

[WWGD]

Hi, everyone:
I apologize if this is out of place. Please let me know if so, and
feel free to remove.
I am trying to get into the world of 4-manifolds. I was wondering how to
get someone to help me get started. I have taken classes in diff. geom.
and hyperbolic geometry, and my analysis , topology, functional analysis
are pretty good (all grad-PHD level). I understand the topic is vast and
complex-- therefore the need for advisement. I am hoping towork more on
the analytic side (i.e, diff. manifolds) , maybe "Functional-Analytic" than
the purely topological one, tho I understand there will be heavy topological
aspects nonetheless. Could someone please suggest how to get advisement?
(my school is not exactly what you'd call the nurturing or supportive type,
and my request for help there was largely ignored, tho I was told--at least so
I understood-- that if I could do the work on my own, that I could go on
and work in the area.).
I understand the topic is complex and very open-ended, so that I do not
expect short -term results. I am also still pretty motivated to put in 8+
hours/day 7 days /week. (I finally found a good energy drink to help me with
it:) )

Thanks for your suggestions.

 

[WWGD]

Sorry, I think I may not hav ebeen clear with the frase :help getting started.
I hav ejust heard the area is very vast and open -ended and that getting results
is difficult. I would just like some ideas on how to approach the topic, understanding
that it is up to me tyo do the work.

Thanks Again.

 

[mathwonk]

most interesting and acessible 4 manifolds are algebraic complex surfaces, so one possibility is to begin the study of complex surfaces, e.g. in the books of beauville

https://www.amazon.com/dp/0521498422/?tag=pfamazon01-20

, or of barth, peters, and van de ven.

https://www.amazon.com/dp/0387121722/?tag=pfamazon01-20

or look up the works of simon donaldson, or those of john morgan and robert friedman.

unfortunately all but the book of beauville are very pricey. that makes beauville's wonderful book a great bargain, but it only discusses complex surfaces and not the new theory of diff'ble 4 manifolds due to donaldson et al...

look up the notion of gauge theory, and works by mrovka.

 

[WWGD]

Thanks for the advice, mathwonk. I am getting started right away.

 

[WWGD]

Follow-up

Mathwonk:

I was wondering if you know of any proffessor working in the area of
smooth/smoothable 4-manifolds. No one in my school is working on it
at the moment, but it is acceptable to have a prof. outside of the system
be the advisor. Thanks for any suggestions, recommendations.

 

[mathwonk]

where are you? what about robert friedman of columbia?

 

[WWGD]

Thanks, Wonk:
I am at CUNY . I will contact Robert Friedman. Ozsvath from Columbia said he was not
working on the area , and J. Morgan did not reply.

 

[mathwonk]

Alexandru Scorpan of Univ of Florida at Gainesville has a well reviewed book on the topic, and Gordana Matic of UGA, may be willing as well, but they are not in your geographic area.

 

[WWGD]

4-mfld. Follow-up

Thanks for helping a Yankee, there, Wonk. I will contact G.Matic ( and her relative
Otto ) .
I wrote to Scorpan, Casson, others, see if he could suggest someone, tho with
no answer yet. Friedman is not working in the area anymore.
Maybe it's the Easter thing and everyone is out or on break.
Please let me know of anyone else in the country, even if not in the northeast,
who may be working in the area, since they may be able to refer me to someone here.

Thanks.

 

[mathwonk]

and that is pronounced "mah - titch".

 

[WWGD]

Sorry. I'll try to come up with another bad joke.