On the Grothendieck ring of lattice configuration categories
This is a continuation of previous talks, although I'll try to make it accessible for any newcomers. We find generators and relations for the commutative Grothendieck ring of the category of weight modules over a bialgebra associated to bosonic lattice configurations on a 1-dimensional line and a 2-dimensional twisted cylinder. This is work in progress joint with Daniele Rosso.
Tensor product decompositions of irreducible representations
I'll discuss how finite-dimensional irreducible representations of the general linear Lie algebra can be decomposed as tensor products of Weyl algebra modules. We will focus on gl_2 and along the way encounter connections to statistical mechanics and noncommutative geometry. This is work in progress joint with Daniele Rosso.
Observing similarities in a series of constructions of algebras and their representations, I will give a sketch of a new entity, out of which all these examples are born. The final form of this entity is yet to be fully revealed, hence the title might be a misnomer. The examples we will study include Kleinian singularities and their noncommutative deformations; enveloping algebras of some classical and affine Lie algebras and their primitive quotients; reduction algebras associated to a Lie algebra and a reductive subalgebra; twisted generalized Weyl algebras; and rational Galois orders attached to complex reflection groups.
This is work in progress, partly joint with Daniele Rosso.
Given a field \(k\) and a finite, simple graph \(G\), one can construct the toric edge ideal \(I_G\) of \(G\) as the kernel of a surjective ring homomorphism \(k[E]\) to \(k[G]\), where \(k[E]\) is the polynomial generated by variables associated to the edges of \(G\) and \(k[G]\) is the subring of the polynomial ring with variable associated to the vertices of G and generators associated to the edges. In the case that \(G = K_{(m,n)}\) is a complete bipartite graph, this corresponds to the Segre embedding of \(\mathbb{P}^{m-1} \times \mathbb{P}^{n-1}\) in \(\mathbb{P}^{nm-1}\). There is a combinatorial description of those bipartite graphs \(G\) such that \(I_G\) is generated by quadrics due to Ohsugi and Hibi. We give a combinatorial classification for bipartite graphs \(G\), of when \(I_G\) has a resolution which is linear for a fixed number of steps. This is joint work with Jason McCullough.
Operads (introduced by May) are gadgets which govern various types of algebraic structures. In the first part of this talk, we'll give an introduction and some basic examples. In the second part, we'll extend the notion of operad in multiple ways. These will include cyclic operads (Getzler–Kapranov) and homotopy-coherent versions of operads (Cisinski–Moerdijk–Weiss).
April 11, 2019
On the s-multiplicity
Lance Miller (University if Arkansas)
The Hilbert-Samuel multiplicity of a local ring is a geometric invariant which may be used to algebraically count points of intersection of complex curves. Working in positive characteristic, a new multiplicity function, the Hilbert-Kunz multiplicity, arise algebraically with subtle and interesting connections to combinatorics. W. D. Taylor introduced a deformation between Hilbert-Samuel and Hilbert-Kunz multiplicities. In this talk, we discuss this deformation concentrating on questions about calculating and estimating it.
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April 4, 2019
On the latest developments in Gelfand-Tsetlin theory
Jonas Hartwig (ISU)
In this talk I will report on recent advances, due to Ben Webster, in the representation theory of principal Galois orders, in particular of the general linear Lie algebra. Besides introducing a new notion, that of principal flag orders, which considerably simplifies and clarifies the subject, he also proves that a very wide class of homological convolution algebras called Coulomb branches (structures appearing in QFT) are examples of principal Galois orders. Mazorchuk's OGZ algebras (and hence the enveloping algebra of the general linear Lie algebra) are examples of such Coulomb branches.
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March 28, 2019
Quantum Affine Wreath Algebras
Daniele Rosso (Indiana University Northwest)
Affine Hecke algebras are an important family of algebras that has been studied by several authors because of their connections to many subjects such as quantized affine Lie algebras, representations of affine Lie algebras, and the Langlands program via algebraic groups over local fields. In this talk I will describe generalizations of Type A affine Hecke algebras, which we call quantum affine wreath algebras, that depend on the choice of a fixed Frobenius (or symmetric) algebra. We obtain results about the structures of these algebras, including a basis and the center, as well as their cyclotomic quotients. This is joint work with Alistair Savage.
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March 14, 2019
A Monstrous(?) Moduli Space
Philip Engel (Harvard University)
Let \(B = \mathbb{C}H^{13}\) be 13-dimensional complex hyperbolic space (a complex ball). There is an arithmetic group \(\Gamma\) in \(PU(13)\) acting on \(B\) generated by order 3 Hermitian isometries \(s_i\) called triflections. Basak and Allcock have studied the geometry of \(X = \Gamma \backslash B\) in detail; it is intimately connected with the finite projective plane \(\mathbb{P}^2F_3\). A conjecture of Allcock states that if one replaces relations \(s_i^3=1\) in \(\Gamma\) with \(s_i^2=1\), the resulting group is the Bimonster---the wreath product of the monster with \(\mathbb{Z}_2\). A resolution of this conjecture likely leads to a resolution of the "Hirzebruch prize question": The existence of a compact 12-complex dimensional manifold with certain topological invariants and an action of the monster. Such a manifold would lead in a known way to a new, geometric proof of monstrous moonshine.
I will discuss three moduli spaces, which might (depending on the status of computations at the time of the talk) be isomorphic to ball quotients of dimensions 13, 7, 4 relating to the projective planes \(\mathbb{P}^2F_3\), \(\mathbb{P}^2F_2\), "\(\mathbb{P}^2F_1\)" = {3 points}. The corresponding finite groups, gotten by replacing 3, 4, 6 with 2, should be the bimonster, an orthogonal group \(O_8(2)\) of a quadratic form on \(F_2^8\), and the symmetric group \(S_6\) respectively. Should these examples work out, they will produce many surprising things: For instance, a formula for the order of the monster group in terms of Hurwitz numbers. This talk is highly speculative and represents joint work with Peter Smillie and Francois Greer.
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March 7, 2019
An Alternating Analogue of \(U(\mathfrak{gl}_n)\) and Its Representations
Erich Jauch (ISU)
The universal enveloping algebra of a Lie algebra \(\mathfrak{g}\) is of utmost importance when studying representations of \(\mathfrak{g}\). In 2010, V. Futorny and S. Ovsienko gave a realization of \(U(\mathfrak{gl}_n)\) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of \(S_1\times S_2\times\cdots\times S_n\) where \(S_j\) is the symmetric group on \(j\) variables. With some connections to Galois Theory, the interesting question is what would a similar object be in the invariant ring with respect to a product of Alternating groups? We will discuss such an object and some results about its representations.
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February 28, 2019
Deriving the Hopf algebra \(U_q(\mathfrak{sl}_2)\) from \(q\)-calculus
Jonas Hartwig (ISU)
The algebra \(U_q(\mathfrak{sl}_2)\) was first defined in 1981 by Kulish and Reshetikhin in the context of quantum integrable systems. This eventually led Drinfeld and Jimbo to independently define \(U_q(\mathfrak{g})\) for an arbitrary symmetrizable Kac-Moody algebra \(\mathfrak{g}\).
In this talk we show that the whole Hopf algebra structure of the quantum group \(U_q(\mathfrak{sl}_2)\) can be derived in a completely elementary and natural way by using \(q\)-differential operators. Such operators were studied by Reverend Frank Hilton Jackson already in 1908. This leads to the intriguing thought that quantum groups could have been discovered much earlier from purely mathematical considerations.
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February 21, 2019
Geometric Inequalities on Riemannian manifolds
Zhifei Zhu (University of Toronto)
I will discuss some upper bounds for the length of a shortest periodic geodesic, and the smallest area of a closed minimal surface on closed Riemannian manifolds of dimension 4 with Ricci curvature between -1 and 1. These are the first bounds that use information about the Ricci curvature rather than sectional curvature of the manifold. (Joint with Nan Wu).
I will also give examples of Riemannian metrics on the 3-disk demonstrating that the maximal area of 2-spheres arising during an ``optimal" homotopy contracting its boundary cannot be majorized in terms of the volume and diameter of the 3-disc and the area of its boundary. This contrasts with earlier 2-dimensional results of Y. Liokomovich, A. Nabutovsky and R. Rotman and answers a question of P. Papasoglu. (Joint with Parker Glynn-Adey).
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February 14, 2019
Geometric Optimal Control of a Class of Quantum Systems
Domenico D'Alessandro (ISU)
In the optimal control of n-level quantum systems there is a class of models which on one hand occur very often and, on the other hand, are amenable of explicit solutions. These are the so-called K-P systems. They include, among others, systems whose energy diagram is a bi-partite graph such as Lambda-systems, double Lambda systems and one quantum bit. For these models, it is possible to find the time optimal control under an energy constraint explicitly. The consideration of the minimum time transfer between two states is a natural requirement in applications such as quantum computing especially in view of the fact that fast transfer is a way to mitigate the influence of the environment. The K-P structure refers to an underlying Cartan-type K-P decomposition of the Lie algebra su(n) such that only the operators corresponding to the P part of the decomposition appear in the Schrodinger equation of the system.
We describe the case of the quantum bit in detail and use it to illustrate the general theory. In particular, we explicitly derive the minimum time trajectories between any two states for this system. This analysis also displays some general features of the optimal synthesis such as: the critical locus, i.e., the locus of points where optimal trajectories lose optimality, the geometry of the set of reachable states at each time and the diameter of the system, i.e., the worst case minimum time. Furthermore, such an analysis leads to the general consideration of the role of symmetries in optimal control problems.
The explicit nature of the solution of the optimal control problem provided lends itself to generalizations to other systems of interest in applications. We shall in particular illustrate the case of N qubits controlled in parallel in minimum time.
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November 15, 2018
Reflection groups and the geometry of polynomial interpolation
Alexandra Seceleanu (University of Nebraska-Lincoln)
The interpolation problem in algebraic geometry asks for the equations of polynomials in several variables passing through a given set of points in the plane with assigned multiplicities. There are many beautiful results and long-standing conjectures regarding the dimensions of the linear spaces formed by the interpolation polynomials and the degrees of these polynomials.
We consider the implications of symmetry on the interpolation problem through the lens of several examples where the interpolation points arise from the action of a reflection group on the complex plane. We use classical invariant theory to show that the ideal of these interpolation points has a surprising property from the point of view of commutative algebra. This is based on joint work with Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Jack Huizenga, and Tomasz Szemberg and also on work of Benjamin Drabkin.
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November 8, 2018
Quantum Groups II: Knot invariants
Jonas Hartwig (ISU)
I'll give a more detailed overview of what quantum groups is all about.
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November 1, 2018
Quantum Groups I: Discrete Calculus
Jonas Hartwig (ISU)
Introductory talk about quantum groups. This is a fascinating subject going back to Euler, with deep connections to knot theory, statistical mechanics, combinatorics and more.
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October 25, 2018
\(Q\)-Systems and Generalizations in Representation Theory
Darlayne Addabbo (University of Notre Dame)
I will discuss hierarchies of difference equations whose solutions, called \(\tau\)-functions, are matrix elements for the action of loop groups, \(\widehat{GL_n}\), on \(n\)-component fermionic Fock space. In the \(n=2\) case, these \(\tau\)-functions are determinants of Hankel matrices and one can see by applying the Desnanot-Jacobi identity that they satisfy a \(Q\)-system. \(Q\)-systems are discrete dynamical systems that appear in many areas of mathematics, so it is interesting to study the more general, \(n>2\) hierarchies. I will discuss these new hierarchies of difference equations and the progress I have made in investigating them within the context of other areas of mathematics. (Joint with Maarten Bergvelt)
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October 18, 2018
Finite Cohen-Macaulay Type for Zero Dimensional Rings
Jason McCullough (ISU)
Following Chapter 3 in the Leuschke-Wiegand book, I'll prove that a zero dimensional commutative local ring has finitely many indecomposable modules if and only if it is an artinian principal ideal ring.
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October 11, 2018
Vertex algebras and Poisson algebras
Jonas Hartwig (ISU)
I'll explain how every vertex algebra produces a Poisson algebra.
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September 27, 2018
Whittaker modules over vertex operator algebras
Jonas Hartwig (ISU)
In the first hour I will give an introduction to vertex operator algebras and Whittaker modules over Lie algebras. In the second hour I will talk about recent joint work with Nina Yu in which we construct certain simple weak modules for the Z_2-orbifold subalgebra of the higher rank free bosonic vertex operator algebra. These modules are Whittaker modules over the Virasoro subalgebra.
September 20, 2018
Lattices and modular forms II
Tathagata Basak (ISU)
This lecture will be largely independent of the previous ones. We will use the basic results of modular forms proved last time as black boxes and give a couple of applications to the theory of lattices/quadratic forms. As last time, we will try to keep prerequisites to a minimum and we will recap the definitions and results we need about modular forms.
Reference: J. P. Serre, A Course in Arithmetics
September 13, 2018
Lattices and modular forms I
Tathagata Basak (ISU)
We will have a gentle introduction to the theory of modular forms and functions. Then, we shall give some examples of their applications in number theory, in particular, to theory of integer lattices. Examples may include
1) counting lattice points on a sphere,
2) finding all even integral self-dual lattices in 24 dimensional Euclidean space. There are exactly 24 such lattices, among them, the famous Leech lattice.
Reference: J. P. Serre, A Course in Arithmetics
September 6, 2018
The Krull-Remak-Schmidt Theorem
Jason McCullough (ISU)
The classical Wedderburn Theorem states that a finitely generated abelian group decompose uniquely (up to isomorphism and reordering) as a direct product of indecomposable groups. It is natural to ask whether a similar theorem holds for modules over a commutative ring. In general, the answer is no, but over complete local rings, there is an analogous statement. The proof involves a detour through non-commutative local endomorphism rings. We'll examine what works for complete local rings and how to fix the situation in the non-complete case. This follows Chapter 1 in the Leuschke-Wiegand book and is our first step toward Cohen-Macaulay Representations. (This talk is starting from scratch and is more elementary than the previous talk.)
August 30, 2018
Cohen-Macaulay rings and modules
Jason McCullough (ISU)
Part 1 will be a somewhat gentle introduction to Cohen-Macaulay rings and modules. There will be an emphasis on examples and geometric interpretations. Part 2 will be a sketch of how big Cohen-Macaulay algebras relate to the famous homological conjectures and why invariant subrings of polynomial rings are Cohen-Macaulay.