Algebra and Number Theory Seminar

Spring 2007

Thursdays, 11:15-12:05, 106 McAllister

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16th January	Please note that Kirsten Eisentraeger (Michigan) is speaking in the Combinatorics and Partitions seminar.
18th January	Mikhail Ershov (University of Chicago)
Title	An introduction to Golod-Shafarevich groups
Abstract	Informally speaking, a finitely generated group is called Golod-Shafarevich if it has a presentation with a ``small'' set of  relators. In 1964, Golod and Shafarevich proved that groups satisfying such condition are necessarily infinite and used this criterion to solve two outstanding problems: the construction of infinite finitely generated periodic groups and the construction of infinite Hilbert class field towers.    An important class of Golod-Shafarevich groups consists of fundamental groups of compact hyperbolic 3-manifolds or, equivalently, torsion-free lattices in $SO(3,1)$. In 1983, Lubotzky used this fact to prove that arithmetic lattices in $SO(3,1)$ do not have the congruence subgroup property. More recently, Lubotzky and Zelmanov proposed a group-theoretic approach (based on Golod-Shafarevich techniques) to an even more ambitious problem, Thurston's virtual positive Betti number conjecture. This approach led to the following question: is it true that Golod-Shafarevich groups never have property $(\tau)$? I will show that the answer to the above question is negative in general and briefly describe examples which are given by lattices in certain topological Kac-Moody groups over finite fields. The existence of Golod-Shafarevich groups with property $(\tau)$ (in fact, property $(T)$) turns out to have interesting applications to several problems in geometric group theory, which I will also discuss.
22nd January	Please note that Chris Francisco (Missouri) is speaking in the Combinatorics and Partitions seminar.
25th January	Yuri Zarhin (PSU)
	Abelian varieties without homotheties
	A celebrated theorem of Bogomolov asserts that the l-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic p: a ``counterexample" is provided by an ordinary elliptic curve defined over a finite field. In this talk we discuss (and explicitly construct) more interesting examples of "non-constant" absolutely simple abelian varieties (without homotheties) over global fields in characteristic $p$.
30th January	Please note that Mihran Papikian (Stanford) is speaking in the Combinatorics and Partitions seminar.
1st February	See above.
8th February	No seminar this week.
15th February	Elena Poletaeva (University of Texas visiting IAS Princeton)
	Superconformal algebras
Cancelled.   Speaker unable to travel due to the weather.	Superconformal algebras are Lie superalgebras of vector fields of Cartan type.They are superextensions of the Virasoro algebra,  and they have many applications in physics. Superconformal algebras were classified by V. Kac. We obtain new realizations of exceptional superconformal algebras as Lie subalgebras of pseudodifferential symbols. This allows us to construct new representations of these superalgebras. We also obtain their realizations as matrices over a Weyl algebra.
20th February	Trevor Wooley (University of Michigan): Note he is speaking in the Combinatorics/Partitions seminar
22nd February	See above
1st March	Sophie Huczynska (University of St. Andrews)
	A new result on generators for finite fields
	Consider a finite field F=GF(q) and its degree n extension E=GF(q^n). The Primitive Normal Basis Theorem (proved in 1987) guarantees the existence of an element of E which is simultaneously primitive and free over F; in other words, an element which is simutaneously a multiplicative and additive generator for the field E. Subsequently, there has been interest in the existence of primitive free elements with various extra properties; answers to such existence questions have combinatorial applications, as well as being interesting in their own right. In this talk, I will discuss a framework for answering such questions, and present a new result of this kind, the "Strong Primitive Normal Basis Theorem". (This is joint work with S.D.Cohen, Glasgow, UK)
8th March	Elena Poletaeva (University of Texas visiting IAS Princeton)
	Superconformal algebras
	Superconformal algebras are Lie superalgebras of vector fields of Cartan type.They are superextensions of the Virasoro algebra,  and they have many applications in physics. Superconformal algebras were classified by V. Kac. We obtain new realizations of exceptional superconformal algebras as Lie subalgebras of pseudodifferential symbols. This allows us to construct new representations of these superalgebras. We also obtain their realizations as matrices over a Weyl algebra.
12-16th March	Spring Break
22nd March	Maosheng Xiong (University of Illinois at Champaign-Urbana)
	Selmer groups and Tate-Shafarevich groups for the congruent number problem
	We study the distribution of the sizes of the Selmer groups arising from the three 2-isogenies and their dual 2-isogenies for the elliptic curve E_n:y^2=x^3-n^2x. We show that three of them are almost always trivial, while the 2-rank of the other three follows a Gaussian distribution. It implies three almost always trivial Tate-Shafarevich groups and three large Tate-Shararevich groups. When combined with a result obtained by Heath-Brown, we show that the mean value of the 2-rank of the large Tate-Shafarevich groups for square-free positive odd integers n not exceeding X is (1/2) loglog X+O(1), as X tends to infinity.
29th March	Robert Vaughan (PSU)
	The generating function in additive number theory for quadratic polynomials
	An essentially best possible estimate is obtained for the additive generating function associated with quadratic polynomials.
5th April	Sinnou David (Institut de Mathématiques de Jussieu, visiting IAS Princeton)
	Baker Theory on group varieties
	A long standing conjecture of Lang on rational functions on elliptic curves is known to follow from sharp lower bounds for linear forms in elliptic logarithms. We shall indicate how to prove such lower bounds. We shall then indicate how these can be generalized to arbitrary commutative group varieties. We shall also take this opportunity to survey  what has been done on this subject since the breakthrough of A. Baker.
12th April	George Andrews (PSU)
	The number of smallest parts in the partitions of n
	There have been variety of studies in the theory of partitions with weighted counts of partitions.  We shall provide some relevant history.  Our prime focus will be spt(n) the total number of appearances of smallest parts in the partitions of n.  For example spt(4) = 10 which can be seen by examining the partitions of 4: 4, 3+1, 2+2, 2+1+1, 1+1+1+1.  Our object will be to show that spt(n) is closely related to the second Atkin-Garvan moment of ranks and from this observation one can deduce that 5|spt(5n+4), 7|spt(7n+5), and (surprisingly) 13|spt(13n+6).
19th April	Leonid Vaserstein (PSU)
	Bounded reduction of invertible matrices over polynomial rings by addition operations
	Every r by r invertible matrix over the polynomial ring  in n variables with integer coefficients (or, more generally,  with coefficients in any Euclidean ring) can be reduced to a two by two matrix by  11n²r+17(n+1)r²  addition operations.  Since this upper bound does not depend on the matrix, it implies the Kazhdan T-property for the group SL_r(Z[x_1,...,x_n])  for any n and any r \ge 3.  We also obtain a more general result where the coefficients are in  any Noetherian ring of finite Krull dimension.  As a corollary, we obtain that every matrix in E_rA,  for any  commutative finitely generated ring A or any finitely generated algebra over any field, can be reduced to  a two by two matrix by  11n²r+17(n+1)r² addition operations, where n is the minimal number of generators and r \ge 3.
26th April	Antun Milas (SUNY at Albany)
	Modular forms and W-algebras
	Rational conformal field theories can be characterized by the property that there are, up to equivalence, finitely many irreducible representations of the vertex operator algebra, and that every representation is completely reducible.       It is tempting to relax the semisimplicity condition and study more general classes of conformal field theories. One such class are rational logarithmic conformal field theories (LCFT), where not all modules are completely reducible and not even L(0) diagonalizable (here L(0) is the degree zero Virasoro generator).  The only known examples of rational LCFT come from certain W-algebras.       In this talk, I will first define several families of W-algebras coming from LCFT. I will discuss their representations and the corresponding (generalized) characters. We shall see how W-algebras and modular forms interact in an unexpected way.       I will assume no knowledge of vertex algebra theory.
3rd May	Gary Mullen (PSU)
	The Extent to Which Subsets Are Additively Closed
	Given a finite abelian group G (written additively), and a subset S of G, the size r(S) of the set may range between 0 and |S|², with the extremal values of r(S) corresponding to sum-free subsets and subgroups of G. In this paper, we consider the intermediate values which r(S) may take, particularly in the setting where G is under addition (p prime).  We obtain various bounds and results.  In the setting, this work may be viewed as a subset generalization of the Cauchy-Davenport Theorem.