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
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 DecFall 2008 Semester begins Tuesday Sept 2; Final Exams begin Monday Dec 15, 2008.
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)
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.
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.
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:
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
(Semyon Alesker, April 21, 2008):
Witt vectors, Lambda-rings, and absolute algebraic geometry
(Jim Borger, April 28, 2008):
Subgroups of Algebraic Groups and Finite Groups
(Richard Lyons, May 5, 2008):
Conformal field theory and Schramm-Loewner evolution
(Benjamin Doyon, Sept. 7, 2007):
An introduction to open-closed conformal field theory
(Liang Kong, Sept. 14, 2007):
Patching subfields of division algebras
(Dan Krashen, Nov. 9, 2007):
Hopf and Lie algebras for renormalizable quantum field theories
(Dirk Kreimer, Dec. 3, 2007):
Vertex operator algebras and recurrence relations
(Bill Cook, March 30, 2007):
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):
G-equvariant modular categories and Verlinde formula
(Vincent Graziano, April 13, 2007):
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.
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.
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.
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.
We will discuss some similarities and differences between the subgroup
structures of connected linear algebraic groups and finite groups.
Fall 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.
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.
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.
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
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.
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.
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.