RUTGERS ALGEBRA SEMINAR - Fall 2009

The Algebra Seminar meets on Mondays, at 4:50-5:50PM in H705 (in the Hill Center, on Busch Campus of Rutgers University).
A more comprehensive listing of all Math Department seminars is available.

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


The seminar will meet at 4:50PM Mondays in H705, and will sometimes be a joint meeting with the Gelfand Seminar. Seminars marked (F) are meetings of the Computational Algebra seminar, which meets on Fridays at 1:00 PM Fall 2009 Semester begins Monday Sept 1; Labor Day is Sept. 7 Final Exams begin Wednesday Dec 16, 2009; Math Group Exams are Dec. 16 (4-7PM).

Spring 2009 Seminars

 
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"

Spring Break is March 14-22, 2009; Final Exams begin Thursday May 7.

Fall 2008 Seminars (at 4:00 Mondays)

 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).

Spring 2008 Seminars (at 4:40 Mondays)

 
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)


Abstracts of seminar talks


Spring 2009

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):
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.

Stability conditions in Physics (Jan Manschot, March 2, 2009):
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 for Toric Surfaces (Elizabeth Gasparim, March 30, 2009):
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.

Differential Quasifields (Bill Keigher, April 13, 2009):
In a recent seminar (2/23), Leon Pritchard talked about partitioned differential quasifields.

Morphisms of cohomological field theories and behavior of Gromov-Witten invariants under quotients
(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.

higher order Hochschild (co)homology (Gregory Ginot, April 27, 2009):
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

Morita Equivalence Revisited (Paul Baum, Sept. 5, 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.

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.
The above is joint work with A.M.Aubert and R.J.Plymen.

Formality theorems for Hochschild (co)chains and their applications (Vasily Dolgushev, Sept. 18, 2008):
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.

Rationality and integrality in dynamical systems (Mike Zieve, Sept. 22, 2008):
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.

Combinatorial identities and Hopf algebras (Earl Taft, October 20, 2008):
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.

Infinite towers of cocompact lattices in Kac-Moody groups (Leigh Cobbs, November 3, 2008):
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

... Γ3 < Γ2 < Γ1 < Γ
of non-conjugate cocompact lattices in G. We give the graph of groups structure of these and other cocompact lattices, and discuss extensions of these infinite towers to rank-3 Kac-Moody groups using complexes of groups.

K-theory of some algebras associated to quivers (Guillermo Cortiñas, November 14, 2008):
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.

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!

Spring 2008

W-algebras, quantum groups and combinatorial identities (Antun Milas, Feb. 5, 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.

The lattice of subfields of K(x) (Mike Zieve, Feb. 8, 2008:
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.

Complex dynamics and shift automorphism groups (Zin Arai, Feb. 11, 2008):
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.

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):
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.

Model categories versus derived categories (Chuck Weibel, march 3, 2008):
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.

Rational points on homogeneous spaces (Parimala, March 10, 2008):
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.

Formal differential representations, Faa di Bruno and the Riordan Group
(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.

Real Groups and Topological Field Theory David Ben-Zvi, March 28, 2008:
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).

The boson-fermion correspondence and one-sided quantum groups (Earl Taft, April 7, 2008):
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.

Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets
(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.

Witt vectors, Lambda-rings, and absolute algebraic geometry (Jim Borger, April 28, 2008):
I'll give an introduction to Witt vectors and Lambda-rings, and I'll explain how they're two different ways of looking at the same concept. Then I'll discuss how these give a "Lambda-equivariant" algebraic geometry, how it relates to usual algebraic geometry, and why one might care about it.

Subgroups of Algebraic Groups and Finite Groups (Richard Lyons, May 5, 2008):
We will discuss some similarities and differences between the subgroup structures of connected linear algebraic groups and finite groups.


Charles Weibel / weibel @ math.rutgers.edu / Aril 28, 2009