Mathematics Dept.
  Events
Ecole Normale Supérieure de Lyon
Thu 27 Apr 2017, 3:30pm
Number Theory Seminar / PIMS Seminars and PDF Colloquiums
ESB 4127
Iterated extensions and p-adic dynamical systems
ESB 4127
Thu 27 Apr 2017, 3:30pm-5:00pm

Abstract

Let K be a field and let P be a polynomial. What can we say about the field generated by the roots of P and of all its iterates? I will discuss some questions motivated by this general problem when K is a p-adic field. Along the way, we'll see Coleman power series, p-adic dynamical systems and a little bit of p-adic Hodge theory.

(This talk is part of the PIMS focus semester on the mod p Langlands program).
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Ph.D. Candidate: Man Shun Ma
Mathematics, UBC
Mon 1 May 2017, 12:30pm SPECIAL
Room 202, Anthropology and Sociology Bldg (ANSOC) 6303 NW Marine Drive, UBC
Geometric properties of the space of Lagrangian self-shrinking tori in R^4
Room 202, Anthropology and Sociology Bldg (ANSOC) 6303 NW Marine Drive, UBC
Mon 1 May 2017, 12:30pm-2:30pm

Details

We prove that any sequence of conformally branched compact Lagrangian self-shrinkers in four dimensional Euclidean space with uniform area upper bound and fixed genus has a convergent subsequence, if the conformal structures do not degenerate. When the genus is one, we can drop the assumption on non-degeneracy the conformal structures. We also show that there is no branched immersion of Lagrangian self-shrinking sphere. When the area bound is small, we show that any such Lagrangian self-shrinking torus is embedded with uniform curvature estimates. For a general area bound, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori.

Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori, along which the Lagrangian condition is preserved, area is decreasing, and the compact type I singularities with a fixed area upper bound can be perturbed away in finitely many steps. This is a Lagrangian version of the construction for embedded surfaces by Colding-Minicozzi.

In the noncompact situation, we derive a parabolic Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on l-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in Euclidean spaces with uniform bounded second fundamental form. This generalizes a result of Wang for compact immersions. We also prove an Omori-Yau maximum principle for properly immersed self-shrinkers.

Note for Attendees

Latecomers will not be admitted.
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MIT
Wed 3 May 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Character maps, free loops, and fusion systems
ESB 4133 (PIMS Lounge)
Wed 3 May 2017, 3:15pm-4:15pm

Abstract

A saturated fusion system associated to a finite group G encodes the p-structure of the group as the Sylow p-subgroup enriched with additional conjugation. The fusion system contains just the right amount of algebraic information to for instance reconstruct the p-completion of BG, but not BG itself. Abstract saturated fusion systems F without ambient groups exist, and these have (p-completed) classifying spaces BF as well.

In a joint project with Tomer Schlank and Nat Stapleton, we combine the theory of abstract fusion systems with the work by Hopkins-Kuhn-Ravenel and Stapleton on transchromatic character maps, and we generalize several results from finite groups to fusion systems.

A main ingredient of this project is studying the free loop spaces L(BG) and L(BF) for groups and fusion systems, and constructing transfer maps from L(BG) to L(BH) when H is a subgroup of G.
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