RUTGERS ALGEBRA SEMINAR - Fall 2005

The Algebra Seminar meets on Mondays, at 4:40-5:40PM 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 4:40PM 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 2008 Seminars

 5 Sep(F)# Paul Baum   Penn State "Morita Equivalence Revisited" Talk is at 2PM in H705
 8 Sep(M)
15 Sep
22 Sep
29 Sep
 6 Oct
13 Oct
20 Oct
27 Oct
 3 Nov
10 Nov Guillermo Cortinas   Buenos Aires  TBA
17 Nov
24 Nov
 1 Dec
 8 Dec
Fall 2008 Semester begins Tuesday Sept 2; Final Exams begin Monday Dec 15, 2008.

Spring 2008 Seminars

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)

Fall 2007 Seminars

 7 Sep* Benjamin Doyon  Durham      Conformal field theory and Schramm-Loewner evolution
14 Sep* Liang Kong      Max Planck  An introduction to open-closed conformal field theory
28 Sep Richard Lyons    Rutgers    Presidential Address and Department Reception
 5 Oct Diane Maclagan  Rutgers-Warwick Starts at 2:15! "Equations for Chow and Hilbert quotients"
12 Oct Rafael Villareal IPN,Mexico  "Unmixed clutters with a perfect matching"
19 Oct  POSTPONED to November 16
2 Nov# Andrea Miller    Harvard   POSTPONED
 9 Nov Dan Krashen       U. Penn   Starts at 2:20!Patching subfields of division algebras
16 Nov Angela Gibney     U. Penn   A new candidate for the nef cone of M0,n 
23 Nov Tom Turkey     Plymouth Colony  ---------Thanksgiving Break-----------
 3 Dec: Dirk Kreimer IHES (France) Monday at 4:40! Hopf and Lie algebras for renormalizable quantum field theories
 7 Dec V. Retakh	 Rutgers   date(s) to change   TK

Fall Classes began September 4, 2007;  Final Exams began Friday, Dec 14, 2007.


Abstracts of seminar talks


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.

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.

Fall 2007

Conformal field theory and Schramm-Loewner evolution (Benjamin Doyon, Sept. 7, 2007):
The scaling limit of two-dimensional statistical models at criticality can be described by two theoretical frameworks: conformal field theory (that is, vertex operator algebras, their modules and representations), and Schramm-Loewner evolution (SLE). The first one has a long history, starting more than 20 years ago with works by both mathematicians and physicists, whereas the second one encompasses recent advances, starting in 2000 with a paper of Schramm until generalisations still under construction. The two frameworks seem quite unrelated in their formulation as well as in their applications. But it is nowadays believed by many that understanding the relation between them will allow us to make important steps in the understanding, both physical and mathematical, of critical regimes of statistical models. I will review the frameworks, advances made in relating them, and the many open problems. This talk will be accessible to non-specialists.

An introduction to open-closed conformal field theory (Liang Kong, Sept. 14, 2007):
Open-closed conformal field theory describes the perturbative open-closed string theory and some critical phenomena in condensed matter physics. It provides a powerful tool to study the still mysterious object called "D-brane", which is important to Kontsevich's homological mirror symmetry program. In this talk, I will outline a mathematical study of open-closed conformal field theory based on the theory of vertex operator algebra. In particular, I will give a tensor-categorical formulation of rational open-closed conformal field theory. I will also briefly discuss what D-branes are in our framework. This talk will be accessible to graduate students who know the definition of category.

Patching subfields of division algebras (Dan Krashen, Nov. 9, 2007):
There has been much work recently in understanding the structure of division algebras whose center is "2-dimensional." For example, in the case that the center is the function field of an algebraic surface, de Jong has shown that every such algebra has a cyclic maximal subfield. In this talk I will describe joint work with Harbater and Hartmann which uses the recent method of "field patching" (related to formal geometry) to understand all possible Galois groups of maximal subfields of division algebras over function fields of certain arithmetic surfaces.

Hopf and Lie algebras for renormalizable quantum field theories (Dirk Kreimer, Dec. 3, 2007):
Physicists have used the combinatorics of renormalization and the renormalization group routinely for a long time. The identification of the underlying algebraic structures in terms of Hopf and Lie algebras is more recent. We explain these algebras and their role in understanding Green functions in quantum field theory.


A new candidate for the nef cone of M0,n (Angela Gibney, Nov. 16, 2007):
There is a well known upper bound $F_{n}$ for the nef cone Nef$(\overline{M}_{0,n})$ of $\overline{M}_{0,n}$. The cone $F_{n}$ is an explicitly defined, polyhedral cone that contains Nef$(\overline{M}_{n})$. The F-conjecture asserts that Nef$(\overline{M}_{n})=F_{g,n}$. In this talk, I will describe a new candidate for the nef cone of $\overline{M}_{0,n}$. This is a polyhedral cone $C_{n}$ that Sean Keel, Diane Maclagan and I have proved is a sub cone of $F_{n}$. We can show that if $F_{n}$ were also contained in $C_{n}$, then it would imply that Nef$(\overline{M}_{0,n})=F_{n}=C_{n}$.

Spring 2007

Vertex operator algebras and recurrence relations (Bill Cook, March 30, 2007):
There are many important classes of examples of vertex operator algebras including Heisenberg VOAs, Virasoro VOAs, lattice VOAs, and the VOAs associated with affine Lie algebras.

We will begin with an introduction to the class of VOAs (along with their modules) associated with affine Lie algebras. Then in the latter part of the talk we will discuss an interesting theorem of Haisheng Li. Applying this theorem to our class of examples, we will obtain recurrence relations among the characters of these Vertex Operator Algebras (and VOA modules).

On a certain family of W-algebras (Antun Milas, April 7, 2007):
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.

G-equvariant modular categories and Verlinde formula (Vincent Graziano, April 13, 2007):
Many features of a conformal field theory can be captured in the language of categories. Modular tensor categories provide the appropriate framework and we will start by discussing the properites of such a category. We will then introduce the Verlinde algebra associated to such a category, the action of the S-matrix, and the Verlinde formula.

Our goal will be to generalize this setup to the case of theories with additional symmetries, such as a vertex operator algebra with a finite group of symmetries. We discuss the extended Verlinde algebra, the S-matrix, and the 'extended' Verlinde formulas.

Vertex-algebraic structure of certain modules for affine Lie algebras underlying recursions
(Corina Calinescu, April 20, 2007):

Many combinatorial identities and recursions have been proved or conjectured via vertex operator constructions of representations of affine Lie algebras.

In this talk we discuss vertex-algebraic structure of the principal subspaces of all the standard A1(1)-modules and we prove suitable presentations for these subspaces. These presentations were used by Capparelli, Lepowsky and Milas for the purpose of obtaining the classical Rogers-Ramanujan and Rogers-Selberg recursions. This is joint work with Jim Lepowsky and Antun Milas.

A Formal Variable Approach to Special Hyperbinomial Sequences
(Tom Robinson, April 27, 2007):

In a nearly self-contained and elementary treatment, we develop the formal calculus used in the theory of vertex algebras to describe certain formal changes of variable. In particular, we extend the logarithmic formal Taylor theorem as found in the work of Y.Z. Huang, J. Lepowsky, and L. Zhang. We apply our results to obtain combinatorial identities concerning generalizations of the Stirling numbers and find that our development leads naturally to a combinatorial definition of the exponential Riordan group which was studied by L.W. Shapiro, S. Getu, W.J. Woan, and L.C. Woodson.


Charles Weibel / weibel @ math.rutgers.edu / May 5, 2008