Graduate Studies - Categorical Logic

[metasoarous]

Hi there.

I'm interested in going back to school for graduate studies and am particularly interested in Categorical Logic, that is, the application of Category Theory towards the study of Logic and Model Theory in particular (at least so far as my specific interests go). I'm also pretty interested in Large Cardinals.

What are some good schools to consider for this area of study? I know that Berkley has a really strong Foundations program which includes plenty of Logic and has some folks interested in Categorial Logic. Carnegie Mellon also seems a great option. Any other recommendations, either in the states or out?

Cheers

Chris

 

[aq1q]

Hi there.

I'm interested in going back to school for graduate studies and am particularly interested in Categorical Logic, that is, the application of Category Theory towards the study of Logic and Model Theory in particular (at least so far as my specific interests go). I'm also pretty interested in Large Cardinals.

What are some good schools to consider for this area of study? I know that Berkley has a really strong Foundations program which includes plenty of Logic and has some folks interested in Categorial Logic. Carnegie Mellon also seems a great option. Any other recommendations, either in the states or out?

Cheers

Chris

CMU! (there might be some bias to that of course.. :) )

 

[vici10]

You can have a look at University of Florida. There are couple of people there who is interested in large cardinals.
For the list of people interested in logic and set theory you can have a look at
http://www.math.ufl.edu/~jal/set_theory.html"

 

[metasoarous]

Thank you both.

CMU does sound pretty awesome. Their list of foundations faculty is impressive and I would have a lot of freedom there methinks.

That set theory folk directory is pretty amazing as well, if not a little overwhelming. Not sure if I'm cut out for life in Florida, but I'll give the program there a look.

Thanks again.

 

[JSuarez]

If you have a chance of being admitted in Berkeley, then go for it. Not only the program is very good, but there is also cooperation with Stanford, which is also very strong in Logic; besides, their department has a lot more variety of people with diverse interests (you may now think that a particular brand of logic is the thing for you, but this may change; it happens a lot and, if it does, you don't want to be trapped in a "narrow" department); furthermore, Berkeley is much more highly ranked than CMU, which greatly increases your chances after finishing the PhD (like it or not, placement is much more important than potential interest in the work of a particular person(s)).

 

[metasoarous]

Thanks for the feedback. Upon looking more carefully, I noted that CMU doesn't have any folks who list large cardinals as a focus. What you are saying about having a bunch of options resonates really strongly. I certainly don't want to box myself in.

 

[vici10]

Thanks for the feedback. Upon looking more carefully, I noted that CMU doesn't have any folks who list large cardinals as a focus.

In CMU, James Cummings deals with large cardinals.
http://www.math.cmu.edu/math/faculty/cummings.html"

 

[metasoarous]

Awesome! Good to know.

Large cardinals are so cool. Seat theory was one of the first "pure math" classes I took and learning about infinite set cardinality just blew my mind. Definitely part of what hooked me on math.

Cheers

Chris

 

[metasoarous]

... Not only the program is very good, but there is also cooperation with Stanford, which is also very strong in Logic...

I took a look at Stanford's site and it didn't have as much information about anything along the lines of a logic program. I did find a couple of professors who are into logic there, but that was it. Can you point me towards anything more on that front? It similarly might be good to apply to Standford to have access to Berkeley's mat department, no?

What about outside of the US? It seems like McGill, University of Ottawa and Dalhousie have strong category theory departments with people interested in logic and foundations. Does anyone have any feedback in these places?

 

[JSuarez]

I took a look at Stanford's site and it didn't ...

It's true, Stanford doesn't have a specific program like the ones at Berkeley or CMU, but it has very prominent logicians, like Solomon Feferman, Gregori Mints and Johan Van Benthem (invited). What you probably found odd is that they all are in the Philosophy Department (Feferman has a joint appointment); this is because Mathematical Logic is a somewhat despised field within Mathematics; in fact, most of it is done in Philosophy or Computer Science Departments.

This is also true in Berkeley and CMU: in the former, their program in Logic and Methodology of Science is an initiative of several departments, but with strong logicians in Mathematics (Woodin is arguably the most eminent set theorist in the world today); in the latter, the better logicians are in the Philosophy department too (there are a few in Mathematics, but they can't compare with the ones in Berkeley). Therefore, if you want to avoid a PhD in Philosophy, I still think the best program is the Berkeley one.

Other options that comes to mind are Madison and Notre Dame: the former has very good logicians in Mathematics and the latter has a joint maths/philosophy program.

The Canadian universities you mention are somewhat insular in this area, and nobody really cares much about what they do. Still in Canada, you might want to consider the University of British Columbia.

In the UK, the strongest logic groups are in Leeds (in mathematics) and Edinburgh (in CS). In Philosophy, there are good groups in Oxford, Cambridge, LSE, Bristol and Manchester.

If language is not much of a problem, Germany, Holland, Denmark and Finland also have very stong groups.

Check this link about Mathematical Logic around the world:

http://world.logic.at/"

 

[vici10]

The Canadian universities you mention are somewhat insular in this area, and nobody really cares much about what they do. Still in Canada, you might want to consider the University of British Columbia.

Regarding Canada, there is a strong set theory group in Fields Institute in Toronto. Although there are no graduate studies in Fields, the members of the group are from University of Toronto or York University.

Regarding universities abroad, it depends on what languages you speak. To add to JSuarez's list Spain (Barcelona) and Israel (with Saharon Shelah)

 

[metasoarous]

Wow! This is all such great information. Thanks so much. Exactly the kind of stuff I wanted to hear.

I looked into Madison (I have a friend studying set theory there) and it looks good, but doesn't have much in the way of categories. Also, my mom had a falling out with a brother who obtained his PhD there, so she asked that I not go there - it's something she could probably get over, but it's a hurdle to some degree.

That's interesting to hear that about the Canadian schools - unfortunate too. I'm glad to hear you mention UBC though. I live in WA state on the San Juan Islands, just a hop skip and a jump from there. I love the region - I have definitely found a sense of home here, so that is a plus. As far as I know, John MacDonald is the only individual in Logic, but he's also interested in categories. Clearly putting a lot of eggs in one basket on that one, but he's nearby so I plan to meet him and get a better sense for what he's doing and what the school has to offer (as well as what directions I could head with Categorical Logic). One of the things I like about the Canadian system is that in essentially every case you get a masters first and then begin a PhD program. This could give me a chance to feel things out a bit and make a better decision about what area I want to get into after knowing more, possibly even switching schools if I realized that there was a better place for me.

I've been looking at Cambridge and am really liking what I see there. There are several Logic/Category folks in the math department, and some working with Topoi in particular, an aspect of categorical logic in which I'm very interested (I've been working through "Topoi : A Categorical Analysis of Logic" - difficult making too much progress without having more of a community and time to dedicate, but it's been a beautiful read so far).

These thought bring me to some questions not directly about schools. One of the things I'm curious about is the extent to which topoi play a role in Categorical Logic on a whole. Is it substantial or minimal? Since topoi behave like abstract versions of Set(the category), I imagine that they play (or could play) a significant role in the use of categories to study the topic of cardinality. To what extent are folks who are working with Categorical Logic pursuing this and to what extent is it pursuable?

As for all of the other schools mentioned, I'll look into them more as time goes on. There is so much to process with all of this! I've spent all day just looking at schools! It's been great, but I need to get back to studying for the math GRE. I'll have to knock it dead to have a chance of getting into half these places.

As for languages, English is the only language I'm fluent in, but I grew up in New Mexico and was exposed to a lot of Spanish. I could probably pick it up fairly quickly. It might fun to go somewhere that forces me to pick up a new language.

Thanks again!