Algebra and Friends

Organizing committee

Veronica Crispin Quiñonez (UU), Nancy Abdallah (Chalmers University of Technology), and Lisa Nicklasson (Mälardalen University).

Location

Ångströmlaboratoriet, Regementsvägen 10. Room: Polhemsalen

Om du kommer från Centralstationen, tag buss 4 från läge B4 till Polacksbacken.

Speakers

• Per Bäck (Mälardalen University),
• Filip Jonsson Kling (Stockholm University),
• Jakob Palmkvist (Chalmers University of Technology),
• Hal Schenck (KTH Royal Institute of Technology and Auburn University).

Registration

There is no fee. Fika is offered to all registered participants. To register to the meeting, please fill the form below before October 20. If you are a doctoral student and wish to apply for financial support, complete the registration form before September 30.

Register here

Programme

11:00 Registration and lunch at Café Ångström

13:30 Per Bäck: Hilbert’s Basis Theorem for Wildebeests

14:30  Hal Schenck: Algebraic Aspects of Homogeneous Kuramoto Oscillators

15:30  Coffee break

16:00  Jakob Palmkvist: Nonlinear realisations of Lie superalgebras

17:00  Filip Jonsson Kling: The Gröbner basis for powers of a general linear form in a monomial complete intersection

18:30  Dinner (at own expense)

Titles and Abstracts

Hilbert’s Basis Theorem for Wildebeests

Per Bäck (Mälardalen University)

In this talk, I will introduce Ore extensions, the principal noncommutative generalization of polynomial rings, and recall how Hilbert’s Basis Theorem extends to them. I will then present generalized nonassociative Ore extensions (GNOEs), or wildebeests, which provide a unifying framework for noncommutative, nonassociative polynomial rings. Finally, I will discuss a Poisson analogue for commutative polynomial rings equipped with a compatible Lie bracket, including examples that are Poisson—Noetherian but not Noetherian. This talk is based on joint work with Masood Aryapoor, Patrik Lundström, Johan Öinert, and Johan Richter.

Algebraic Aspects of Homogeneous Kuramoto Oscillators

Hal Schenck (KTH Royal Institute of Technology and Auburn University)

We investigate algebraic characteristics of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks). We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. Furthermore, we describe a construction of networks on an arbitrary number of vertices having linearly stable states that are not twisted stable states. This talk is based on a joint work with Heather A. Harrington and Mike Stillman.

Nonlinear realisations of Lie superalgebras

Jakob Palmkvist (Chalmers University of Technology)

Let g be a Lie algebra with a decomposition into a direct sum of subspaces g=h+e, where h is a subalgebra. The Lie algebra of formal vector fields on e can be extended to a semidirect sum including also the tensor product of h with the algebra of formal power series in coordinates on e. I will construct a homomorphism from g to this semidirect sum of Lie algebras. The construction generalises a well known formula that associates a vector field on a Lie group to any element in its Lie algebra, and can be further generalised to Lie superalgebras. There are many applications, of which I will mention a few. The talk is based on 2012.10954 and work in progress with Jonas T. Hartwig.

The Gröbner basis for powers of a general linear form in a monomial complete intersection

Filip Jonsson Kling (Stockholm University)

In this talk, we will study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. The main result is an explicit description of the reduced Gröbner bases for these ideals under any term order. The approach is primarily combinatorial, focusing on the structure of the initial ideal. To do this, we associate a lattice path to each monomial in the vector space basis of an Artinian monomial complete intersection and introduce a reflection operation on these paths, which enables a key counting argument.

As a consequence, we will see a new proof that Artinian monomial complete intersections possess the strong Lefschetz property over fields of characteristic zero. If time permits, we will also see something about how the number of Gröbner basis elements in each degree is connected to several well-known sequences, including the (generalized) Catalan, Motzkin, and Riordan numbers.

Funders

• Stiftelsen Längmanska kulturfonden
• Stiftelsen för Vetenskaplig Forskning och Utbildning i Matematik
• Department of Mathematics, Uppsala University