24 Jan Daniel Erman Michigan "Equations, syzygies, and vector bundles" 30 Jan David Anderson U. Paris "Equivariant Schubert calculus: positivity, formulas, applications" 6 Feb Chuck Weibel Rutgers "What is a Derivator?" 13 Feb V. Retakh Rutgers "A geometric approach to noncommutative Laurent phenomenon" 20 Feb Tatiana Bandman Bar-Ilan "Dynamics and surjectivity of someSpring 2013 Semester starts Jan. 22, Classes end Monday May 6,
word maps on SL(2,q)" 27 Feb Bob Guralnick USC and IAS "Strongly Dense Subgroups of Algebraic Groups" 13 Mar Mina Teicher Bar-Ilan "The 3 main problems in the braid group" 20 Mar no seminar -------------- Spring Break ------------- 3 Apr Joe Ross USC "Intersection theory on singular varieties" 10 Apr Lev Borisov Rutgers "Hilbert modular threefolds of discriminant 49" 17 Apr Charlie Siegel (IPMU Japan) "Cyclic Covers, Prym Varieties and the Schottky-Jung Relations" 24 Apr Freya Pritchard CUNY "Implicit systems of differential equations" 1 May Alexei Stepanov (St.Petersburg State University) "Structure of Chevalley groups over rings"
19 Sept Chuck Weibel Rutgers "Binary codes and Galois covers of varieties" 10 Oct Anders Buch Rutgers "Curve neighborhoods" 17 Oct Dan Grayson IAS "Computations in intersection theory" 24 Oct Justin Lynd Rutgers "Fusion systems with prescribed involution centralizers" 31 Oct Lev Borisov Rutgers "On Hilbert modular threefolds of discriminant 49" 7 Nov Oliver Rondigs Osnabruck, Germany "On the slice filtration for hermitian K-theory" 14 Nov Howie Nuer Rutgers "surfaces on Calabi-Yao 3-folds" 21 Nov no seminar, Friday classes (Thanksgiving week) 28 Nov Susan Durst Rutgers "Universal labelling algebras for directed graphs" 5 Dec Anastasia Stavrova U.Essen "Injectivity property of etale H^1, non-stable K_1, and other functors" 12 Dec Joe Ross USC "Presheaves with oriented weak transfers"Fall 2012 Semester starts Sept.4; (Wednesday Nov. 21 will be Friday classes).
25 Jan Vasily Dolgashev Temple Univ. "Exhausting quantization procedures" 8 Feb Chuck Weibel Rutgers "Shift Equivalence and Z[t]-modules" 15 Feb Pablo Pelaez Rutgers "An introduction to weights" 22 Feb Anders Buch Rutgers "K-theory of miniscule varieties" 29 Feb Julia Plavnik U.Cordoba "From algebra to category theory: a first approach to fusion categories 7 Mar Anastasia Stavrova U.Essen "On the unstable K_1-functors associated to simple algebraic groups" 14 Mar no seminar -------------- Spring Break ------------- 21 Mar Mark Walker U. Nebraska "Invariants of Matrix Factorizations" 28 Mar Lev Borisov Rutgers "Combinatorial aspects of toric mirror symmetry" 5 Apr Joe Ross USC "Cohomology Theories with Supports" Thursday 11:00AM in Hill 425 11 Apr V. Retakh Rutgers "Noncommutative Laurent Phenomena" 18 Apr Ben Wyser U.Georgia "Symmetric subgroup orbit closures on flag varieties as universal degeneracy loci" 25 Apr Ling Bao Chalmers U. (Sweden) "Algebraic symmetries in supergravity"Spring 2012 Semester starts Jan. 17, Classes end April 30,
Computations in intersection theory
(Dan Grayson, Oct. 17, 2012):
On the slice filtration for hermitian K-theory
(Oliver Rondigs, November 7, 2012):
Injectivity property of etale H^1, non-stable K_1, and other functors
(Anastasia Stavrova, December 5, 2012):
Exhausting quantization procedures
(Vasily Dolgashev, Jan. 25, 2012):
Shift Equivalence and Z[t]-modules (Chuck Weibel, February 8, 2012):
K-theory of minuscule varieties (Anders Buch, February 22, 2012):
From algebra to category theory: a first approach to fusion categories
The idea of this talk is to introduce and motivate the
notion of fusion category. We shall give some basic definitions
and examples that help us understand this structure.
We shall introduce the ideas of gradings, solvability and
nilpotency for fusion categories and we shall connect it to the
corresponding ideas for groups. We shall also discuss some
results concerning to the structure of fusion categories with
restrictions on the Frobenius-Perron dimensions of its simple
objects.
Invariants of Matrix Factorizations
(Mark Walker, March 21, 2012):
Combinatorial aspects of toric mirror symmetry
(Lev Borisov, March 28, 2012):
Equivariant Schubert calculus: positivity, formulas, applications
(David Anderson, Jan. 30, 2013):
Schubert's enumerative calculus is the subject of Hilbert's 15th
problem. It is a technique for solving problems of enumerative geometry;
for example, how many conics are tangent to five given conics? In its
modern formulation, Schubert calculus concerns computations in the
cohomology rings of Grassmannians, flag varieties, and related spaces.
These spaces carry large group actions, which can be used to both
refine and simplify the computations. The cohomology calculations can
be modeled by multiplication of polynomials, and a central role is
played by these polynomial representatives. Formulas for these
polynomials are of both theoretical and computational interest.
In this talk, I will survey recent developments in this subject,
including some new formulas and applications.
What is a Derivator? (Chuck Weibel, Feb. 6, 2013):
As the name implies, this is an introductory talk.
Derivators were introduced in 1983 by Grothendieck in a 600-page manuscript,
and refined in his 2000-page manuscript in 1991. They are designed to
enhance triangulated categories, and have recently been used in the study
of non-commutative algebraic geometry.
A geometric approach to noncommutative Laurent phenomenon"
(V. Retakh, Feb. 13, 2013):
A composition of birational maps given by Laurent polynomials need not be
a Laurent polynomial. When it does, we talk about the Laurent
phenomenon. A large variety of examples of the Laurent phenomena for
commuting variables comes from the theory of cluster algebras. Much
less is known in the noncommutative case. I will present a number of the
noncommutative Laurent phenomenoma of a "geometric origin." This is a
joint work with A. Berenstein.
Dynamics and surjectivity of some word maps on SL(2,q)
Tatiana Bandman, Feb. 20, 2013):
I will speak about a geometric approach, based on the
classical trace map, for investigating dynamics, surjectivity and
equidistribution of word maps on groups PSL(2,q) and SL(2,q).
It was also used for a characterization of finite solvable groups
by two-variable identities.
Strongly Dense Subgroups of Algebraic Groups
(Robert Guralnick, Feb. 27, 2013):
Let G be a simple algebraic group. A free finitely generated
subgroup H of G is called strongly dense in G if every nonabelian
subgroup of H is Zariski dense in G. We will discuss joint work
with Breuillard, Green and Tao which shows that strongly dense
subgroups exist (over sufficiently large fields) and some recent
improvements on this by Brueillard, Guralnick and Larsen.
This has applications to finding Cayley graphs of finite simple
groups of Lie type and some results on generation of finite
simple groups of Lie type. Using these ideas, we can also
improve on results of Borel and Deligne-Sullivan related to
the Hausdorff-Banach-Tarski paradox.
The 3 main problems in the braid group
(Mina Teicher, March 13, 2013):
These are: The Word Problem, The Conjugacy Problem and the
Hurwitz Equivalence Problem. I shall present the questions,
some answers and, time permitted, also an application to Cryptography.
Intersection theory on singular varieties
(Joe Ross, April 3, 2013):
Whereas algebraic cycle classes may be multiplied on a smooth variety,
this is not in general possible on a singular variety. In topology,
the intersection homology groups of Goresky-MacPherson provide
interesting invariants of singular spaces. Intersection homology sits
in between singular cohomology and homology, and admits natural
pairings generalizing the product structure on the singular homology
of a smooth manifold.
I will propose an algebraic analogue of intersection homology which
sits in between the algebraic cocycles of Friedlander-Lawson and the
classical Chow groups. In some special cases these "intersection Chow
groups" admit pairings. This is joint work with Eric Friedlander.
Implicit systems of differential equations
(Freya Pritchard, April 24, 2013)
We will consider implicit systems that are given by polynomial
relations on the coordinates of the indeterminate function and the
coordinates of the time derivative of the indeterminate function. For such
implicit system of differential-algebraic equations, we will be concerned
with algebraic constraints such that on the algebraic variety determined by
the constraint equations the original implicit system of differential
equations has an explicit representation.
Our approach to such systems is algebraic. Although there have been a
number of articles that approach implicit differential equations
algebraically, all such approaches have relied heavily on linear algebra.
The approach that we will consider is different in that we have no
linearity requirements at all, instead we shall rely on classical algebraic
geometry. In particular we will use birational mappings to produce an
explicit system of differential equations and an algebraic variety of
possible initial values.
Structure of Chevalley groups over rings
(Alexei Stepanov, May 1, 2013)
Let G be a Chevalley group scheme with elementary group E.
Using a localization procedure to reduce to the well understood
case of local rings,
we study the following problems over a commutative ring R:
a) Normality of E(R) and commutator formulas;
b) Nilpotent structure of K1=G(R)/E(R)
c) bounded word length in E(R); and
d) normal subgroups of G(R).
Fall 2012
Given a generalized flag manifold X = G/P, a Schubert variety X(w),
and a degree d, consider the set of points that can be reached from
X(w) by a rational curve of degree d, i.e. the union of all rational
degree d curves through X(w). It turns out that the Zariski closure
of this set is a larger Schubert variety, which is important for many
aspects of the quantum cohomology of X, including the quantum
Chevalley formula and the smallest q-degree in the quantum product of
two Schubert classes. I will give a very explicit description of this
"curve neighborhood" of the Schubert variety in terms of the Hecke
product of Weyl group elements, and use it to give a simple proof of
the (equivariant) quantum Chevalley formula. This is joint work with
Leonardo Mihalcea.
This is joint work with Alexandra Seceleanu and Michael E. Stillman.
We describe Groebner bases for the ideals of relations between the
Chern classes of the tautological bundles on partial flag bundles,
and show how the result can be used to enable practical computation
of intersection numbers in the "Macaulay2" package "Schubert2".
We also generalize the result to cover isotropic flag bundles.
Let F be a perfect field of characteristic different from 2. In joint
work with Paul Arne Ostvaer, we describe the slices of hermitian K-theory
and higher Witt-theory in the motivic stable homotopy category of F.
Applications include computations of hermitian K-groups and Witt groups
for number fields and projective spaces, as well as a different perspective on
Milnor's conjecture on quadratic forms.
The first term of Gersten conjecture for K-theory claims the injectivity
of the map K_i(R)→K_i(K) for any regular semilocal ring R with field
of fractions K. The same statement with K_i replaced by the etale
cohomology functor H^1(-,G), where G a reductive algebraic group,
is known as the Grothendieck-Serre conjecture. The latter conjecture was
recently settled by I. Panin et al. under the assumption that R contains
an infinite perfect field. We discuss how essentially the same argument
carries over to non-stable K_1 and similar functors.
Spring 2012
Deformation quantization is a procedure which
assigns a formal deformation of the associative algebra of
functions on a variety to a Poisson structure on this variety.
Such a procedure can be obtained from Kontsevich's
formality quasi-isomorphism and, it is known that, there are
many homotopy inequivalent formality quasi-isomorphisms.
I propose a framework in which all homotopy classes of
formality quasi-isomorphisms can be described. More precisely,
I will show that homotopy classes of "stable" formality quasi-isomorphisms
form a torsor for the group exp(H°(GC)), where GC denotes
the full graph complex. The group exp(H°(GC)) is isomorphic
to the Grothendieck-Teichmueller group which is, in turn, related to
moduli of curves and to theory of motives.
Shift equivalence is an equivalence relation on nxn matrices (say over Z).
Such a matrix T may be regarded as defining a Z[t]-module structure on a
free abelian group, and shift equivalence translates into the assertion that
the modules become isomorphic over Z[t,1/t]. This talk is a description of
a weaker equivalence relation related to class groups of number fields.
Thomas and Yong have conjectured a Littlewood-Richardson
rule for the K-theory of any minuscule homogeneous space, based on
counting tableaux that rectify to a certain superstandard tableau.
This conjecture has been proved for Grassmannians of type A and
maximal orthogonal Grassmannians, but it fails for the Freudenthal
variety of type E7. I will speak about a fix that replaces the
superstandard tableaux with minimal increasing tableaux. These
tableaux have several other combinatorial advantages, for example they
make it possible to recognize which tableaux should be counted without
rectifying them. This is joint work with Matthew Samuel.
(Julia Plavnik, February 29, 2012):
A good way of thinking about category theory is that it is a
refinement (or "categorification") of ordinary algebra. In other
words, there is a dictionary between these two subjects, such
that usual algebraic structures are recovered from the corresponding
categorical structures by passing to the set of isomorphism classes
of objects (Etingof, Gelaki, Nikshych and Ostrik).
Given, for example, a polynomial f(x_1,...,x_n) with
complex coefficients, a matrix factorization of f is a pair of r x r
matrices of polynomials (A, B) satisfying AB = f I_r = BA. Introduced
over 30 years ago by Eisenbud in the context of studying projective
resolutions of modules over hyper-surfaces, there has been a revival
of interest in matrix factorizations lately, as connections with
mathematical physics and knot theory have emerged. I will discuss some
recent progress in understanding certain fundamental invariants of
matrix factorizations.
I will review the combinatorial aspects of toric mirror
symmetry. In particular, I will focus on the new phenomena one encounters
when dealing with complete intersections as opposed to hypersurfaces.