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GEAR Panel Discussion Notes

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Transcript of GEAR Panel Discussion Notes

GEAR RETREAT PANEL DISCUSSION
First GEAR Retreat, University of Illinois
August 2012

Geometric Structures
Special Representations
Dynamics on Moduli Spaces
3-manifolds
Higgs bundles

Danciger
GEAR Panel Discussion

Take are you're in, find area you're interested in, connect them. I work in geometric structures, rich and well appreciated connections between that and all but Higgs bundles, want to explore these connections.

Specifically:

(1) What are geometric structures corresponding to the Hitchin component- build on Guichard-Wienhard

(2) What is geometric meaning of Anosov representations in complex Lie groups?

(3) Understand better the Non-Abelian Hodge theorem.

(4) Understand Hitchin's Riemannian metric on Teichmuller space

(5) Special loci of character variety leads to special loci of Higgs bundles. Can understand this for real reps. What about other reps?
Dumas
Best math comes from interactions of two fields

Personally, hyperbolic 3-manifolds and special representations

Apply `low-tech' ideas from hyperbolic 3-manifolds to special representations

Bring ideas from dynamics into hyperbolic 3-manifolds (Minsky-Gelander)

Find really smart co-authors, ride coattails.
CANARY
Coming from geometric structures and 3-manifolds

All are geometry

Associating geometric structures to other objects.

Coming together and discussing leads to progress
Figure out what other categories can do for your area. In my case, 3-manifolds.

Low-d topology/hyp geometry and relation to Heegard-Floer and gauge theory.

Eg: every integer homology 3-sphere that's not S^3 has a representation into SU(2) or SL(2, R).

Look at character variety, action.

Torsion invariants
Dunfield
Again, being helped by others, trying to help other fields

coming from Higgs bunldes, one particular area is special representations, in particular representations into groups of Hermitian type, maximal Toledo type.

What is the role of Hitchin equations? What do they say about geometric structures?

What do subvarieties of moduli spaces of Higgs Bundles (perhaps with geometric meaning) correspond to in special reps.

Sp(4, R) program- potential WP-metric on moduli space of Sp(4, R) reps, in particular components where image is Zariski dense.
Fock-Goncharov: higher mapping class group- what is it geometrically?

Guichard-Wienhard- is there an analogy with Anosov reps to the 3-manifold story of hyperbolic manifolds and the conformal structure on the boundary.

Labourie
Lukyanenko: what physics?

Labourie: General background, for example, what is relationship of Higgs bundles to physics? Perhaps a summer school on TQFT, etc.?

Bradlow: what opportunities do you see, Andy?

Neitzke: My work started by trying to understand math, in particular, Kontsevich-Soibelman. Continue to interact. Kevin Costello has produced great math/physics.

Labourie: use physics as a `sieve' for good questions.

Dumas: what prevents us from talking to physicists?
Me; different languages

Wienhard: we're not the usual group of mathematician who talks to physicists.

Garcia-Prada: Higgs bundles arose from physics, from instantons, so it's already benefited a lot from physics. Physics is a source of inspiration, good problems.

Neitzke: more physicists would be interested in attending.

Danciger: how do we find them?

Bradlow: never been at a meeting with dynamicists/low-d topologists, how do we break down more barriers? Which talks fired you up?

Me: Nietzke's talk connected, through physics?

Bradlow: What fills in the middle of the diagram? quadratic differentials? What do quadratic differentials mean to me?

Kerckhoff: how can we set up meeting to encourage more interactions? One topic/day is good for first meeting, but how to get more interactions?

Canary: there's plenty of bumping and self-bumping, the circles are not really disjoint.

Wentworth (via email): Good to have survey talks on topics, but might be better to focus on the edges instead of vertices.

Wienhard: again, mixing topics/changing structure for retreat may be useful.

Wright: for junior speakers, one topic/day was very useful
Audience Questions:
Resources at `edge, not nodes'
Practical Matters
Current programs:
Short term visits
exchanges
workshops
Lots of resources for young people, visits to other places

Can GEAR help students go on long-term visits/eliminate teaching?

Wienhard: semesters are shifted, so can arrange visits during breaks

Create a GEAR overflow 'Forum' for questions/references/jobs, managed by grad student/postdoc
Labourie
Concentrate on `edges' of subjects, rather than nodes.

Focus courses by having two experts in related subjects give a joint course, for example.

Small mini-workshops (8-10 people), young people involved, but getting into the guts of something, not necessarily motivated by an immediate collaboration.

Sp(4)-lab, for example!

Wienhard: exchanges can be people coming together to meet at a different (non-GEAR node) place. Small workshops can be funded via applications, can be very speculative, can be area-based, can be geography based. Applications can come from all levels.
Dunfield
Used GEAR resources, again not sure what, maybe visits/exchanges.

Apply and use money!

Can be intimidating to ask for money to faculty directly, use GEAR!

Wienhard/Bradlow: can't buy out teaching, but can provide stipend to postdoc to visit other places
Danciger
I've used a lot, feel like a grad student again.

use the money!

Senior people like to talk about their work.
Canary
Work against the tendency to cluster on the things you're doing.

Bring people from different areas together in informal settings.

Make it easy to find out who's doing what, adding a paragraph to each directory entry- souped up member's list, with links to websites.

Make it like a department faculty list.

Keep sending emails about funding opportunities.
Dumas:
If a department has a GEAR member, you are at a GEAR node, and any student/postdoc there is also eligible for GEAR funding.

If you are a faculty, you can be nominated by a current GEAR member.
Wienhard
Roger: how long in advance do we need to plan events?
Audience Questions:
Sur-Wei Fu: keep advertising GEAR on a regular basis.