A '(C)' marks a related Colloquium Talk at 4:00 PM on a Friday.
A '#' marks a joint meeting with the LIE GROUPS seminar.
Click here for the algebra seminars in
previous semesters
1 Feb Max Karoubi Univ. Paris 7 "Periodicity in Hermitian K-groups" 15 Feb Chuck Weibel Rutgers Exceptional objects (after Polishchuk) 22 Feb 1 Mar Ray Hoobler CCNY tentative, about Brauer & Witt groups 8 Mar 15 Mar no seminar -------------- Spring Break ------------- 22 Mar Earl Taft Rutgers "Recursive sequences and Hopf algebras" 29 Mar 5 Apr Carlo Mazza (Univ. Genoa) TBA 12 Apr 19 Apr 26 Apr 3 May Spring Break is March 13-21, 2010; Final Exams begin Thursday May 6.
14 Sep no seminar
28 Sep no seminar Yom Kippur
5 Oct Lourdes Juan Texas Tech Differential Central Simple Algebras and Picard-Vessiot representations
12 Oct Bob Guralnick USC Derangements in Finite and Algebraic Groups
19 Oct Ken Johnson Penn State-Abington Mathematics arising from a new look at the
Dedekind-Frobenius group matrix and group determinant
2 Nov Chloe Perin Hebrew Univ. "Induced definable structure on cyclic subgroups of the free group"
9 Nov Paul Ellis U. Connecticut "The classfication problem for finite rank dimension groups"
16 Nov Ravi Srinivasan RU-Newark "Picard-Vessiot Theory"
23 Nov Vladimir Retakh Rutgers "Noncommutative algebra and combinatorial topology"
30 Nov Chuck Weibel Rutgers "homotopy model structures as tools for homogical algebra"
7 Dec no seminar cancelled due to Gelfand Memorial
Fall 2009 Semester begins Tuesday Sept 1; Labor Day is Sept. 7
Final Exams begin Wednesday Dec 16, 2009; Math Group Exams are Dec. 16 (4-7PM).
2 Feb Chuck Weibel Rutgers "Stability conditions for triangulated categories" 9 Feb Luis Caffarelli U. Texas Special Colloquium talk at this time 16 Feb Vladimir Retakh Rutgers "Lie algebras over noncommutative rings" 23 Feb Leon Pritchard CUNY "Partitioned differential quasifields" 2 Mar Jan Manschot Rutgers-Physics "Stability conditions in physics" 16 Mar no seminar -------------- Spring Break ------------- 30 Mar Elizabeth Gasparim Edinburgh The Nekrasov Conjecture for Toric Surfaces" 6 Apr Vladimir Retakh Rutgers "Noncommutative Laurent phenomenon" 13 Apr Bill Keigher Rutgers-Newark "Differential quasifields" 20 Apr Chris Woodward Rutgers "Morphisms of cohomological field theories and behavior of Gromov-Witten invariants under quotients" 27 Apr Gregory Ginot Univ.Paris "higher order Hochschild (co)homology"
Spring Break is March 14-22, 2009; Final Exams begin Thursday May 7.
5 Sep(F)# Paul Baum Penn State "Morita Equivalence Revisited" Talk is at 2PM in H705 15 Sep(M) no seminar MSMF Reception 18 Sep(Th) Vasily Dolgushev UC Riverside "Formality theorems for Hochschild (co)chains and their applications" Talk at 2PM in H425 22 Sep Mike Zieve Rutgers "Rationality and integrality in dynamical systems" 29 Sep no seminar Rosh Hoshanna 6 Oct Chuck Weibel Rutgers "The de Rham-Witt complex of R[t]" 13 Oct Anders Buch Rutgers "Quantum K-theory" 20 Oct Earl Taft Rutgers "Combinatorial Identities and Hopf Algebras" 27 Oct Siddhartha Sahi Rutgers "Interpolation and binomial identities in several variables" 3 Nov Leigh Cobbs Rutgers "Infinite towers of co-compact lattices in Kac-Moody groups" 10 Nov Jarden Logic Seminar "The absolute Galois group of subfields of the field of totally S-adic numbers" 14 Nov(F) Guillermo Cortiñas Buenos Aires "K-theory of some algebras associated to quivers" Talk is at 2PM in H425 17 Nov no seminar ------- ------------------------------ 24 Nov Robert Wilson Rutgers "Splitting Algebras associated to cell complexes" 1 Dec Roozbeh Hazrat Queens Univ. Belfast "Reduced K-theory of Azumaya algebras" 9 Dec(T) Steven Duplij Kharkov Univ. "Quantum Enveloping Algebras and the Pierce Decomposition " Talk is Tuesday, 2PM in H425 Fall 2008 Semester begins Tuesday Sept 2; Final Exams begin Monday Dec 15, 2008 and Math Group Exams are Dec. 15 (4-7PM).
Periodicity of hermitian K-groups
(Max Karoubi, Feb. 1, 2010):
This is joint work with Jon Berrick and Paul Arne Ostvaer.
It has been known for a few years,
essentially by the work of Voevodsky and Rost,
that the algebraic K-theory of a commutative
ring A with finite coefficients is periodic above
the etale cohomological dimension of A. In this lecture,
we show that such a ring A
has also a periodic hermitian K-theory in the same range.
This essentially means that theorems about the general (infinite) linear
group, such as the one proved by Rost and Voevodsky,
imply similar ones for the orthogonal and symplectic groups.
Differential Central Simple Algebras and Picard-Vessiot representations
(Lourdes Juan, Oct. 5, 2009):
A differential field is a field K with a derivation, that is,
an additive map D:K → K satisfying D(fg)=D(f)g+fD(g)
for f,g in K. The field of constants C of K are the zeros of D.
A differential central simple algebra (DCSA) over K is a pair
(A,\mathcal D) where A is a central simple algebra and $\mathcal D$
is a derivation of A extending the derivation D of its center.
Any DCSA, and in particular a matrix differential algebra over K,
can be trivialized by a Picard-Vessiot (differential Galois) extension
E of K. In the matrix algebra case, there is a correspondence between
K-algebras trivialized by E and representations of the differential
Galois group of E over K in PGLn(C) that can be
interpreted as cocycles equivalent up to coboundaries. I will start with
a brief introduction to differential Galois theory.
Derangements in Finite and Algebraic Groups
(Bob Guralnick, Oct. 12, 2009):
Mathematics arising from a new look at the
Dedekind-Frobenius group matrix and group determinant
(Ken Johnson, Oct. 19, 2009):
Induced definable structure on cyclic subgroups of the free group
The classfication problem for finite rank dimension groups
(Paul Ellis, Nov. 9, 2009):
Picard-Vessiot Theory (Ravi Srinivasan, Nov.16, 2009):
Stability conditions for triangulated categories
(Chuck Weibel, Feb. 2, 2009):
Partitioned Differential Quasifields
(Leon Pritchard, Feb. 23, 2009):
Stability conditions in Physics (Jan Manschot, March 2, 2009):
The Nekrasov Conjecture for Toric Surfaces
(Elizabeth Gasparim, March 30, 2009):
Differential Quasifields (Bill Keigher, April 13, 2009):
Morphisms of cohomological field theories and
behavior of Gromov-Witten invariants under quotients
higher order Hochschild (co)homology
(Gregory Ginot, April 27, 2009):
Morita Equivalence Revisited (Paul Baum, Sept. 5, 2008):
This talk studies an equivalence relation
between k-algebras which is a weakening of Morita
equivalence. If A and B are equivalent in the new equivalence
relation, then A and B have isomorphic periodic cyclic
homology, and Prim(A) is in bijection with Prim(B). However, the
bijection between Prim(A) and Prim(B) might not be
a homeomorphism. Thus the new equivalence relation permits a tearing
apart of strata in the primitive ideal spaces
which is not allowed by Morita equvalence. An application to the
representation theory of p-adic groups will be briefly
indicated. This talk is intended for non-specialists. All the basic
definitions will be carefully stated.
Formality theorems for Hochschild (co)chains and their applications
(Vasily Dolgushev, Sept. 18, 2008):
Rationality and integrality in dynamical systems
(Mike Zieve, Sept. 22, 2008):
Combinatorial identities and Hopf algebras
(Earl Taft, October 20, 2008):
Infinite towers of cocompact lattices in Kac-Moody groups
(Leigh Cobbs, November 3, 2008):
A permutation on a set is called a derangement if it has
no fixed points. The study of the proportion of derangements
in finite transitive groups has a long history and the problem has
many applications. We will discuss this as well as the analogous
problem for algebraic and show the connection between the two.
In particular, we will discuss recent results (joint with Fulman)
about conjugacy classes in finite Chevalley groups and the solution
of a conjecture made independently by Aner Shalev and Nigel Boston.
Frobenius invented group character theory in order to solve
the problem of the factorization of the group determinant. His papers
are hard to understand and when the modern methods for group
representation theory were introduced his initial work was largely
forgotten. To each representation of a (finite) group there is
associated a polynomial which is a factor of the group determinant,
and Frobenius introduced "k-characters" to describe this
polynomial. Professor Gelfand has commented that perhaps physicists
might benefit from looking at these polynomials. Among other places
these k-characters have occurred in work of Buchstaber and Rees and
also are related to work of Wiles and Taylor on
"pseudocharacters" of finite dimensional representations of
infinite groups.
I will describe the early work from an elementary
point of view and give an account of some of the new ideas coming from
it, and also indicate some of the connections with probablity.
(Chloe Perin, Nov. 2, 2009):
Let C be a cyclic subgroup of a finitely generated free group F. We
show that the intersection of a definable set D in F^n with C^n is in
the Boolean algebra of cosets of subgroups of C^n. In other words, the
definable structure induced by the embedding of C in F is no richer
than the definable structure on C. We make extensive use of Sela's
geometric techniques for studying the first-order theory of the free
group, in particular of his construction of "formal solutions" to an
equation.
An unperforated partially ordered abelian group A is a
dimension group if A satises the Riesz interpolation property
(given a,a' ≤b,b' there is a c with a,a' ≤ c ≤b,b').
These are related to "Bratteli diagrams". Paul will discuss
the difficulty of classifying them when the rank is at least 3,
and show that the problem for a given rank cannot be reduced to the
classification problem for a smaller rank.
Let F be a characteristic zero differential field with an
algebraically closed field of constants C. I will describe the construction
of a Picard-Vessiot Extension (PVE) for a linear homogeneous
differential equation over F. The group of differential automorphisms
of a PVE fixing F is called the differential Galois group;
there is a Galois correspondence between its algebraic subgroups
and intermediate differential subfields. Examples of PVEs for F=C(x)
with the usual derivation will be discussed, and we will also compute
the differential Galois group for our examples.
Spring 2009
This is an introductory survey talk.
There is a complex topological manifold, called the Stability Space,
associated to any triangulated category D. It was conceived by Mike
Douglass as an aspect of string theory, and made mathematical by
Tom Bridgeland. Subspaces correspond to t-structures, and the stability
space of the projective line is the affine complex plane.
A differential quasifield is a natural generalization of a differential
field in characteristic p>0. Elementary properties of
differential quasifields are considered, and a generalized version
of the theorem on the connection between linear independence over
constants and the Wronskian is presented.
In a recent seminar (2/2/09), C. Weibel discussed recent developments on
stability in (triangulated) categories. These developments are inspired by
physics, in particular string theory. This introductory talk will explain
the notion of stability in string theory, and how it is connected to
stability in mathematics.
The Nekrasov conjecture predicts a relation between the partition
function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten
prepotential. For instantons on ℝ4, the conjecture
was proved, independently and using different methods, by Nekrasov-Okounkov,
Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of
the conjecture for instantons on noncompact toric surfaces.
In a recent seminar (2/23), Leon Pritchard talked about partitioned
differential quasifields.
(Chris Woodward, April 20, 2009):
I will talk about a "quantum non-abelian localization" conjecture that
relates Gromov-Witten invariants of GIT quotients with equivariant
Gromov-Witten invariants of the total space. Some special cases are
proved. A key notion in the conjecture is the notion of morphism of
cohomological field theories, which "complexifies" the notion of
A-infinity morphism.
We will explain how one can define Hochschild (co)chain complex
associated in a functorial way to any space X, CDG algebra A and
A-module M. We will give several examples and applications to Adams
operations and (if time permits) Brane topology.
Fall 2008
Notation: k denotes a unital algebra over the complex numbers which is
commutative, finitely generated, and nilpotent-free,
i.e., k is the coordinate algebra of a complex affine variety. A
k-algebra is an algebra A over the complex numbers
which is a k-module such that the algebra structure and the k-module
structure are compatible in the evident way.
Note that A is not required to be commutative. Prim(A) denotes the
set of primitive ideals in A. Prim(A) is topologized
by the Jacobson topology.
The above is joint work with A.M.Aubert and R.J.Plymen.
I will start my talk with a review of the algebraic
operations on the pair Hochschild cochain complex
and Hochschild chain complex of an associative algebra.
Then I will speak about the formality theorems
for these complexes. Finally I will discuss applications
of these formality theorems to deformation quantization,
computation of Hochschild (co)homology and
the Kashiwara-Vergne conjecture.
I will present various results about the arithmetic of dynamical systems
given by iterating a polynomial mapping over a ring. Sample topics
include: describing the minimal N for which the backward orbit of a point
under a given polynomial over a number field K contains infinitely many
points of degree N over K; and determining the possible lengths of
periodic and preperiodic forward orbits of a point under a polynomial
mapping of a ring. I will also discuss connections with torsion in
abelian varieties, Sen's theorem (Grothendieck's H^1 conjecture), and the
Nottingham group.
R. G. Larson and E. J. Taft showed that the space of linearly
recursive sequences is a bialgebra. A coproduct formula for such a
sequence can be interpreted as a quadratic identity on the coordinates
of the sequence. This was extended by C. A. Futia, E. F. Mueller and
E. J. Taft[CMT] to D-finite sequences. This means that from some point
on, each coordinate is a linear combination of previous coordinates
with variable(polynomial) coefficients. These D-finite sequences form a
topological bialgebra, i.e., the coproduct is an infinite sum of tensor
products of such sequences. Such a coproduct formula can still be
interpreted as a quadratic identity on the coordinates, often of a
combinatorial nature. In [FMT], we obtained such formulae and
identities for the sequences (n!) and (n(n!)). Here we extend this to
the sequences whose n-th term is ((n/k)(n!)) for each k=2, 3, 4,....
Here (n/k) is the binomial coefficient.
Let G be a locally compact Kac-Moody group of affine or hyperbolic type
over a finite field Fq; G admits an action on its
Tits building X. In the setting rank(G)=2, X is a locally finite,
homogeneous tree. We can then use the combinatorial tools of
Bass-Serre theory, namely graphs of groups, to construct discrete
subgroups of G. We show that if q=2 then G contains a cocompact lattice
Γ whose quotient Γ\X equals G\X, a simplex. We then give
two distinct constructions of infinite towers