The Number Theory Seminar

Fall 2009

PLACE and TIME:The seminar meets on Tuesdays in LN 2205 at 4:15 p.m.
Before the talks, there will be refreshments in the Anderson Reading Room at 4:00 p.m.

ORGANIZERS: Marcin Mazur and Adrian Vasiu

FALL 2009: 16 talks.

• September 1: Ding Ding, Binghamton University.
  Title: Affine group schemes over perfect fields of characteristic p.
  Abstract: In this talk we first review basic definitions and properties of the category of affine group schemes over a perfect field k of characteristic p>0. We will show that certain subcategories of it are abelian categories. Then I will present an important construction on finite affine group schemes over k: Cartier dual. We will end up by introducing p-divisible groups over k.
  
  
• September 3 (CROSS LISTING WITH THE COLLOQUIUM; SPECIAL TIME 4:30 p.m.): Thomas Zink, Bielefeld University (Germany).
  Title: Witt vectors cohomology of algebraic varieties in characteristic p>0.
  Abstract: Let X be a smooth affine variety over the field C of complex numbers. By a theorem of Grothendieck the cohomology groups $H^i(X,C)$ may be computed by algebraic differential forms. Let now X be a smooth algebraic variety over an algebraically closed field k of characteristic p >0. Monsky and Washnitzer introduced a complex of differential forms which lead to the correct cohomology groups for affine varieties X. We globalize their approach using the de Rham-Witt complex of Deligne--Illusie. There is a comparison with the rigid cohomology of Berthelot.
  
  
• September 8: Thomas Zink, Bielefeld University (Germany).
  Title: Cartier theory of curves of formal commutative groups. Part I.
  Abstract: We present basic theorems of Cartier theory over arbitrary commutative rings. The ideas of this theory are related to the Riemann--Roch algebra (universal $\Lambda$-rings and Chern classes). No a priori knowledge of formal groups is required.
  
  
• September 15: Thomas Zink, Bielefeld University (Germany).
  Title: Cartier theory of curves of formal commutative groups. Part II.
  Abstract: The talk will continue the one of the previous week. The main two theorems of Cartier theory will be presented.
  
  
• September 22: Thomas Zink, Bielefeld University (Germany).
  Title: Cartier theory of curves of formal commutative groups. Part III.
  Abstract: Several corollaries of the main theorems will be presented. In particular, the Cartier ring will be introduced.
  
  
• September 29: Thomas Zink, Bielefeld University (Germany).
  Title: Cartier theory and its relations to displays and crystalline theories.
  Abstract: This is the completion of the 4 weeks course. It will present several relations between the Cartier theory, the displays associated to formal groups, and the crystalline theories.
  
  
• October 6: Xiao Xiao, Binghamton University.
  Title: F-crystals of K3 type.
  Abstract: Let k be an algebraically closed field of characteristic p>0. The F-crystals of K3 type over k are the natural generalizations of the F-crystals associated to surfaces of K3 type over k. In this talk we present several old and new results on F-crystals of K3 type.
  
  
• October 13: Alex Feingold, Binghamton University.
  Title: An overview of Lie algebras, their representation theory, and some of their applications in physics. Part I.
  Abstract: Lie algebras are non-associative algebras with a beautiful structure and representation theory. The classification of the finite dimensional simple Lie algebras over the complex numbers by Killing and Cartan was an inspiration for the classification of the finite simple groups. Lie theory is a vast subject with many important generalizations, for example, in the direction of the infinite dimensional Lie algebras (Kac-Moody, Heisenberg, Virasoro and vertex operator algebras) and their representations. These have applications in physics to conformal field theory and string theory, and connections to other parts of mathematics, for example, sporadic simple groups, modular functions, combinatorics, differential equations, and many other topics. I will try to give a broad overview of this field and some of its applications and techniques, leaning towards those topics in which I have the greatest interest.
  
  
• October 20: Alex Feingold, Binghamton University.
  Title: An overview of Lie algebras, their representation theory, and some of their applications in physics. Part II.
  Abstract: See the abstract of Part I.
  
  
• October 27 (CROSS LISTING WITH ALGEBRA SEMINAR; SPECIAL TIME 2:50 p.m.): Martin Kassabov, Cornell University.
  Title: Subspace arrangements and property T
  Abstract: I will mainly talk about (my viewpoint at) a method for proving property T started by Dymara and Januszkiewicz. Their original motivation came from groups acting on dimensional buildings, but the refined idea does not use anything more than angles between subspaces in an finite dimensional Euclidian space. Parts of the talk are based on a work of M. Ershov and A. Jaikin.
  
  
• November 3: Benjamin Lundell, Cornell University.
  Title: New Parts of Hecke Rings.
  Abstract: In 1970s, Barry Mazur studied certain completed Hecke algebras and related their ring-theoretic properties to deep arithmetic results. In this talk, we will discuss recent progress towards answering a modified version of a question of Mazur's about the rank of such a Hecke algebra and some of the arithmetic corollaries.
  
  
• November 10:Rebecca Torrey, Cornell University.
  Title: An Introduction to Serre's Conjecture.
  Abstract: This important conjecture, which has influenced much of the research in number theory over the past twenty years, is now (due to Khare and Wintenberger) a theorem. I will explain the statement of the conjecture, talk about why it is so important and discuss recent work on generalizations.
  
  
• November 17 (CROSS LISTING WITH COMBINATORICS SEMINAR; SPECIAL TIME 1:15 p.m.): Justin Lambright, Lehigh University.
  Title: A combinatorial interpretation for computations in the quantum polynomial ring.
  Abstract: A Hopf algebra called the quantum coordinate ring of SL(n,C) is often studied in terms of a related noncommutative ring called the quantum polynomial ring in n2 variables. Various bases of these rings and their representation-theoretic applications lead to the study of transition matrices whose entries are commutative polynomials having nonnegative integer coefficients. Examples of such polynomials include Brenti's modified R-polynomials. I generalize Brenti's work to give combinatorial interpretations for coefficients in a larger class of transition matrices. As an application, I simplify somewhat the previous formulation of the dual canonical basis of the quantum polynomial ring.
  
  
• November 24: Adrian Vasiu, Binghamton University.
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• December 4 (SPECIAL TIME INSTEAD OF DECEMBER 1): Ilir Snopce, Binghamton University (Ph.D. thesis defense).
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• December 8: Marcin Mazur, Binghamton University.
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  Spring 2010

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• January 26:
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• February 2:
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• February 9: Ravi Ramakrishna, Cornell University.
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• February 16:
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• February 23:
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• March 2: Adrian Vasiu, Binghamton University.
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• March 9: Xiao Xiao, Binghamton University.
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• March 16:
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• March 23:
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• April 6:
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• April 13: Ding Ding, Binghamton University (Admission to Candidacy Exam, Part I).
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• April 20: Ding Ding, Binghamton University (Admission to Candidacy Exam, Part II).
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• April 29 (SPECIAL TIME): Florian Pop, University of Pennsylvania
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• April 30 (SPECIAL TIME): Florian Pop, University of Pennsylvania (tentative speaker)
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