University of Warwick, Coventry

Tuesday

23/03/10

Speakers: John Greenlees (Sheffield), Oscar Randal-Williams (Oxford), Andrew Ranicki (Edinburgh).

- 11.00 am, Tea and coffee
- 11.30 am, B3.03:
**John Greenlees**

"Homotopy invariant notions of complete intersections in algebra and topology." - 12:30 pm, Lunch break
- 2.30 pm, B3.03:
**Oscar Randal-Williams**

"Resolutions of moduli spaces." - 3.30 pm, Tea and coffee
- 4.15 pm, B3.03:
**Andrew Ranicki**

"The algebraic eta invariant."

Tea and coffee will be served in the common room, all lectures will take place in room B3.03 (top floor). After the last talk we expect to take a bus and go for dinner at some place with easy access to the railway and bus stations.

- John Greenlees,
**Homotopy invariant notions of complete intersections in algebra and topology**The talk will consider three well-known characterizations of the notion of complete intersection in commutative algebra and construct homotopy invariant versions of each. This allows them to be applied to homotopy theory (eg C*(X) for spaces X). It turns out these coincide (for classical rings) with the usual notions, and they coincide with each other in rational homotopy theory. The relation between them in characteristic p is under investigation. (Joint work with DJ Benson, K.Hess, S.Shamir).

- Oscar Randal-Williams,
**Resolutions of moduli spaces**I will explain how to approximate moduli spaces of topological surfaces of genus g (possibly equipped with a tangential structure) by moduli spaces of strictly less genus, and how this often implies that the homology of these moduli spaces stabilises with g. This extends the famous stability theorem of J. Harer on the homology of the oriented mapping class groups to new families of mapping class groups preserving a tangential structure. I will discuss some new examples if there is time.

- Andrew Ranicki,
**The algebraic eta-invariant**The signature a 4k-dimensional Riemannian manifold with boundary (P,\partial P) was shown by Atiyah, Patodi and Singer (1973) to be the sum of the Hirzebruch L-genus and the \eta-invariant of \partial P. Many authors (Neumann, Meyer, Cappell-Lee-Miller, Bunke, Nemethi, ...) have subsequently given a more algebraic treatment of the \eta-invariant, defining it in particular for a closed (4k-1)-dimensional manifold N with a separating hypersurface M \subset N and a complex structure J on H^{2k-1}(M). The talk will describe an even more algebraic treatment of the \eta-invariant, using notions of the algebraic theory of surgery. The algebraic \eta-invariant is related to the Wall non-additivity of the signature, the Maslov index, the Witt group of the function field of the reals, the Witt group of symplectic automorphisms, etc. The von Neumann \rho-invariant of a knot is the high-dimensional knot concordance invariant defined by the average of the Tristram-Levine knot signatures. (It plays an important role in the Cochran-Orr-Teichner calculations of the classical knot concordance group). The von Neumann \rho-invariant turns out to be the sum of the algebraic \eta-invariant and half the signature of the knot.

See the departmental website for the location of the maths department and more detailed maps of the campus.

- From the main bus station (Pool Meadow) take bus No. 12 (stand N) toward University/Leamington (the bus starts there, so there is no issue of which direction to choose). The DaySaver ticket for 2.90 is recommended as it also covers return (single fare is 1.50). The bus driver sells tickets for exact change. Get off at University Road bus stop, after approx. 30 minutes ride.
- From the train station: exit the platform to the west and take the bus No. 12 towards University/Leamington from the bus stop at Warwick Rd, located on the Rail Station Bridge. From there the ride takes approx. 20 minutes. Other bus instructions as above.
- If you plan to use Megabus (this probably only makes sense from Manchester), it has its stop at De Montfort Way, within 5 minutes walk from the campus.

TTT meetings are partially supported by an LMS Scheme 3 grant. Anyone who wishes to participate is welcome: we shall operate the usual arrangements for assistance with travel expenses.

For further information email John Greenlees j.greenlees@sheffield.ac.uk, (or John Hunton jrh7@mcs.le.ac.uk or Nige Ray nige@ma.man.ac.uk) if you are interested, so we can make approximately the right amount of tea and coffee.

For local questions concerning this TTT, e-mail Ana Lucia Garcia Pulido (a.l.garcia-pulido AT warwick.ac.uk), Michal Adamaszek (m.j.adamaszek AT warwick.ac.uk) or Rupert Swarbrick (r.j.swarbrick AT warwick.ac.uk).

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