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Algebraic Geometry

Automorphism groups in algebraic geometry

Speaker: 
Michel Brion
Date: 
Fri, Mar 10, 2017
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 
The talk will first present some classical results on the automorphisms of complex projective curves (or alternatively, of compact Riemann surfaces). We will then discuss the automorphism groups of projective algebraic varieties of higher dimensions; in particular, their "connected part" (which can be arbitrary) and their "discrete part" (of which little is known).

Signs of abelian varieties and representations

Speaker: 
Matthew Greenberg
Date: 
Thu, Oct 15, 2015
Location: 
PIMS, University of Calgary
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
The sign is a fundamental invariant of an abelian variety defined over a local (archimedian or p-adic) or global (number or function) field. The sign of an abelian varieties over a global field has arithmetic significance: it is the parity of Mordell-Weil group of the abelian variety. The sign also appears in the functional equation of the L-function of abelian variety, determining the parity of its order of vanishing at s=1. The modularity conjecture says that this L-function coincides with the L-function of an automorphic representation, and the sign can be expressed in terms of this representation. Although we know how to compute this sign using representation theory, this computation does not really shed any light on the representation theoretic significance of the sign. This representation theoretic significance was articulated first by Dipendra Prasad (in his thesis), where he relates the sign of a representation to branching laws — laws that govern how an irreducible group representation decomposes when restricted to a subgroup. The globalization of Prasad’s theory culminates in the conjectures of Gan, Gross and Prasad. These conjectures suggest non-torsion elements in Mordell-Weil groups of abelian varieties can be obstructions to the existence of branching laws. By exploiting p-adic variation, though, one can hope to actually produce the Mordell-Weil elements giving rise to these obstructions. Aspects of this last point are joint work with Marco Seveso.

Algebraic Stacks and the Inertia Operator

Speaker: 
Kai Behrend
Date: 
Fri, Mar 27, 2015
Location: 
PIMS, University of British Columbia
Conference: 
CRM-Fields-PIMS Prize Lecture
Abstract: 

Motivated by subtle questions in Donaldson-Thomas theory, we study the spectrum of the inertia operator on the Grothendieck module of algebraic stacks. We hope to give an idea of what this statement means.  Along the way, we encounter some elementary, but apparently new, questions about finite groups and matrix groups.  Prerequisites for this talk: a little linear algebra, and a little group theory. 

 

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Algebraic groups and maximal tori

Speaker: 
Vladimir Chernousov
Date: 
Mon, Mar 23, 2015
Location: 
PIMS, University of British Columbia
Conference: 
Geometry and Physics Seminar
Abstract: 

We will survey recent developments dealing with characterization of absolutely almost simple algebraic groups having the same isomorphism/isogeny classes of maximal tori over the field of definition. These questions arose in the analysis of weakly commensurable Zariski-dense subgroups. While there are definitive  results over number fields (which we will briefly review), the  theory over general fields is only emerging. We will formulate the  existing conjectures, outline their potential applications, and  report on recent progress. Joint work with A. Rapinchuk and  I. Rapinchuk.

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The Work of Misha Gromov, a Truly Original Thinker

Speaker: 
Jean-Pierre Bourguignon
Date: 
Fri, Apr 5, 2013
Location: 
PIMS, University of British Columbia
Conference: 
Special Lecture
Abstract: 

The work of Misha Gromov has revolutionized geometry in many respects, but at the same time introduced a geometric point of view in many questions. His impact is very broad and one can say without exaggeration that many fields are not the same after the introduction of Gromov's ideas.I will try and explain several avenues that Gromov has been pursuing, stressing the changes in points of view that he brought in non-technical terms.Here is a list of topics that the lecture will touch:

  1. The h-Principle
  2. Distance and Riemannian Geometry
  3. Group Theory and Negative Curvature
  4. Symplectic Geometry
  5. A wealth of Geometric Invariants
  6. Interface with other Sciences
  7. Conceptualizing Concept Creation

On the Sylvester-Gallai Theorem

Speaker: 
Ben Green
Date: 
Wed, Sep 26, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 
The Sylvester-Gallai Theorem states that, given any set P of n points in the plane not all on one line, there is at least one line through precisely two points of P. Such a line is called an ordinary line. How many ordinary lines must there be? The Sylvester-Gallai Theorem says that there must be at least one but, in recent joint work with T. Tao, we have shown that there must be at least n/2 if n is even and at least 3n/4 - C if n is odd, provided that n is sufficiently large. These results are sharp

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Frozen Boundaries and Log Fronts

Speaker: 
Andrei Okounkov
Date: 
Mon, Oct 16, 2006
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS 10th Anniversary Lectures
Abstract: 
In this talk, based on joint work with Richard Kenyon and Grisha Mikhalkin, Andrei Okounkov discusses a binary operation on plane curves which
  1. generalizes classical duality for plane curves and
  2. arises naturally in probabilistic context,
namely as a facet boundary in certain random surface models.

Expanders, Group Theory, Arithmetic Geometry, Cryptography and Much More

Speaker: 
Eyal Goran
Date: 
Tue, Apr 6, 2010
Location: 
University of Calgary, Calgary, Canada
CRG: 
Number Theory (2010-2013)
Abstract: 
This is a lecture given on the occasion of the launch of the PIMS CRG in "L-functions and Number Theory". The theory of expander graphs is undergoing intensive development. It finds more and more applications to diverse areas of mathematics. In this talk, aimed at a general audience, I will introduce the concept of expander graphs and discuss some interesting connections to arithmetic geometry, group theory and cryptography, including some very recent breakthroughs.
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