Ottawa-Carleton Joint Algebra Seminar

In this talk I will first introduce the two Hecke actions of the formal affine Demazure algebra (defined by Hoffnung–Malagón-Lopez–Savage–Zainoulline) on equivariant oriented cohomology of flag varieties. Using this, I will recover a resolution of Deodhar. I will then focus on the cohomology theory corresponding to the so-called hyperbolic formal group law, and define the Kazhdan–Lusztig Schubert class. Such class is closely related with smoothness of Schubert varieties. This is joint work with C. Lenart and K. Zainoulline.

Open genus zero Gromov-Witten invariants are numbers that count pseudoholomorphic maps from a disk into a symplectic manifold, with Lagrangian boundary conditions, satisfying various constraints. Related invariants were defined by Welschinger in the early 2000's for real symplectic manifolds of complex dimensions 2 and 3, and the problem of finding a definition for higher dimensions remained open until recently (Georgieva, 2013, and Solomon-Tukachinsky, 2016).

An obstruction to invariance is bubbling of disks at the boundary. In this talk I will explain the problem, and define an A_∞ algebra associated to a Lagrangian submanifold, following ideas of Fukaya-Oh-Ohta-Ono (2009) and Fukaya (2011). This A_∞ algebra serves as a language for tracking pseudoholomorphic disks and bubbling. We then introduce an associated Maurer-Cartan equation, whose solutions allow a definition of invariants in a suitable sense (an idea demonstrated by Joyce, 2006).

This is a continuation of the talk by R. Devyatov on computing intersections of invariant ideals in representation rings; it is based on the joint recent preprint by Baek, Devyatov and myself. We introduce a new techniques that allows to compute various subgroups of cohomological invariants of degree 3 of semisimple algebraic groups, hence, extending previous results by Merkurjev and others.

Permutation-Hermite (PH) equivalence on integer matrices, or equivalently on subgroups of Zⁿ, is an equivalence relation intermediate between the usual Smith equivalence (invertible row and column operations) and Hermite equivalence (invertible row operations)—invertible row operations as well as column permutations are allowed. It comes up in a few places, e.g., classification of lattice simplices up to the action AGL(n,Z), but for me, it arose from classification of dense subgroups of Rⁿ that are free of rank n+1 (the minimum possible) as partially ordered abelian groups.

Relatively easily computed invariants for PH-equivalence are given that together are pretty good, and a notion of duality (previously observed in the lattice simplex situation) yields still more. This is a subject still in its naissance, so numerous examples and conjectures will be discussed.

The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory for elliptic curves. The finiteness of the Shafarevich-Tate group of an elliptic curve E over the field of rational numbers is included in the Birch and Swinnerton-Dyer conjectures, and is still an open question.

We will present an overview of the Shafarevich-Tate and Selmer groups of an elliptic curve in the framework of group cohomology. Known results on the finiteness of the Shafarevich-Tate group will be mentioned, including results of Rubin and Kolyvagin.

We will then discuss the vanishing of the p-component of the torsion subgroup of the Shafarevich-Tate group for almost all primes p, under the assumption that the elliptic curve E has non-integral j-invariant. This is original joint work of the speaker with Dmitry Malinin.

In this talk I will give an introduction to the concept of A(rthur)-packet by showing that, the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups. If time allows me I will end the talk with a short exposition of the work of J. Adams, D. Barbasch and D. Vogan on the local Arthur conjecture.

We introduce noncrossing tree partitions which are certain noncrossing collections of curves on a tree embedded in a disk. These generalize the classical type A noncrossing partitions, and, as in the classical case, they form a lattice whose partial order is given by refinement. In this talk, we will interpret noncrossing tree partitions in terms of the representation theory of a family of gentle algebras called tiling algebras. In particular, we show that the lattice of noncrossing tree partitions is isomorphic to the lattice of wide subcategories in the module category of a tiling algebra, and we use noncrossing tree partitions to classify all 2-term simple-minded collections in the bounded derived category of a tiling algebra. This is joint work with Thomas McConville.

The Steinberg group St(C) of a Chevalley group C over a field F is defined by generators and relations involving commutators of root subgroups of C. A classical result of Steinberg says that St(C) is the universal central extension of C except when the rank of C and the size of F is small. In particular, St(C) is centrally closed (= its own universal central extension).

Several natural generalizations of this situation have been considered and will be discussed in the talk. One replaces C by a group G with commutator relations with respect to a family of subgroups and shows that the canonically defined Steinberg group associated with G is centrally closed, although it may in general no longer be a universal central extension of G. For example, this is so for G the elementary group of n x n matrices over a ring. . In characteristic 0, these generalizations are special cases of groups associated with the elementary automorphism group of a root-graded Lie algebra.

The Mirkovic-Vybornov isomorphism is an isomorphism of certain algebraic varieties which are of interest in geometric representation theory: between slices to nilpotent orbit closures in gl(N) on the one hand, and slices to spherical Schubert varieties in PGL(n) on the other (they also connected these to Nakajima quiver varieties of type A, but we will not discuss this). I will discuss joint work with Ben Webster and Oded Yacobi, which lifts this to an isomorphism between deformation quantizations: between (parabolic) finite W-algebras, and truncated shifted Yangians, respectively. As a consequence, we see that the original MV isomorphism is Poisson.

Affine Lie algebras burst onto the mathematical scene in the late 1960s as the most important infinite-dimensional examples of the newly discovered Kac-Moody algebras. From the beginning, it was understood that they have a powerful alternative interpretation as extensions of loop algebras, families of functions from the circle to finite-dimensional Lie algebras. By replacing the circle with other smooth manifolds or algebraic varieties, we obtain a large class of Lie algebras, known as current algebras. They appear in geometry and many applications, including singularity theory, gauge theory, soliton equations, and exactly solvable models.

After introducing these algebras, we will discuss the classification of simple weight modules (with finite-dimensional weight spaces) for current algebras. The modules are constructed using parabolic induction from admissible representations of Levi subalgebras, and their classification reduces to Mathieu's results on admissible weight modules for reductive Lie algebras.

Geodesics play an essential role in studying Cayley graphs of groups. I will define the notion of a uniquely labelled geodesic and study such geodesics in Cayley graphs of certain Coxeter groups. I will give a formula for the maximal length of such a geodesic. More specifically, I will present a generating function which describes the number of uniquely labelled geodesics with each label, from which we can deduce results about the overall number of such geodesics. This is joint work with Kirill Zainoulline.

A lot of interesting arithmetic information is contained in congruences between l-adic representations of the absolute Galois group of a p-adic field, for l and p rational primes. One approach to studying when this can happen is via deformation theory, which allows one to use methods of algebraic geometry, and particularly intersection theory, to study such congruences. A conjecture of Breuil and Mézard predicts that the intersection theory of the associated geometric “deformation spaces” can be described in terms of the representation theory of GL(n,Z_p), and this conjecture is intimately related to central problems in algebraic number theory – for example, Kisin’s proof of the 2-dimensional Fontaine—Mazur conjecture follows from establishing the Breuil—Mézard conjecture for 2-dimensional representations when l=p. More recently, Shotton has proved the Breuil—Mézard conjecture in complete generality in the case that l and p are distinct. I will describe work extending Shotton’s result to fit in with the wider context of the local Langlands programme, leading to a description of congruences between the “tame regular semisimple elliptic Langlands parameters” of DeBacker and Reeder in terms of the representation theory of certain p-adic integral group schemes.

This is a report on joint work with Seidon Alsaody (Lyon). Let C be an octonion algebra defined over a ring R. Informally speaking, an isotope C' of C is a kind of deformation of C defined by Albert and which has been investigated by McCrimmon and Knus/Parimala/Sridharan. The aim of this talk is to describe and to classify the isotopes of C by means of triality and cohomological methods.

We will present the basic notions of locally conformal symplectic geometry and explain why this might be relevant for studying contact manifolds. We will show that the neighbourhood of a Lagrangian L in such manifolds is conformally equivalent to the cotangent bundle of L with the structure deformed by a closed one form b. In this context we will then relate the intersection points between the zero section and one of its Hamiltonian perturbations with the free Betti numbers of the Morse-Novikov homology of b. This is joint work with Emmy Murphy.