p-Adic Geometry and Homotopy Theory

Conference at Hotel Alexandra, Loen, Norway, August 2nd-8th 2009

The Research Council of Norway funds a research program on Brave New Rings in the period 2005 - 2009. In this framework, there will be a Conference on p-Adic Geometry and Homotopy Theory at Hotel Alexandra at Loen in the Nordfjord region of western Norway.

The aim of the conference is to develop the current interplay between arithmetic algebraic geometry and stable homotopy theory. Using structured ring spectra one can form topological structure sheaves for moduli stacks of elliptic curves or other abelian varieties, whose ring spectra of global sections define powerful new homology theories. Using sheaves of infinity-categories one can form topological crystalline cohomology, with associated de Rham-Witt complexes closely related to topological cyclic homology and p-adic algebraic K-theory. For complete local rings of mixed characteristic, or for structured ring spectra of mixed chromatic types, there are log (= logarithmic) geometric extensions of these theories, leading to de Rham-Witt complexes with log poles, log topological cyclic homology and log K-theory. The hope is that both algebraic geometers and homotopy theorists will have something to learn from the modern developments in these adjacent fields.

Here is a list of titles and abstracts. Talks marked (L) are in Loensal 3 on the first floor, while talks marked (F) are in Fjordsal 1+2 on the ground floor. Lecture notes for some of the talks are available below.

Program:

Sunday August 2nd

WF158 arrival
Widerøe flight WF158 Oslo-Sandane, departure OSL 11:55, arrival SDN 12:58.
Fjord1 chartered bus Sandane-Loen, boarding from 13:00, arrival ca 14:30.
WF153 arrival
Widerøe flight WF153 Oslo-Sandane, departure OSL 16:10, arrival SDN 17:13.
Fjord1 chartered bus Sandane-Loen, boarding from 17:15, arrival ca 18:45.
19:00 - 21:00
Dinner buffet in Restaurant Charlotte 19:00 - 21:00. Drinks are bought separately.

Monday August 3rd

07:00 - 10:00
Breakfast.
10:00 - 11:00 (L)
Jean-Marc Fontaine (Orsay): "p-adic cohomologies and comparison theorems" (lecture notes)
11:30 - 12:30 (L)
Arthur Ogus (Berkeley): "Logarithmic geometry" (lecture notes)
13:00 - 14:00
Lunch in Restaurant Charlotte.
15:00 - 16:00 (L)
Christian Schlichtkrull (Bergen): "Symmetric spectra and diagram spaces" (lecture notes)
16:30 - 17:00
Coffee/tea break in Loensal salon.
17:00 - 18:00 (L)
Steffen Sagave (Münster): "Graded log symmetric ring spectra" (lecture notes)
19:00 - 21:00
Dinner in Restaurant Charlotte.

Tuesday August 4th

07:00 - 10:00
Breakfast.
10:00 - 11:00 (L)
Luc Illusie (Orsay): "Review of de Rham-Witt theory" (lecture notes)
11:30 - 12:30 (L)
Kiran Kedlaya (MIT): "Rigid cohomology and its coefficients" (lecture notes)
13:00 - 14:00
Lunch in Restaurant Charlotte.
15:00 - 16:00 (F)
Teena Gerhardt (Indiana): "Fixed points of THH and the de Rham-Witt complex"
16:30 - 17:00
Coffee/tea break in Loensal salon.
17:00 - 18:00 (L)
Christian Ausoni (Bonn): "Algebraic K-theory of the fraction field of topological K-theory" (lecture notes)
19:00 - 21:00
Dinner in Restaurant Charlotte.

Wednesday August 5th

07:00 - 09:30
Breakfast. Niels Henrik Abel's birthday.
09:30 - 10:30 (L)
Amnon Besser (Ben Gurion): "Syntomic regulators and special values of p-adic L-functions" (lecture notes)
10:30 - 11:00
Coffee/tea break in Loensal salon.
11:00 - 12:00 (F)
Wiesława Niziol (Utah): "p-adic comparison morphism via K-theory"
12:00 - 13:00
Lunch in Restaurant Charlotte.
Conference excursion
Meet outside hotel at 13:05. Bus departure to Sande, MS Kjendal boat at 13:30 across Loenvatnet. Bus and easy walk to Kjenndalsbreen glacier arm. Return from Kjenndalsanden at 16:30, reaching hotel ca 17:30.
Alternative hike
Meet outside hotel at 13:05. Minibus transport to Oppheim. Mountain hike to Lofjellet (1379 meters, six hours, "magnificent view over Loen and Olden valley").
Other alternatives
  • Mountain hike to Årheimsfjellet (three hours easy walk, "good mountain track with beautiful views").
  • Mountain hike to Auflemåsen (four to five hours, "breathtaking views from the summit").
  • 19:00 - 21:00
    Dinner in Restaurant Charlotte.

    Thursday August 6th

    07:00 - 10:00
    Breakfast.
    10:00 - 11:00 (F)
    Tyler Lawson (Minnesota): "Topological automorphic forms"
    11:30 - 12:30 (F)
    Clark Barwick (Harvard): "Equivariant derived algebraic geometry and crystalline realizations of K-theory, I" (lecture notes)
    13:00 - 14:00
    Lunch in Restaurant Charlotte.
    15:00 - 16:00 (F)
    John Rognes (Oslo): "Logarithmic THH and TC" (lecture notes)
    16:30 - 17:00
    Coffee/tea break in Loensal salon.
    17:00 - 18:00 (F)
    Wiesława Niziol (Utah): "Logarithmic K-theory"
    19:00 - 21:00
    Dinner in Restaurant Charlotte.

    Friday August 7th

    07:00 - 10:00
    Breakfast.
    10:00 - 11:00 (L)
    Markus Szymik (Bochum): "Brave new rings from K3 surfaces" (lecture notes)
    11:30 - 12:30 (F)
    Clark Barwick (Harvard): "Equivariant derived algebraic geometry and crystalline realizations of K-theory, II" (lecture notes)
    13:00 - 14:00
    Lunch in Restaurant Charlotte.
    15:00 - 16:00
    Mike Hill (Virginia): "On the slice filtration and the Kervaire invariant one problem" (lecture notes)
    16:30 - 17:00
    Coffee/tea break in Loensal salon.
    17:00 - 18:00 (F)
    Jack Morava (Johns Hopkins): "A theory of base motives" (lecture notes)
    19:00 - 21:00
    Dinner in Restaurant Charlotte.

    Saturday August 8th

    07:30 - 09:45
    Breakfast. Please settle hotel bills before departure.
    WF165 departure
    Fjord1 chartered bus departure from hotel Saturday at 09:45, for
    Widerøe flight WF165 Sandane-Oslo, departure SDN 12:00, arrival OSL 12:57.
    Alternative ending
    Climb to, or towards, the summit of Skåla at 1848 meters above sea level. Allow five hours for ascent and three hours for descent. Good physical condition and proper hiking shoes required. Please check the weather forecast before walking in the mountains.
    19:00 - 21:00
    Dinner in Restaurant Charlotte 19:00 - 21:00.

    Sunday August 9th

    07:30 - 10:00
    Breakfast. Please settle hotel bills before departure.
    WF159 departure
    Fjord1 chartered bus departure from hotel Sunday at 11:00, for
    Widerøe flight WF159 Sandane-Oslo, departure SDN 13:13, arrival OSL 14:10.

    Participants:

    John Rognes / August 21st 2009