Abstract: Matrix factorizations were introduced by Eisenbud in 1980 to study hypersurface singularities. Since their introduction they have been used in a wide range of fields including representation theory, knot theory and string theory. In my talk I will give an overview of this construction, talk briefly about how they are used, and then discuss recent work using matrix factorizations to study complete intersection singularities.
Abstract: In this talk, I will provide an overview of recent work pertaining to Hopf algebra actions on (noncommutative) regular algebras. Many examples of such actions will be presented. I will also list several open questions for future work. The results discussed here are from joint works with Kenneth Chan, Pavel Etingof, Ellen Kirkman, Yanhua Wang, and James Zhang.
Abstract: It is a well understood story that one can extract link invariants associated to simple Lie algebras. These invariants are called Reshetikhin-Turaev invariants and the famous Jones polynomial is the simplest example. Kauffman showed that the Jones polynomial could be described very simply by replacing crossings in a knot diagram by various smoothings. In this talk we will explain Cautis-Kamnitzer-Licata's simple new approach to understanding these invariants using basic representation theory and the quantum Weyl group action. Their approach is based on a version of Howe duality for exterior algebras called skew-Howe duality. Even the graphical (or skein theory) description of these invariants can be recovered in an elementary way from this data. The advantage of this approach is that it suggests a `categorification' where knot homology theories arise in an elementary way from higher representation theory and the structure of categorified quantum groups.
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Abstract: Over the last decade, much work has been undertaken toward understanding the prime and primitive spectra of the quantized coordinate ring of m x n matrices and related algebras. Two approaches have been successful: one representation theoretic, the other ring theoretic. The first half of this talk will be an overview of the known results in this area. The second half will focus on my recent work that takes a combinatorial point of view of the ring theoretic approach. Here we interpret these algebras as arising from paths in a certain directed graph.
Abstract: I will explain how 2-categories naturally arise in the study of two-dimensional topological field theories with defect lines. An interesting example is the 2-category whose objects are isolated hypersurface singularities and whose 1-morphisms are matrix factorisations; I will discuss a result about adjoints in this 2-category worked out in recent joint work with Nils Carqueville.
Abstract: We will see that covolume of a lattice in a semisimple Lie group says a lot about their structures. I will survey some of the related problems and results. In particular, we will discuss the following problems:
Abstract: For every element w in the Weyl group of a simple Lie algebra g, De Concini, Kac, and Procesi defined a subalgebra U_q^w of the quantized universal enveloping algebra U_q(g). The algebra U_q^w is a deformation of the universal enveloping algebra U(n_+\cap w.n_-), where n_+ and n_- are the nilradicals of a pair of dual Borel subalgebras. I will describe a smash product construction of certain finite-type De Concini-Kac-Procesi algebras to obtain ones of affine-type. Next, I will describe how the algebras arising from this smash product construction can be twisted by a cocycle to produce certain subalgebras related to the corresponding Faddeev-Reshetikhin-Takhtajan algebras. This is joint work with Chris Nowlin.
Abstract: In this talk I will describe the following obstruction result for functors extending the Zariski spectrum: every contravariant functor from rings to sets whose restriction to the full subcategory of commutative rings is isomorphic to Spec must assign the empty set to complex matrix algebras of order 3. The proof relies crucially on the Kochen-Specker ``no-hidden-variables'' theorem of quantum mechanics. I will also mention a recent generalization of the result due to van den Berg and Heunen.
Abstract: We will first give some basic review and a survey of how coalgebra structures occur from different areas of mathematics, such as representation theory, graded rings and modules, category theory and homological algebra or topology. We show how the language of coalgebras and comodules (rational modules) can be used to give a unifying interpretation of both (co)homology and the newer q-(co)homology, and how it could potentially lead to the introduction of new cohomology-like theories. In the second part, we use this language to generalize Frobenius algebras to the infinite dimensional case, and define an integral theory for such algebras. This generalizes the theory from Hopf algebras with nonzero integrals, which in turn, can be thought as a quantization of the topological compact groups. We show how our methods apply to new proofs of known fundamental results of Hopf algebras and allow new analogous results for (co)quasi-Hopf algebras (bijectivity of the antipode). They also give further new insight into the parallelism between Hopf algebras with nonzero integral and compact groups.
Abstract: I will give a very brief introduction to the Morel-Voevodsky A^1-homotopy theory. The main goal will be to explain some connections between A^1-homotopy theory and questions regarding existence of rational points on smooth projective varieties over a field (based on joint work with Christian Haesemeyer).
Abstract: Determinantal varieties are of course a well understood set of objects, but the k-th order jet scheme of these varieties (for example, taking k=1, their "algebraic tangent bundle") are much less understood.
In this talk, we will describe some results about these jet schemes: irreducibility for some values of natural parameters, description of the components for some other values, Grobner bases for some special cases, conjectures about connections to invariant theory, etc.
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Abstract: I will introduce Lawson homology for complex varieties and reduced Lawson homology for real varieties and discuss their connection with motivic cohomology and the Milnor conjecture proved by V.Voevodsky. I then prove the analogue of the Friedlander-Mazur conjecture for reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety X vanishes in homological degrees larger than the dimension of X in all weights. This result is a joint work with J. Heller.
Abstract: In this talk I describe an unusual sequence of groups and an unusual sequence of spaces on which they act. A very rough description is that the groups are "braided versions" of the real orthogonal groups. More concretely, there is an algebraic procedure (that I call "pulling a group apart") that takes a symmetric groups as input and produces Artin's braid groups as output. When the same procedure is applied to the group O(n,R), the results are interesting, pretty and (as I hope to convince you by the end of the talk) useful.
Abstract: Symplectic reflection algebras were introduced by Etingof and Ginzburg around 2000. They are a family of algebras closely related to Lie algebras, quivers, and Cherednik's double affine Hecke algebras. I will give an overview of the connections between these algebras and their generalizations.
Abstract: I will introduce cycle complexes of Bloch and Suslin-Voevodsky and discuss their applications to duality for etale cohomology, class field theory, and a conjecture of Kato.
Abstract: Spaces of orderings were introduced by Murray Marshall in the 1970's and provide an abstract framework for studying orderings on fields and the reduced theory of quadratic forms over fields. Numerous important notions in this theory, such as isometry, isotropy, or being an element of a value set of a form, make an extensive use of positive primitive formulae in the language of special groups. Therefore, the following question, which can be viewed as a type of very general local-global principle, is of great importance: is it true that if a positive primitive formula holds in every finite subspace of a space of orderings, then it also holds in the whole space? This problem is now known as the pp conjecture. The answer to this question is affirmative in many cases, although it has always seemed unlikely that the conjecture has a positive solution in general. In this talk we shall discuss first counterexamples for which the pp conjecture fails, as well as address some implications following from these examples.
Abstract: A permutation of a finite set is called a derangement if it has no fixed points. In the first half of the talk, we survey some applications of derangements in number theory and generation of groups, and describe some basic results (due to Jordan, Cameron and Cohen, and others) in the subject. We then discuss some joint work with Diaconis and Guralnick in which we show that except for the action of the symmetric group on k-sets, almost all permutations have no fixed points in any action of the symmetric group. Work with Guralnick extending this to actions of finite classical groups and settling a conjecture of Boston and Shalev is also described.
Abstract: I'll briefly recall what the Witt group of quadratic forms is and how the so-called "triangular Witt groups" come in the picture. Then, I'll explain how to compute (ordinary) Witt groups of Grassmann varieties, which is a joint work with Baptiste Calmes. Our result will be compared to what classically happens in other theories, like Chow groups or K-theory, and we shall try to understand where the main differences arise.
Abstract: The main objective of this talk is to introduce new quantum objects - braided symmetric and exterior algebras. Their construction is motivated by the following classical (open) problem: Decompose the symmetric and exterior powers of a finite-dimensional module over a semisimple Lie algebra $g$ into simple $g$-modules. In a joint paper with Arkady Berenstein we approach the problem by associating to each finite-dimensional $g$-module the braided symmetric and exterior algebras - module algebras over the quantized enveloping algebra of $g$.
We can ask for which $g$-modules the braided symmetric algebra can be realized as a (flat) deformation of the (classical) symmetric algebra. I will show how the solution of this problem opens a surprising connection to the theory of multiplicity-free representations via a classification of Poisson structures.
Finally, I will outline connections to canonical bases and quantum cluster algebras and discuss open questions and conjectures.
Abstract: Let A be a finite-dimensional algebra over a field k. The stable module category of A is defined as the quotient of the category of all finite-dimensional A-modules by the subcategory of projective modules. Such categories arise naturally in the study of representations of finite groups in prime characteristic, where it may happen that the stable category of G-modules is equivalent to that of H-modules for a proper subgroup H of G. While such equivalences of stable categories are generally quite hard to come by, when they do exist they imply a great deal of homological information is shared by the two algebras. Perhaps most notably, two stably equivalent algebras have the same stable AR-quiver, which can be thought of as a "skeleton" of the category of indecomposable modules. We will give some examples of this phenomenon and talk about recent joint work with Martinez-Villa on stable categories of graded modules over (possibly nonartinian) graded algebras.
Abstract: A typical infinitesimal group scheme is an infinitesimal p-th power neighborhood of the identity of an algebraic group over a field of characteristic p. Infinitesimal group schemes constitute a class of finite group schemes whose representation theory remains less well understood than the modular representation theory of finite groups. This talk will discuss joint work with Julia Pevtsova following earlier work of many others, especially Suslin-Friedlander-Bendel. Constructions specific to the representation theory on infinitesimal group schemes are presented: the "universal p-nilpotent operator", vector bundles associated to special representations, and "generalized support varieties". The general theory is made more concrete by considering four families of examples.
Abstract: Let X be an infinite set, M the monoid of all endomaps of X, G the group of all permutations of X, and R the ring of endomorphisms of a vector space with basis X. I'll discuss some unusual properties shared by M, G, and R. For instance, every countable subset of one of these objects can be embedded in a subobject generated by two elements. In each case, one can also obtain an interesting classification of subobjects, upon introducing an equivalence relation under which two submonoids (resp. subgroups, subrings) A and A' are equivalent if and only if there exists a finite subset U of M (resp. G, R) such that the submonoid (resp. subgroup, subring) generated by A and U is equal to that generated by A' and U.
Abstract: Most approaches to the inverse Galois problem have depended on Riemann's existence theorem to produce a "regular" realization of a given group $G$: a curve cover of the projective line whose automorphism group is $G$ and that is defined over the rational numbers. Unfortunately, for any finite group $G$ and integer $r$, there is a finite group epimorphism $H \twoheadrightarrow G$ such that every regular realization of $H$ has at least $r$ branch points.
A conjecture of Fried generalizes Mazur's theorem on rational torsion points in elliptic curves to curves of higher genus. Viewing Mazur's theorem as classifying regular realizations of dihedral groups with four branch points and having reflections as inertia group generators, Fried posed that $H$ can be chosen to be a $p$-Frattini cover of $G$. The group theory of said objects is ideal for negative questions on regular realizations and is intriguing in its own right. The place of modular curves in Mazur's setup is taken by Fried's analogous construction of modular towers; these also support further generalizations of elliptic curve arithmetic like Serre's theorem that the absolute Galois group acts on the Tate module as an open subgroup of the automorphism group (if the elliptic curve does not have complex multiplication).
I will explain these connections and the current approach to Fried's conjecture for $r=4$.
Abstract: A-infinity algebras and L-infinity algebras are generalizations of associative and Lie algebras, respectively. They naturally appear in deformation theory and algebraic topology. One expects that every L-infinity algebra K should have a universal enveloping algebra U(K). Several constructions were earlier proposed by different authors but none of them reduces to the usual universal enveloping algebra if the L-infinity structure on K reduces to a Lie structure. We suggest a definition of U(K) which does possess this property (which at present works only under a certain condition on K) and outline its application to the study of sheaves on complete intersections in toric varieties.
Abstract: I will give an introduction to some of the ideas of noncommutative projective geometry, focusing on the problem of classifying surfaces. My point of view tends to emphasize the ring-theoretic aspect of this subject; namely, I like to study noncommutative graded algebras using the help of tools from commutative algebraic geometry. I will end by describing some recent work, joint with Toby Stafford, which solves one slice of the classification problem in terms of noncommutative blowing up.
Abstract: We show how probabilistic methods and algebraic group methods can be used to answer some questions about generating finite simple groups. It is known (using the classification of finite simple groups) that every finite simple group can be generated by two elements. Indeed, it follows that the probability that a random pair of elements generate the simple groups tends to 1 as the order goes to infinity. We will mention some stronger results and show how these characterize the solvable radical of a finite group.
Abstract: In this talk we shall recall several classical constructions of generic splitting varieties and their applications. Then we discuss the use of generic splitting varieties in Voevodsky's proof of the Milnor Conjecture and in the proposed proof of the Bloch-Kato Conjecture.
Abstract: I will discuss several possible definitions for a dynamical quantum group in the super case. I will begin by talking about the nongraded definitions, and then give some possible extensions to the super case and see which one makes more sense under various perspectives. The talk will be based on the arxiv preprint, ("Dynamical Quantum Groups - The Super Story" - math.QA/0508556), a review paper on dynamical quantum groups in the super case. I will incorporate into the talk all the comments and feedback I have received so far about that paper.
Abstract: Let d be a prime. The Nakayama conjecture (for symmetric groups) states that two complex irreducible characters of the symmetric group lie in the same d-block if and only if the characters are labelled by partitions with the same d-core. This conjecture was first proved by Brauer and Robinson. Recently, Kulshammer, Olsson, and Robinson proved a deep generalization of this theorem for an arbitrary integer d (not necessarily prime) at least 2. In this talk I will state an even further generalization of the Nakayama conjecture. The proof - which we will outline - is similar to the argument of Kulshammer, Olsson, and Robinson, but it is shorter and more elementary.
Abstract: We consider the problem of determining when a Hopf algebra is PI, i.e., satisfies a polynomial identity as an algebra. The simplest examples are group algebras, universal enveloping algebras, and, in prime characteristic, restricted enveloping algebras. For these classes of Hopf algebras, explicit necessary and sufficient conditions are given in the works of Passman, Latyshev, Bahturin, and Petrogradsky. We generalize their results and solve the problem for cocommutative Hopf algebras - completely in characteristic zero and partially in prime characteristic. Time permitting, we will also consider the dual problem of determining when a Hopf algebra is coPI, i.e., satisfies a coidentity as a coalgebra.
Abstract: The problem of classifying central, finite dimensional division algebras is vast. One strategy is to identify all of the possible constructions of division algebras over a particular field. Cyclic division algebras form a particularly nice class of division algebras. I will talk about a criterion for determining the cyclic maximal subfields of the tame division algebras over the Laurent series field $Q_p((t))$. Knowing the cyclic maximal subfields not only proves cyclicity, it gives us a concrete description of the multiplication and a useful way to analyze its structure.