Asterisks (*) mark meetings of the
QUANTUM MATH SEMINAR, which occasionally replaces the Algebra Seminar.
The marks '* :' refers to a joint ALGEBRA/QUANTUM meeting.
A '#' marks a joint meeting with the LIE GROUPS seminar.
A '(C)' marks a related Colloquium Talk at 4:00 PM on a Friday.
Click here for the algebra seminars in
previous semesters
26 Jan no seminar -------------- Job Talks? ------------- 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"
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 H425Fall 2008 Semester begins Tuesday Sept 2; Final Exams begin Monday Dec 15, 2008 and Math Group Exams are Dec. 15 (4-7PM).
25 Jan(F) W. Vasconcelos Rutgers The Chern numbers of a local ring (I) 28 Jan: Vladimir Retakh Rutgers "Obstructions to formality and obstructions to deformations" 4 Feb: Chuck Weibel Rutgers "Generation of Galois cohomology by symbols" 5 Feb(T)* Tony Milas SUNY Albany "W-algebras, quantum groups and combinatorial identities" 8 Feb(F) M. Zieve Rutgers "The lattice of subfields of K(x) 11 Feb: Zin Arai Kyoto Univ "Complex dynamics and shift automorphism groups" 18 Feb: Andrzej Zuk Univ Paris "Automata Groups" 25 Feb: Mike Zieve Rutgers "Automorphism groups of curves" 29 Feb(F) Laura Ghezzi CUNY "Generalizations of the Strong Castelnuovo Lemma" 3 Mar: Chuck Weibel Rutgers "Model categories versus derived categories" 10 Mar: R Parimala Emory Univ. "Rational points on homogeneous spaces" 14 Mar#* Tom Robinson Rutgers "Formal differential representations" 11:55 AM Friday in Hill 425 17 Mar: no seminar -------------- Spring Break ------------- 28 Mar#* David Ben-Zvi IAS & U.Texas "Real Groups and Topological Field Theory" 28 Mar(F) Jooyoun Hong Purdue "Homology and Elimination" 31 Mar: Siddhartha Sahi Rutgers "Tensor categories and equivariant cohomology" 4 Apr(C) David Saltman CCR and U.Texas "Division Algebras over Surfaces" 7 Apr: Earl Taft Rutgers "The boson-fermion correspondence and one-sided quantum groups 14 Apr: Colleen Duffy Rutgers "Graded traces and irreducible representations of graph algebras" 21 Apr: Semyon Alesker Tel-Aviv U. "Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets" 28 Apr: Jim Borger Australia Natl Univ. "Witt vectors, Lambda-rings, and absolute algebraic geometry" 5 May: Richard Lyons Rutgers "Subgroups of Algebraic Groups and Finite Groups" Spring 2008 Semester begins Tuesday Jan 22; Spring Finals are May 8-14, 2008 (Spring Break is March 15-23, 2008)
Stability conditions for triangulated categories
(Chuck Weibel, Feb. 2, 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.
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):
K-theory of some algebras associated to quivers
(Guillermo Cortiñas, November 14, 2008):
Reduced K-theory of Azumaya algebras
(Roozbeh Hazrat, December 1, 2008):
The theory of Azumaya algebras developed parallel to the theory of
central simple algebras. However the latter are algebras over fields
whereas the former are algebras over rings. One wonders how the K-theory
of these objects compare to each other. We look at higher K-theory and
reduced K-theory of these objects. We ask nice questions!
W-algebras, quantum groups and combinatorial identities
(Antun Milas, Feb. 5, 2008):
The lattice of subfields of K(x)
(Mike Zieve, Feb. 8, 2008:
Complex dynamics and shift automorphism groups
(Zin Arai, Feb. 11, 2008):
In this talk, we consider the monodromy homomorphism for the complex
Henon map, a 2-dimensional analog of the quadratic map. We need the
shift space of bi-infinite sequences in this case, and the automorphism
group of this space is much more complicated than that of the one-sided
shift space. We propose a computer-assisted method to compute the
monodromy homomorphism and show that automorphisms of the shift space
can be used to determine the dynamics of the real Henon map.
Automorphism groups of curves
(Mike Zieve, Feb. 25, 2008):
Model categories versus derived categories
(Chuck Weibel, march 3, 2008):
Rational points on homogeneous spaces
(Parimala, March 10, 2008):
Formal differential representations, Faa di Bruno and the Riordan Group
Real Groups and Topological Field Theory
David Ben-Zvi, March 28, 2008:
The boson-fermion correspondence and one-sided quantum groups
(Earl Taft, April 7, 2008):
Plurisubharmonic functions on the octonionic plane and
Spin(9)-invariant valuations on convex sets
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
Given a quiver Q and a field k, it is possible to associate several
k-algebras. Best known among them is the path algebra, PQ.
Localizing PQ one obtains a new algebra, the Leavitt algebra LQ.
This algebra is equipped with an involution. If k is the field of
complex numbers, LQ may be view as an algebra of operators in Hilbert
space; its completion in the operator norm gives a C*-algebra, the
Cuntz-Krieger algebra of the quiver. The topological K-theory of the
Cuntz-Krieger algebra was computed in a now classical paper of Cuntz.
In the talk we will discuss recent joint results with Pere Ara and
Miquel Brustenga concerning the algebraic K-theory of LQ and its
relation with the topological K-theory of the Cuntz-Krieger algebra.
Spring 2008
I will discuss a conjectural relationship between certain quantum
W-algebras (vertex algebras) and finite-dimensional quantum groups
associated to $sl_2$ (Hopf algebras). In the process we shall
encounter interesting multisum identities.
I will present various results about the lattice of fields between K and K(x),
where K is a field. These include classical results of Ritt, Schinzel,
Fried, et al., as well as new results. I will also give some applications,
for instance a recent joint result with Ghioca and Tucker describing all
pairs of complex polynomials having orbits with infinite intersection.
Symbolic dynamics is a standard and powerful tool to understand
chaotic dynamics. For example, we can identify the Julia set of
quadratic polynomials with the one-sided shift space, the space
of infinite sequences of 0 or 1, provided the parameter of the
map is outside the Mandelbrot set. Furthermore, via the monodromy
homomorphism, the topological structure of the Mandelbrot set is
also captured by the automorphism group of the shift space.
Hurwitz proved that a complex curve of
genus g>1 has at most 84(g-1) automorphisms.
In case equality holds, the automorphism group
has a quite special structure. However, in a
qualitative sense, all finite groups G behave the
same way: the least g>1 for which G acts on a
genus-g curve is on the order of (#G)*d(G), where
d(G) is the minimal number of generators of G.
I will present joint work with Bob Guralnick on
the analogous question in positive characteristic.
In this situation, certain special families of
groups behave fundamentally differently from
others. If we restrict to G-actions on curves
with ordinary Jacobians, we obtain a precise
description of the exceptional groups and curves.
Quillen invented the notion of a model category in order to do
homotopical algebra. We will consider these structures on the categories
of R-modules, presheaves and sheaves, and show how localization works.
We discuss the following open concerning rational points
on homogeneous spaces under connected linear algebraic groups.
If a homogeneous space under a connected linear algebraic group
has a zero cycle of degree one, does it admit a rational point?
We explain the arithmetic case and some recent progress
concerning this question for more general fields.
(Tom Robinson, March 14, 2008):
First I will show explicitly how a calculation in
Frenkel-Lepowsky-Meurman's book on vertex operator algebras, which I
will in its essentials redo, can be viewed as an application of a
formal representation of exponentiated derivations. The outcome of
the calculation is Faa di Bruno's formula for the higher derivatives
of a composite function. Then building on this result I will show how
another application of an easy class of formal differential
representation leads to the Riordan Group. No prerequisites
necessary.
I will explain current joint work with David Nadler, in which the
representation theory of real reductive Lie groups is examined through
the lens of topological field theory and the geometric Langlands
program. Our main results show how to recover the representation
theory of real forms of a complex group G from the representation
theory of G, and how to deduce a Langlands dual description of the
representation theory (a form of Soergel's conjecture, generalizing
results of Vogan and Langlands).
Recent quantizations of the boson-fermion correspondence of classical
physics use one half of the relations for the bialgebra of quantum
matrices. Using this philosophy, A.Lauve, S. Rodriguez and myself have
independently constructed certain one-sided qauntum groups, i.e.,
there is a left antipode which is not a right antipode. We will
explain the connections between these two quantizations.
(Semyon Alesker, April 21, 2008):
We introduce a class of plurisubharmonic functions on the
octonionic plane O² and establish basic results about it. Then we apply
these results to produce new examples of continuous valuatons on convex
subsets of O²=R^{16}, in particular valuations invariant under the group
Spin(9). The constructions use the determinant of octonionic hermitian
matrices of size 2.