## Cascade Topology Seminar

- Start Date: 04/28/2012
- End Date: 04/29/2012

Please note that due to unavoidable circumstances Mladen Bestvina will be unable to attend. We are happy to announce that Ian Hambleton will be able to speak in his place.

Ian Hambleton (McMaster)

"Finite group actions on Kervaire manifolds"

Abstract:

The Kervaire manifolds are closed, oriented, almost framed PL manifolds in dimension 4k+2 with Arf invariant 1. The usual examples are constructed by "plumbing", and the resulting manifolds are 2k-connected with the same homology as a product of spheres. In certain exceptional dimensions (2, 6, 14, 30, 62 and perhaps 126) there exist smooth Kervaire manifolds. The talk will be about the existence of free smooth or PL actions of finite groups on Kervaire manifolds, starting with the existence of free involutions. In contrast with a product of spheres $S^{2k+1} \times S^{2k+1}$, it turns out that the Kervaire manifolds have very restricted symmetry. This is joint work (in progress) with Diarmuid Crowley.

Ian Biringer (Yale)

"Random samplings of locally symmetric spaces"

Abstract:

Suppose that Mi is a sequence of closed locally symmetric spaces modeled on a fixed symmetric space X, e.g. a sequence of hyperbolic n-manifolds. We introduce a tool to understand how the geometry of Mi near a random sample point develops as i tends to infinity. Applications include a control on the growth of the Betti numbers of Mi when X has higher rank.

Joan Birman (Barnard-Columbia)

"The dilatation of a pseudo-Anosov maps and its invariant polynomials"

Abstract:

Let S be a fixed orientable surface. The dilatation D(F) of a pseudo-Anosov diffeomorphism of S is a real number >1 and an invariant of the conjugacy class [F] of F in Mod(S). The work we will discuss was motivated by the question: What is the minimum value D(F), as F ranges over all possible diffeomorphisms of S? It's known that $D_{min}$ exists, is > 1, and is realized by some F in Diff(S), but the actual value has proved to be elusive except in special cases. We will discuss our efforts, in recent joint work with Brinkmann and Kawamuro, to study this problem from a new point of view, asking about the minimum polynomial of $D(F)$ over the integers. In the course of our work we turned up two new polynomials, both invariants of [F].

Alas, there are cases when they are both reducible over Z, so we didn't answer the question that had motivated our work!

Jack Morava (Johns Hopkins)

"Theories of Anything"

Abstract:

Many theories of classification -- conic sections are a classical example -- can be formulated in terms of the quotient of an action of a nice group on a nice topological space; which might however have a terrible (e.g., non-Hausdorff) quotient. Thom's theory of structurally stable forms elaborates this idea, and I claim [arXiv:1202.0684] that the categorical knowledge representation systems [ologs] of D Spivak conveneniently encode the structure of these bad quotients as databases.

Arnold's classification of isolated singularities of holomorphic functions provides a good example of this claim, as do knot tables; and there are many others, such as the phase diagrams of condensed matter physics; but fitting them into a nice framework calls for a categorification of the functor \pi_0 which assigns to a space, its set of components. This is pretty easy to do for global group actions, but finding a more general construction, applicable to topological groupoids and stacks, seems an interesting problem in geometric topology.

Dev Sinha (Oregon)

"Cohomology of symmetric groups"

Abstract:

We have recently made progress on understanding the cohomology of symmetric groups by fully using the geometry of the Fox-Neuwirth cell structure and the algebra of Strickland-Turner's Hopf ring structure. We share progress including a full description mod-p and an almost complete description two-locally. There is still perhaps some interesting geometry which needs to be revealed at odd primes. We also touch on related topics such as invariant theory, characteristic classes for covering maps, and the cohomology of alternating groups.

Juan Souto (UBC)

"Metrics on manifolds with large volume and spectral gap"

Abstract:

I will explain how to construct, for d > 2, metrics on d-dimensional manifolds with bounded geometry, arbitrarily large volume, and spectral gap bounded from below away from zero. As a consequence we obtain that there are hyperbolic knot complements M_i whose volume tends to infinity and whose Cheeger constant is larger than some e>0$ for all i. This is joint work with Marc Lackenby.

University of British Columbia

This event will celebrate Dale Rolfsen's 70th birthday.

There will be a special conference dinner.

The Cascade Topology Seminar is a bi-annual gathering of topologists from the US Pacific Northwest and Southwestern Canada.

**SATURDAY, April 28:**

9:30 am Coffee and pastries

10:00 am Ian Hambleton (McMaster)

11:15 am Dev Sinha (Oregon)

12:15 pm Lunch

2:00 pm Ian Biringer (Yale)

3:00 pm Coffee Break

4:00 pm Joan Birman (Barnard-Columbia)

6:30 pm Dinner for Dale Rolfsen on occasion of his 70th

Birthday

**SUNDAY, April 29:**

9:30 am Coffee and pastries

10:00 am Jack Morava (Johns Hopkins)

11:15 am Juan Souto (UBC)

Check back for further information or visit: http://www.mth.pdx.edu/cts/.

**ACCOMMODATIONS:**

Participants should arrange their own accommodations.

Please contact Oriana Bella (orianab@pims.math.ca) if you need assistance.

**List of on-campus hotels**

1. Carey Centre

Rooms from $71/night and up.

Booking Request Form here.

2. TRIUMF House

(604) 222-1062

Rooms from $110/night and up.

Contact: TRIUMF House Reservation Office

Email: housing@triumf.ca

Booking Request form here.

3. UBC Conferences and Accommodations

(604) 822-1000

West Coast Suites and Walter Gage Towers only. Rates vary.

(Pacific Spirit Hostel is summer only, not available for this conference.)

E-Mail: reservations@housing.ubc.ca

4. Green College (Guest rooms/Student res.) Rooms from $83/night and up.

Contact:

(604) 822-8660

Email: greencol@interchange.ubc.ca

Reservation booking form here.

5. St. John's College

(604) 822-6522

Email: allam@interchange.ubc.ca

contact: Alice Lam for rates and reservations. Please mention you are attending PIMS Cascade Topology Seminar.

6. Vancouver School of Theology (VST)

6000 Iona Drive, Vancouver, BC V6T 1L4

Phone: 604-822-6398

Rate: $105.80/night (15% tax)

Booking Request form here.

**LOCATION:**

The Cascade Topology Seminar will take place at the Pacific Institute for the Mathematical Sciences (WMAX) Room 216, 1933 West Mall, Vancouver, BC

Please help PIMS to improve the quality of its events and plan for the future by filling out this quick and painless survey.