Every profinite group is a Galois group, but which one is also an ${\textit{absolute}}$ Galois group? The cohomological implications of the Bloch-Kato conjecture -- positively solved by M.~Rost and V.~Voevodsky --
allows us to define ${\bf{Bloch-Kato}}$ ${\bf{pro-}}$$p$ ${\bf{groups}}$, which play a crucial role, since they arise naturally
as maximal pro-$p$ quotients and Sylow pro-$p$ subgroups of absolute Galois groups.
In this seminar I will present the state of the art of the research on Bloch-Kato groups,
with a particular mention of the 'Elementary Type Conjecture' of maximal pro-$p$ Galois groups.
Yet, there's still a lot of work to do: indeed every maximal pro-$p$ Galois group is equipped with an
${\textit{orientation}}$ $G_F(p)\rightarrow\mathbb{Z}_p^\times$, arising from the action on the group of
the roots of unity of $p$-power order. The study of such orientation for Bloch-Kato groups will provide hopefully new results.
Let k be a field and G be an algebraic group. If char k=0 Birger Iversen showed in 1976 that the pull-back Br(k)$\to$ Br (G) is an isomorphism if $G$ is simply connected. If fact, he proved this using topological methods for $k$ the field of complex numbers from which the general case follows by Galois cohomology. In my talk I will present one or two (if time permits) algebraic proofs of Iversen's result which show more: If $G$ is simply connected and $k$ arbitrary then the pull-back $n$-torsion Br(k)$\to$ $n$-torsion Br (G) is an isomorphism as long as $n$ is prime to the characteristic of $k$. If $k$ is not perfect of char $p$ I will show then that the pull-back $p$-torsion Br(k)$\to$ $p$-torsion Br(G) is not surjective for any semisimple isotropic connected linear algebraic group over $k$.
Recently Positselski proved that an unbounded derived category of quasi-coherent D-modules on a smooth algebraic variety X is equivalent to a so-called coderived category of quasi-coherent DG-modules over the de Rham algebra of X. I will explain how to work with this coderived category.
Starting with an action $G\times X\to X$
we analyze the maximal $G$-rational subalgebra
$\mathscr{O}_K$ of $k(X)$ and use it to obtain the
action $G_K\times U_K\to U_K$ where $K = k(X)^G$, and $U_K$ is a certain quasi-affine variety over
$K$ with $\mathscr{O}(U_K) = \mathscr{O}_K$. This gives us a generic "homogeneous" picture of the original action.
We also analyze the maximal $G$-rational
subalgebra of $k[X]_\mathfrak{p}$, where
$\mathfrak{p}$ is a height-one $G$-prime of $k[X]$.
We use these results to assess the behavior of
the canonical map $\pi : U\to U/G$ for
a sufficiently small $G$-invariant, open subset $U$ of $X$.
Finally we use ${\textit{observable}}$ $G$-actions over $k$ to construct the functor $K\mapsto H^1(K,G/H)$,
from finitely generated fields over $k$ to ${\textit{Sets}}$. From there we define the ${\textit{essential dimension}}$ of a homogeneous space $G/H$, whenever $H\subset G$ is a
pair of connected, reductive groups.
We study the linear syzygies of a homogeneous ideal
$I$ in a polynomial ring $S = k[x_0..x_n]$, focussing on the graded betti numbers
\[
b_i = {\textrm{dim}}_k {\textrm{Tor}}_i(S/I, k)_{i+1}.
\]
For any projective variety $X$ in $P^n$ and divisor $D$, what conditions on $D$ ensure that $b_i$ is nonzero? Eisenbud has shown that a decomposition $D=A+B$ such that $A$ and $B$ have at least
two sections give rise to determinantal equations (and
corresponding syzygies) in $I_X$ and conjectured that if
the quadratic component of $I$ is generated by quadrics of
rank at most four, then the last nonvanishing $b_i$ is a
consequence of such a decomposition. We describe obstructions
to the conjecture and prove a variant. The obstructions arise
from toric specializations of the Rees algebra of Koszul cycles,
and we give an explicit construction of toric varieties with
minimal linear syzygies of arbitrarily high rank. This leads
to a number of interesting open questions.
(joint work with M. Stillman).
A linear algebraic group is called a Cayley group if it is equivariantly birationally isomorphic to its Lie algebra. It is stably Cayley if the product of the group and some torus is Cayley. Cayley gave the first examples
of Cayley groups with his Cayley map back in 1846.
In joint work with Blunk, Borovoi, Kunyavskii and Reichstein, we classify the simple stably Cayley groups over an arbitrary field of characteristic $0$.
Little is known about the transcendence of special values of the Gamma function at rational points. In this talk we examine the Gamma function at points from an imaginary quadratic field. As a corollary of our analysis, we gain knowledge about values of infinite products of rational functions.
Multiple zeta values (MZV) are the numbers defined by the
convergent series of the form
$$\zeta(s_1,s_2,...,s_k)=\sum_{n_1>n_2>...>n_k>0}^\infty
\{1/(n_1^{s_1} >... n_k^{s_k})\}$$
for $s_i$ positive integers. For these real numbers there are some beautiful relations, some of them due to Euler, like $\zeta(2,1) = \zeta(3)$ or $\zeta(2n) = q\pi^{2n}$ for $q$ a rational number. In this lecture I will present some of the most famous conjectures about MZV and its relations. I will show how we try to see the truthfulness of this conjecture by looking at them until a small
degree bounded by the capacity of the actual computers.
Quillen’s derived functor notion of homology provides
interesting and useful invariants in a wide variety of homotopical algebraic contexts. For instance, in Haynes Miller’s proof of the Sullivan conjecture on maps from classifying spaces, Quillen homology of commutative algebras (Andre-Quillen homology) is a critical
ingredient. Working in the topological context of structured ring spectra, this talk will introduce several recent results on localization and completion with respect to topological Quillen homology of commutative ring spectra (topological Andre-Quillen homology), $E_n$ ring spectra, and operad algebras in spectra. This includes homotopical analysis of a completion construction and strong
convergence of its associated homotopy spectral sequence. The localization and completion constructions for structured ring spectra are precisely analogous to Sullivan's localization and completion of
spaces (for which he recently won the Wolf prize), and Bousfield-Kan's version of Sullivan's localization and completion called the $R$-completion of a space with respect to a ring $R$. This is joint work with Michael Ching.
If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.
Let $G \times X \rightarrow X$ be an action of the algebraic group $G$ on
the affine, algebraic variety $X$. There are two quite different
notions of quotient associated with this situation.
If we embrace the conventional approach, we just accept
the object $Y =$ Spec$(k[X]^G)$, along with the natural map
$\pi : X \rightarrow Y$, as the inevitable thing to study. If $G$ is reductive
then this "quotient" is a variety and it has the anticipated universal
property, even if $Y$ is not an orbit space. Similar results hold if
$X$ is a projective variety. Many important moduli spaces have been
constructed using this approach.
But maybe there is another approach, where the emphasis is on orbits
of maximal dimension rather than on closed orbits. In this scenario we consider sufficiently small open, $G$-subsets $U$ of $X$ such that each $G \times U \rightarrow U$ has as many desirable properties as the situation will tolerate. If we define $U/G$ by the equation $k[U/G] = k[U]^G$, then we can ask for the following.
(1) $k[U/G]$ is finitely generated.
(2) $k[U/G]$ is a regular ring.
(3) $\pi : U \rightarrow U/G$ is surjective.
(4) $\pi : U \rightarrow U/G$ separates orbits of maximal dimension.
(5) $\pi : U \rightarrow U/G$ has no exceptional divisors. (6) $\pi : U \rightarrow U/G$ is flat. We do this so as to discard only a small portion of $X$. We then try to glue all these $U/G$'s together to get a separated quotient variety, without the help of semi-invariants.
Recently an analogue of the Gauss-Bonnet theorem has been proved by Connes-Tretkoff and Fathizadeh-Khalkhali for noncommutative two torus. The idea is based on the direct computation of the value at origin of the zeta function associated to the corresponding Laplacian.
In this talk we will briefly discuss the above theorem and explain a related problem.
We are now witnessing exciting interactions between computer science and
mathematics on one side, and the natural sciences on the other. While the
natural sciences are rapidly absorbing notions, techniques and
methodologies intrinsic to computer science and mathematics, theoretical
computer science is adapting and extending its traditional notion of
computation and computational techniques, to account for computation taking
part in nature around us.
This talk will outline several of the fruitful directions of theoretical
computer science research that originated from the study of DNA. I will
describe comma-free codes inspired by the studies into the genetic code,
splicing systems, optimal encodings for DNA Computing, sticker systems,
Watson-Crick automata, combinatorics on DNA words, cellular computing, and computing by DNA self-assembly.
In his seminal 1961 paper, Auslander gave a beautiful characterization of flatness of a finite module over a regular local ring $R$ in terms of torsion in tensor powers of the module. Almost 40 years later, Vasconcelos conjectured a generalization of this criterion to the category of finite type $R$-algebras. I will survey a recent development in this area, leading to the establishing of Vasconcelos' conjecture (and then some) in characteristic zero, by local analytic methods. These are joint works with E. Bierstone and P.D. Milman, and with Hadi Seyedinejad.
A connected graded algebra is called Koszul if the ground field has a linear resolution, i.e. differentials are defined by matrices that only have linear entries. This condition has less than a million equivalent descriptions. In this survey talk, I will mention a few of these characterizations and examine the resulting homological behavior. As a motivation, I start off by showing the LCS formula for the pure braid group. This is an instance of a more general result about the cohomology ring of a nice class of hyperplane arrangements. I am also planning to describe more examples with origins in quantum groups and show a quick proof for the classical PBW theorem. If there is time left, I will say a few words about the interaction of the Koszul property with the Bloch-Kato conjecture. At last but not least, I will mention the biggest open problem of this area which asks for the correct pronunciation of the word Koszul.
Roughly speaking, the Hall algebra $H(A)$ of a (small) Abelian category $A$ is the algebra of finitely supported functions on the moduli space of objects of $A$ (i.e. the set of isoclasses of objects of $A$ with the discrete topology). Interest in Hall algebras exploded in the early 1990's when Ringel discovered that the Hall algebra associated to the category of $F_q$-representations of a Dynkin quiver $Q$ provides a realization of the positive part of the (quantized) enveloping algebra of the (simple) complex Lie algebra associated to the same Dynkin diagram.
To\"{e}n and Bergner have used the theory of model categories to obtain Hall algebras on triangulated categories. In this talk we will survey these constructions and, time permitting, explain some open problems in this area which are being studied via homotopy theory.
Solving non-linear algebraic problems is one of the major challenges
in scientific computing. In several areas of engineering sciences,
algebraic problems encode geometric conditions on variables taking
their values over the reals. Thus, most of the time, one aims to
obtain some informations on the real solution set of polynomial
systems. The resolution of these problems often has a complexity which
is exponential in the number of variables.
In this talk, I will review some geometric and algebraic techniques
which enable to obtain fast practical algorithms meeting the best
known complexity bounds. These algorithms are implemented in the
maple package (RAGlib: The Real Algebraic Geometry library) which has
the feature to provide algorithms of asymptotically optimal algorithms
in real geometry. Its practical performances will be discussed and
some applications will be presented.
This talk is based on joint work with J.C. Faugere, A. Greuet, E. Schost, PJ Spaenlehauer and L. Zhi.
In this talk, I will continue the lecture given by Masoud Khalkhali on our recent joint work on the Gauss-Bonnet theorem and scalar curvature for the noncommutative two torus, in the context of Alain Connes' noncommutative differential geometry. I will first construct the Connes-Tretkoff spectral triple encoding the metric information on this $C^*$-algebra so that we view it as a noncommutative Riemannian manifold equipped with a general metric. Then I will recall a spectral definition for its scalar curvature, and will illustrate the process of finding a local expression for the curvature by employing a special case of Connes' pseudodifferential calculus for $C^*$-dynamical systems by means of which one can pursue the heat kernel scheme of elliptic differential operators and index theory. I should mention that recently Connes and Moscovici also found precisely the same formula independently. At the end I will explain how this formula fits into our earlier work which extends the Gauss-Bonnet theorem of Connes and Tretkoff to general conformal structures on noncommutative two tori.
I describe the theory of the quantum Hall effect in monolayer and bilayer graphene based on the magnetic catalysis effect. The role of the symmetry and its breakdown in this phenomenon is discussed.
Let $\Delta$ be a triangulation of a connected region in the real plane. Let $C(r,d,\Delta)$ be the space of piecewise polynomial functions of degree $\leq d$ and smoothness $r$. A major question in Approximation Theory is to find the dimension of this space, which is not known even for the case when $d=3$ and $r=1$. Alfeld and Schumaker give a formula for this dimension, when $d\geq 3r+1$ and any $\Delta$. Using homological algebra, this problem can be translated into finding the Hilbert function of a graded module (the ``homogenization'' of $C(r,d,\Delta)$). I will discuss about this approach and about the Schenck-Stiller conjecture that says that Alfeld-Schumaker formula holds for any $d\geq 2r+1$. I will present a very recent project with Jan Minac where we prove this conjecture for a triangulation that is not trivial, in the sense that the formula does not hold if $d=2r$.
We consider a natural holomorphic structure on the quantum projective space $\mathbb{C}P^l_q$ already presented in the literature and define holomorphic structures on canonical quantum line bundles on it. The space of holomorphic sections of these line bundles then will determine the quantum homogeneous coordinate ring of $\mathbb{C}P^l_q$. We define bimodule connections on canonical line bundles and this enables us to identify the quantum homogeneous coordinate ring of $\mathbb{C}P^l_q$ with the ring of twisted polynomials. We also introduce a twisted positive Hochschild $2l$-cocycle on $\mathbb{C}P^l_q$, by using the complex structure of $\mathbb{C}P^l_q$, and show that it is cohomologous to its fundamental class which is represented by a twisted cyclic cocycle. This certainly provides further evidence for the belief that holomorphic structures in noncommutative geometry should be represented by (extremal) positive Hochschild cocycles within the fundamental class. Finally we verify directly that the main statements of the Riemann-Roch formula and Serre duality theorem hold for $\mathbb{C}P^1_q$ and $\mathbb{C}P^2_q$. This is joint work with Masoud Khalkhali.
For any given field $F$ there is a well known parametrizing
space for elementary p-abelian Galois extensions of $F$; for example,
if $K$ contains a primitive pth root of unity, Kummer theory provides
this parametrizing space for us. By putting additional structure on
these parametrizing spaces, we are able to give a parametrizing space
for solutions to any given embedding problem where the quotient is a
cyclic $p$-group and the kernel is an elementary $p$-abelian group. This
allows us to give an explicit count to the number of such solutions,
and in particular we can make certain universal statements about the
number of solutions to such embedding problems. For instance, we use
our results to show that $p$-groups have unbounded realization
multiplicity.
A theorem of Beauville implies that the general smooth quartic surface in $P^3$ may be expressed as the zerolocus of a Pfaffian associated to an 8 by 8 skew-symmetric matrix of linear forms. In this talk, I will discuss how the recent work of Aprodu-Farkas on the Green conjecture for curves on K3 surfaces may be used to generalize this statement to all smooth quartic surfaces in $P^3$. This is joint work with Emre Coskun and Rajesh Kulkarni.
Darboux transformation methods for integrable equations can be classified as differential equations, algebra, analysis, geometry of surfaces, mathematical physics or theoretic computer algebra. In my talk I shall give some introdution into the area intertwining some classical results with some of my own.
In order to describe linear representations of a group, it is sufficient to find all its indecomposable representations. It is known that indecomposable algebraic representations of G=SL(2) correspond to irreducible subrepresentations of G in the ring R of polynomials in two variables x and y. Given a derivation ' on the ground field, R extends to a G-representation R' by adding variables x', y', x", y", etc. We will investigate indecomposable subrepresentations of R' and discuss their relation to description of all differential representations of G. If time permits, I will describe the category of differential representations of tori.
Accurate computer recognition of handwritten mathematics offers to provide a natural interface for mathematical computing, document creation and collaboration. Mathematical handwriting, however, provides a number of challenges beyond what is required for the recognition of handwritten natural languages. For example, it is usual to use symbols from a range of different alphabets and there are many similar-looking symbols. We present a geometric theory that we have found useful for recognizing mathematical symbols. Characters are represented as parametric curves approximated by certain truncated orthogonal series. This maps symbols to the low-dimensional vector space of series coefficients. The beauty of this theory is that a single, coherent view provides several related geometric techniques that give a high recognition rate and do not rely on peculiarities of the symbol set.
I will review the classification of representations of the symmetric and unitary groups, and how they are related to each other. In particular, I will describe the Young projection operators whose images give the irreducible representations. Then I will give new formulas which use
the Young projection operators to construct a family of orthogonal projections which are convenient for computations. Finally, I will describe how computations of traces of maps of symmetric group
representations can be used to compute traces of maps of $U(n)$ representations for all $n$ at once. If I have time, I hope to package this up in the language of traced monoidal categories.
Integration is sometimes said to be a solved problem in computer algebra, but integration problems
are the source of a significant percentage of bug reports and complaints to Mathematica and Maple. The reasons for this are discussed and some remedies described.
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. We propose adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems.
We show that any such system can be decomposed into finitely many so-called "regular semi-algebraic systems". We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time with respect to the number of variables. We have implemented our algorithms and the experimental
results illustrate their effectiveness. A software demonstration will conclude this talk.
This is a joint work with Changbo Chen (UWO), James H. Davenport (Bath U.), John P. May (Maplesoft), Bican Xia (Peking U.) and Xiao Rong (UWO).
The corresponding article is published in the Proceedings of the 2010 International Symposium of Symbolic and Algebraic Computation (ISSAC'10) and available at
ww.csd.uwo.ca/~moreno/Publications/118_paper.pdf
The Kunneth spectral sequence for abelian sheaf cohomology is displayed and discussed. Computational applications of this spectral sequence for the etale cohomology of classifying spaces of algebraic groups will also be displayed.
If $R$ is a field then the conjugacy class of $x\in M_n(R) = End(V)$ is determined by its rational
canonical form using the theory of modules over
the PID $R[T]$. If $R$ is a discrete valuation ring
then the situation is more complicated, even if the characteristic polynomial of $x\in M_n(R)$ is
irreducible over the quotient field $K$ of $R$.
We discuss the following questions.
(1) What further assumptions on $x$ and $R$ are useful?
(e.g. $x$ semisimple, $R$ Henselian)
(2) How do we sort out non-conjugate elements
of $M_n(R)$ that become conjugate in $M_n(K)$?
(3) Are some conjugacy classes of $M_n(R)$ better than others?
(4) To what extent can $x$ be measured against a canonical form?
Let $G$ be the absolute Galois group of a field that contains a primitive $p$th root of unity. This is a very mysterious profinite group which is a central object of study in Galois theory. In joint work with Ido Efrat and Jan Minac we have shown that a remarkably small quotient of this big group determines the entire Galois cohomology of $G$. As application of this surprising result, we give new examples of profinite groups that are not realisable as absolute Galois groups of fields. I will present an overview of this work.
A $G$-cover of a smooth projective curve $X$ over some algebraically closed field of characteristic zero ramified at a finite set $D$ of points, may be identified with a tensor functor from the category of finite representations of $G$ into bundles with parabolic structure along $D$, by work of Nori. Now when $X$ is the sphere and $D= \{0,1,\infty\},$ the parabolic bundle is of the form of $\oplus\mathcal{O}(s_i)$ for some integers $s_i$. These constants are very difficult to determine in general, but Ajneet Dhillon devised a clever method of bounding them using group theoretic data, and this is the subject of my talk.
There is a general cohomology defined by Sweedler for co-commutative Hopf algebras, generalizing the usual cohomology of a group or a Lie algebra. Recently it was discovered that low-dimensional groups could be defined without the co-commutativity requirement. In joint work with Christian Kassel, we have given the first few examples of computations with these, in the case of algebras of functions on groups. These turn out to be related to torsors in algebraic geometry, and Drinfeld twists in quantum groups theory.
In this talk, we introduce the notion of the essential dimension of an algebraic structure and discuss some recent results on the essential dimension of certain classes of central simple algebras. We also relate these results to the essential dimension of split simple groups of type $A_n$.
Let $X$ be a complex affine variety with an action of a torus $T$, and an attractive fixed point $x_0$. We say that $X$ is a rational cell if
$H^{2n}(X,X-\{x_0\})=\mathbb{Q}$ and $H^{i}(X,X-\{x_0\})=0$ for $i\neq 2n$, where $n={\rm dim\,}_{\mathbb{C}}(X)$.
These objects appear naturally in the study of group embeddings.
A fundamental result in equivariant cohomology asserts that
the transgression ${\bf Eu}_T \in H^{2n}(BT)$ of a generator of
$H^{2n}(X,X-\{x_0\})$
splits into a product of singular characters,
${\bf Eu}_T={\chi_1}^{k_1}\ldots {\chi_m}^{k_m}$.
This characteristic class is by definition the Equivariant Euler class of $X$ at $x_0$. Loosely speaking, one could think of $X$ as a sort of $T$-vector bundle over a point. My goal in this talk is to make this claim precise, and to show why one could hope to build similar elements in equivariant $K$-theory, i.e. Bott classes, by using localization and completion techniques. This is work in progress.
I will survey what is known about pro-$p$ groups that contain a subgroup of finite cohomological dimension, including results of Serre, Scheiderer, and aiming at a description of the structure of pro-$p$ groups that are virtually free pro-$p$.
We will give an introductory talk to the theory of operads, giving basic definitions and examples. We will focus on examples in topology, category theory, and computer science.
Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. In the past 15 years
this notion has been investigated in several contexts by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry. I will survey some of this research in my talk.
I will discuss how stability conditions and well adapted autoequivalences can be used to understand geometric information in derived categories. Following this discussion, I will provide an example of the usefulness of these techniques. In particular, I will show how to classify all compactifications of stable bundles on a class of genus 0 singular reducible curves.
In the first part of my talk I will give an introduction to Chow motives and explain their most important properties. In the second part I will discuss some applications. In particular I will present some recent results about the relation between motives and canonical dimension. (This Algebra Seminar talk is Stefan's second talk, which follows Stefan's first talk, which is a Colloquium talk.)
The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of R is, in general, not large enough to contain the diagrams of all subsystems of R, the answer to this question is negative. We introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow one to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in R.
Let $\mathcal C$ be an $[n,k,d]-$linear code with generating matrix $A$; this is assumed to be a rank $k$, $k\times n$ matrix with entries in a field $\mathbb K$. Computing $d$, the minimum distance, is in general an NP-hard problem and finding a good lower bound has been a major question in algebraic coding theory. If the matrix $A$ has no proportional nor zero columns, consider $\Gamma\subset\mathbb P^{k-1}$ the set of $n$ distinct points with homogeneous coordinates the entries of each column of $A$. In this talk we present a good lower bound for $d$ in terms of a certain shift in the last free module of the graded minimal free resolution of $\mathbb K$ $[x_1,\ldots,x_k]/I(\Gamma)$. We also present the De Boer-Pellikaan method to compute $d$. As a consequence of this method one can obtain a symbolic description of the variety of minimal codewords in terms of the top $Ext$ module of a certain ideal generated by products of linear forms.
Classically, if some manifold $M$ is equipped with an action of a subgroup $G$ of $PSL(2, Z)$ under which a certain space $F$ of functions on $M$ transforms via a $1$-cocycle, the latter is referred to as an automorphy factor, and the functions $F$ are said to be $G$-modular. In this talk, I will produce an injective $1$-cocycle of $PSL(2, Z)$ into a certain group of formal power series which extends the well-known identification of the fundamental group of $P^1\{0,1,\infty\}\;$ with associated Chen series. This cocycle may be regarded as a quasi-automorphy factor for sections of the universal prounipotent bundle with connection on $PSL(2, Z)$ - in particular for the polylogarithm generating series $Li(z)$. I will go on to show that the quasi-modularity of $Li(z)$ may be used to give a family of proofs of the analytic continuation and functional equation for the Riemann zeta function. Moreover, under this cocycle, the involutive generator of $PSL(2, Z)$ maps to the Drinfeld associator, while the infinite cyclic generator maps to an $R-matrix$, in Drinfeld's formal model of the quasi-triangular quasi-Hopf algebras, thereby producing a representation of $PSL(2, Z)$ into tensor products of certain $qtqH$ algebras. Underlying the whole story is a path space realization of $PSL(2, Z)$ using Deligne's idea of tangential basepoint.
A strong generating hypothesis for the stable module category
June 03, 14:0 - 14:0, MC 107
A $kG$-linear map between $kG$-modules is called a strong ghost map if it induces the zero map in Tate cohomology when restricted to each subgroup of $G$. We formulate the strong generating hypothesis as the statement that every strong ghost between finitely generated $kG$-modules factors through a projective module, i.e., it is trivial in the stable module category. In recent joint work with Jon Carlson and Jan Minac, we have identified the class of groups for which this strong generating hypothesis holds. I will present an overview of this work by focusing on some concrete examples.
It is now more than a decade that Hopf algebras established
themselves as an integral part of Noncommutative Geometry via the work of Connes and Moscovici on the computation of the index of hypoelliptic operators on manifolds. The latest Hopf algebras constructed were those associated to Cartan-Lie pseudogroups. In this talk we canonically associate a Hopf algebra to any bicrossed sum Lie algebras. This construction covers all known cases in type II and also type III. The constructed Hopf algebra is naturally equipped with a modular pair in involution which is the
coefficients for the Hopf cyclic cohomology of the Hopf algebra. At the end we show how to compute the Hopf cyclic cohomology of these Hopf algebras.
Betti numbers of a rationally smooth toric variety
April 16, 12:30 - 13:30, MC 108
Consider an irreducible representation of a semisimple algebraic group with $\lambda$ its highest weight and look at the action of the Weyl group $W$ on the rational vector space spanned by the roots. Take the convex hull of the $W$-orbit of $\lambda$ and obtain the polytope $P_{\lambda}= {\textrm{Conv}}(W.\lambda)$. We are interested in describing the Betti numbers of the toric variety $X(J)$ associated to the polytope $P_{\lambda}$ when the Weyl group is the $n$ symmetric group and $X(J)$ is a rationally smooth variety which doesn't depend on the highest weight $\lambda$ but on the set of reflections that fix $\lambda$ called $J$. The main result is a recursion formula for the Betti numbers of $X(J)$ in terms of Eulerian polynomials. The theory of algebraic monoids developed by Renner and Putcha is effectively used in our computations.
Given a valuation $v$ on a division algebra $D$, there is an associated graded division algebra $\textrm{gr}(D)$, which is often much easier to work with than $D$ itself, but which retains surprisingly much of the structure of $D$. We illustrate this with $SK_1(D)$. If $v$ on the center of $D$ is Henselian and $D$ is tame, then $SK_1(D)$ is isomorphic to $SK_1(\textrm{gr}(D)).$
A geometric realization of extreme components of the tensor product of modules over algebraic groups
March 12, 14:30 - 14:30, MC 108
In this talk I will explain how the celebrated theorem of Borel-Weil-Bott provides a natural realization of some extreme components of the tensor product of two irreducible modules of simple algebraic groups. I will also discuss a number of connections of our construction with problems coming from Representation Theory, Combinatorics, and Geometry, including questions about the Littlewood-Richardson cone related to Horn's conjecture, settled by Knutson and Tao in the late 1990's.
The talk is based on joint work with Mike Roth.
The (stable) chromatic spectral sequence has had a significant impact on our understanding of the stable homotopy groups of the spheres. I will talk about preliminary attempts to construct an unstable version. I will try to describe a filtration of the stable chromatic spectral sequence induced by the Hopf rings for the odd spheres. There are natural questions that arise in the unstable world (e.g. an unstable version of the Morava stabilizer algebra) and a chromatic interpretation of the Hopf invariant.
Molecular algorithmic self-assembly: theoretical foundations and open problems
March 05, 14:30 - 14:30, MC 108
We review a formal model of molecular self-assembly known as the abstract Tile Assembly Model (aTAM). The aTAM which models the interaction of artificial biochemical macromoleclues known as "DNA tiles", which are capable of binding to each other in specific and surprising ways. The goal of this and other models of self-assembly is to study the feasibility of engineering nanoscale structures through a bottom-up approach, through the "programming" of molecules to automatically assemble themselves, in contrast to top-down approaches such as lithography.
After presenting the aTAM and a few basic results that illustrate its power and its limitations, we survey some theoretical conjectures. These conjectures share the properties of being easy to state, easy to understand, "obviously true", and unresolved. A primary goal is to frustrate the audience with the simplicity of these problems, in the hopes that one of them will step in and solve them.
Invariants for the modular cyclic group of prime order via classical invariant theory
February 26, 14:30 - 14:30, MC 108
Let $F$ be any field of characteristic $p$ and let $C_p$ denote the cyclic group of order $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,\dots,V_p$ of $C_p$ defined over $F$. It is also well-known that there is a unique
(up to equivalence) $d+1$ dimensional irreducible complex representation of $\textrm{SL}_2(\mathbb{C})$ given by the action on the space $R_d$ of $d$ forms. In this talk I will describe my recent result which reduces the computation of the ring of $C_p$-invariants of a $C_p$-representation $V=\oplus_{i=1}^k V_{n_i+1}$ to the computation
of the classical ring of invariants (or covariants) $C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\textrm{SL}_2(\mathbb{C})}$.
This allows us to compute for the first time the ring of invariants for many representations of $C_p$.
Local splitting behaviour of modular Galois representations
February 12, 14:30 - 14:30, MC 108
This talk will discuss the local splitting behaviour of ordinary modular Galois representations and relate them to companion forms and complex multiplication. Two modular forms (specifically $p$-ordinary, normalized eigenforms) are said to be "companions" if the Galois representations attached to them satisfy a certain congruence property. Companion forms modulo $p$ play a role in the weight optimization part of (the recently established) Serre's Modularity Conjecture. Companion forms modulo $p^n$ can be used to reformulate a question of Greenberg about when a normalized eigenform has CM (complex multiplication).
Using spaces of homomorphisms and the descending central series of the free
groups, simplicial spaces are constructed for each integer $q>1$ and every
topological group $G$, with realizations $B(q,G)$ that filter the classifying space
$BG$. In particular for $q=2$ this yields a single space $B(2,G)$ assembled from all
the $n$-tuples of commuting elements in $G$. Homotopy properties of the $B(q,G)$ will
be described for finite groups, and cohomology calculations provided for
compact Lie groups. Recent results on understanding both the number and
stable homotopy type of the components of related spaces of
representations will also be discussed.
Let $G$ be a reductive group. A $G\times G$-variety $X$ is called an equivariant compactification of $G$ if $X$ is normal, projective, and contains $G$ as
an open and dense orbit. Regular compactifications and reductive embeddings are the main source of examples. In the first case, the equivariant cohomology ring has been explicitely described by Bifet, de Concini, Procesi and Brion. Loosely speaking, it depends mostly on the torus embedding part and the structure of the $G\times G$-orbits. As for the second class, Renner has found that they have a canonical cell decomposition based on underlying monoid data. My goal in this talk is to give an overview of the theory of group embeddings, putting more emphasis on the monoid approach, and to describe the structure of the so called rational cells. Finally, I will explain how such cellular decompositions could lead to a further application of GKM theory to the study of reductive embeddings.
Consider the following questions (coming for instance from
motion planning problems): given two points on a
real algebraic set $S$, do they belong to the same connected
component? If so, how can we connect them?
Canny introduced "roadmaps" as a way to reduce such
problems to computations with curves only. Given $s$ polynomial
equations with rational coefficients, of degree $d$ in $n$ variables,
Canny's algorithm, and its generalizations by Basu, Pollack and
Roy, have a cost polynomial in $(s D^n)^n$.
This is depressingly high; as a result, none of these algorithms
is practical for realistic instances. Indeed, one would rather
expect a cost polynomial in $s D^n$. I will present ongoing work
with Mohab Safey El Din toward this goal.
Generalizations of the notion of "Rank of a Matrix"
January 15, 14:30 - 14:30, MC 108
The notion of rank for matrices (or $2$-tensors) is well understood but the generalizations to higher order tensors is much less well understood and is an active area of research with applications in Biology, Statistics, Computing and Signal Processing.
I will explain the nature of the problem, the notions of "tensor rank" and "border rank" and how problems in this area have been studied through the lens of Higher Secant Varieties of Segre Varieties and the cohomology of non-reduced varieties whose support is a union of linear spaces. I will mention some recent progress on the problems that have been done by me and my collaborators M. V. Catalisano and A. Gimigliano and also some conjectures and open problems. If time permits I will also explain some recent work of Landsberg, Weyman and ourselves on the defining ideals of these higher order secant varieties.
Natural computing is the field of research that investigates models and computational techniques inspired by nature and, dually, attempts to understand the world around us in terms of information processing. It is a
highly interdisciplinary field that connects the natural sciences with mathematical and computational science, both at the level of information technology and at the level of fundamental research. As a matter of fact, natural computing areas and topics come in many flavours, including pure theoretical research, algorithms and software applications, as well as biology, chemistry and physics experimental laboratory research.
In this talk, we describe models and computing paradigms abstracted from natural phenomena as diverse as self-reproduction, the functioning of the brain, Darwinian evolution, group behaviour, the immune system, the
characteristics of life, cell membranes, and morphogenesis. These paradigms can be implemented either on traditional electronic hardware or on alternative physical media such as biomolecular (DNA, RNA) computing, or trapped-ion quantum computing devices. Dually, we briefly describe
several natural processes that can be viewed as information processing, such as gene regulatory networks, protein-protein interaction networks, biological transport networks, and gene assembly in unicellular organisms.
In the same vein, we list efforts to understand biological systems by engineering semisynthetic organisms, and to understand the universe from the point of view of information processing.
The talk is based on the review article "The Many Facets of Natural Computing", L. Kari, G. Rozenberg, "Communication of the ACM", October 2008.
Let $G$ be an affine algebraic group and let $X$ be an irreducible, affine variety.
Assume that $G$ acts on $X$ via $G \times X \to X$. The action is called stable if there
exists a nonempty, open subset $U\subseteq X$ consisting entirely of closed $G$-orbits. The action
is called observable if for any proper, $G$-invariant, closed subset $Y\subseteq X$ there
is a nonzero invariant function $f\in k[X]^G$ such that $f|_Y = 0$. It is easy to prove that
"observable implies stable" but the two notions are not the same for general groups.
We discuss a useful geometric characterization of observability. We then discuss some
of the following questions and illustrate them with the appropriate examples.
(1) When is the action $H \times G\to G$, by left translation, observable?
(2) Does the characterization simplify if G is unipotent? solvable? reductive?
(3) What happens if $X$ is factorial? reducible?
(4) Is $int : G\times G\to G$, $(g,x)\mapsto gxg^{-1}$, always observable?
(5) Can we generalize to the case where $X$ is projective and $G \times X \to X$ is linearizable?
We review slope stability for coherent sheaves on algebraic curves and then discuss Tom Bridgeland's generalization of stability to triangulated
categories. For a certain triangulated category associated to a Kleinian singularity, Bridgeland conjectured that a connected component of the space of
stability conditions should be the universal cover of a $K(G,1)$ for $G$ a generalized braid group and showed this is the case for $G$ the classical braid
group. We generalize this to braid groups of types ADE. This is joint work with Hugh Thomas.
This is the first talk in a learning seminar about the Steenrod algebra. See the
seminar web page
for more information.
After the talk, we will discuss the schedule for future talks in this seminar.
In 1969, Quillen introduced the notion of "model category" as an
axiomitization of homotopy theory. In the intervening four decades
model categories have been proven to be remarkably useful in homotopy
theory. In this talk, I will discuss a structure theorem giving a
notion of a presentation of a model category, refining a 2001 paper of
Dan Dugger; and I will introduce a monoidal structure on the category
of model categories that simplifies many "generic" arguments in model
category theory.
On the homological structure of modules over regular local rings
October 23, 14:30 - 15:30, MC108
Homological structure of finite modules over regular local rings is fairly well understood. The classical results date back to Serre, Auslander
and Buchsbaum. On the other hand, little is known, in general, about the structure of infinite modules. In this talk, we will consider a class of such
modules most important from the geometric point of view, namely those that arise as stalks of coherent sheaves over the source of a morphism with regular
target. We will sketch the idea how to generalize the classical theory to the case of those modules, by a kind of fibre dimension reduction argument.
Counting invariants for $O+O(-2)$-quiver representations
October 16, 14:30 - 15:30, MC 108
In this talk we prove a wall-crossing formula of counting
invariants in the derived category of $O+O(-2)$-quiver
representations. We verify the GW/DT/PT/NCDT-correspondence for the
counting invariants.
The geometry of the functional equation of Riemann's zeta function
October 02, 14:30 - 15:30, MC108
In a seminal 1859 paper, Riemann gave two proofs of the analytic continuation and functional equation of his
zeta function. The ideas behind his theta function proof were later developed into a powerful theory of Fourier analysis on
number fields, in work of Hecke, Tate and others. In this talk, I will focus instead on the contour integral proof, and
based on the ideas therein, will present two infinite families of new proofs of the analytic continuation and functional
equation. The proofs are facilitated by geometric data coming from the fact that the polylogarithm generating function is a
flat section of the universal unipotent bundle with connection over $\mathbb{P}^{1} \backslash \{0,1,\infty\}$.
The aim of the talk will be to look at non-abelian analogues of
Kummer theory for function fields of Riemann surfaces and relate them
to bundles. We will trace the ramifications of Weil's paper
"Generalizations des fonctiones abeliennes".
This is a report on efforts to classify the endotrivial modules over
the modular groups algebras of groups which are not $p$-groups. A
classification of the endotrivial modules over $p$-groups was
completed by the speaker and Th\'evenaz a few years ago, building on
the work of many others, notably Dade and Alperin. The endotrivial
modules form an important part of the Picard group of self
equivalences of the stable category of modules over the group
algebra. For groups which are not $p$-groups, the problem of
determining the endotrivial modules often reduces to discovering when
the Green correspondent of an endotrivial module is endotrivial. This
investigation often involves a detailed study of the representation
theory of the groups in question.
Watts' theorems in homological algebra and algebraic topology
September 22, 15:0 - 16:0, MC 107
The classical Watts' theorems identify functors which are tensor
products or Hom functors by internal properties. We extend these
theorems to homological algebra and algebraic topology. So, in the
easiest case, we characterize all functors from the unbounded derived
category $D(R)$ of a ring $R$ to $D(S)$ which are given by the derived
tensor product with a complex of bimodules (recovering a result of
Keller's in this case). We draw conclusions about Brown
representability of homology and cohomology functors.
Note room change: MC107.
CS Department colloquium
It would be nice to have a mathematical understanding of basic
biological concepts and to be able to prove that life must evolve in
very general circumstances. At present we are far from being able to do
this. But I'll discuss some partial steps in this direction plus what I
regard as a possible future line of attack.
Given a fixed binary form $f(u,v)$ of degree $d$ over a field $k$, the associated Clifford algebra is the $k$-algebra $C_f=k\{u,v\}/I$, where $I$ is the
two-sided ideal generated by elements of the form $(\alpha u+\beta v)^{d}-f(\alpha,\beta)$ with $\alpha$ and $\beta$ arbitrary elements in $k$. All representations
of $C_f$ have dimensions that are multiples of $d$, and occur in families. In this article we construct fine moduli spaces $U=U_{f,r}$ for the $rd$-dimensional
representations of $C_f$ for each $r \geq 2$. Our construction starts with the projective curve $C \subset \mathbb{P}^{2}_{k}$ defined by the equation $w^d=f(u,v)$,
and produces $U_{f,r}$ as a quasiprojective variety in the moduli space $\mathcal{M}(r,d_r)$ of stable vector bundles over $C$ with rank $r$ and degree $d_r=r(d+g-1)
$, where $g$ denotes the genus of $C$.
The Fine Moduli Space of Representations of Clifford Algebras, Part 1
September 11, 14:30 - 15:30, MC108
Given a fixed binary form $f(u,v)$ of degree $d$ over a field $k$, the associated Clifford algebra is the $k$-algebra $C_f=k\{u,v\}/I$, where $I$ is the
two-sided ideal generated by elements of the form $(\alpha u+\beta v)^{d}-f(\alpha,\beta)$ with $\alpha$ and $\beta$ arbitrary elements in $k$. All representations
of $C_f$ have dimensions that are multiples of $d$, and occur in families. In this article we construct fine moduli spaces $U=U_{f,r}$ for the $rd$-dimensional
representations of $C_f$ for each $r \geq 2$. Our construction starts with the projective curve $C \subset \mathbb{P}^{2}_{k}$ defined by the equation $w^d=f(u,v)$,
and produces $U_{f,r}$ as a quasiprojective variety in the moduli space $\mathcal{M}(r,d_r)$ of stable vector bundles over $C$ with rank $r$ and degree $d_r=r(d+g-1)
$, where $g$ denotes the genus of $C$.
The University of Western Ontario
