Johan Öinert (Wednesday, 13:30–14:30)
Rank conditions for Ore extensions
Ore extensions were introduced by Øystein Ore in 1933 and naturally generalize classical polynomial rings. For such a ring extension an important question reads:
When is an algebraic property inherited from the base ring R to its Ore extension R[x; σ, δ]? This question has been answered for several algebraic properties. The rank condition and strong rank conditions for a ring have their roots in linear algebra and module theory. In this talk, we will show that an Ore extension R[x; σ, δ] satisfies the rank condition if and only if the base ring R does. For the left and right strong rank conditions the situation becomes more involved. Still, with the appropriate assumptions on the maps σ and δ, we are able to establish analogous results for the strong rank conditions. If time permits we will also show analogous results for skew power series rings when it comes to the properties of being directly finite resp. stably finite. This talk is based on joint work with Karl Lorensen (Penn. State Uni., USA).
|
Jonathan Nilsson (Wednesday, 14:50–15:50)
Lie algebra representations and distinguished subalgebras A central idea in the representation theory of Lie algebras is to single out a distinguished subalgebra and study modules according to how this subalgebra acts. The classical example is the theory of weight modules, where a Cartan subalgebra acts diagonally, giving rise to familiar decompositions into weight spaces. In recent years, new families of modules have been explored where a distinguished subalgebra instead acts freely. This shift of perspective has uncovered novel simple modules and enriched the landscape of representation theory. In this talk, I will survey these developments and present my own contributions to the study of so-called U(h)-free modules, which can be seen as a natural counterpart to weight modules. |
Alfilfgen Sebandal (Wednesday, 16:00–17:00)
Leavitt path algebras: Representations and K-theory In the 1960s, W. Leavitt studied a class of universal algebras which do not have a well-defined dimension, later called the Leavitt algebras. In two simultaneous but independent studies by G. Abrams and G. Pino, and P. Ara et. al., an algebra arising from a directed graph has been introduced called the Leavitt path algebra which turned out to be the generalization of a certain type of Leavitt algebra. In this talk, we shall see the K_0 and K_0^{gr} groups of Leavitt path algebras extracted directly from the geometry of the graph and use them as classification tools for its representations. |
Jens Wittsten (Thursday, 10:00–11:00)
Bohr–Sommerfeld Rules for Matrix-Valued Systems
The Bohr–Sommerfeld quantization rules are among the earliest bridges between classical and quantum mechanics, linking closed classical trajectories in phase space to discrete quantum energy levels. Although the scalar case is well understood, important modern models in condensed matter physics lead naturally to matrix-valued Hamiltonians, where new challenges arise. In joint work with S. Becker and S. Fujiié, we establish a complete formulation of Bohr–Sommerfeld quantization for semiclassical self-adjoint 2 x 2 (matrix-valued) systems on the real line. We identify the geometric phase contributions that appear in this setting and give explicit formulas that make them transparent and clarify when they become quantized, i.e., take discrete values. I will take some time in the talk to explain what this means in accessible terms, beginning with an overview of Bohr–Sommerfeld rules before turning to our new results for matrix-valued systems. I will also illustrate our results with numerical comparisons to spectral computations in physically relevant models, which show strong agreement with the predictions of our Bohr–Sommerfeld rule. |
Markus Klintborg (Thursday, 11:20–12:20)
Generalized Harmonic Functions The idea of a generalized harmonic function is a loosely defined concept that has been applied across many different contexts. It commonly refers to a function that satisfies an equation which, in one way or another, resembles that of Laplace. A growing number of examples arising in the harmonic analysis of complex such functions point to recurring structural features; however, a more systematic framework is still lacking. We aim to establish common ground for these functions by imposing certain symmetry constraints on the ring of differential operators A2 = ⟨z, z¯, ∂, ∂¯⟩ of Weyl. This leads to a notion that retains the essential analytical aspects, and that allows us to consider such functions within the framework of non-commutative algebra. To lend a bit more substance to these abstractions, we also touch on the topic of Almansi decompositions, and discuss the cellular decomposition for polyharmonic functions. |
Eskil Rydhe (Thursday, 13:30–14:30)
Laplace–Carleson embeddings Embeddings of Laplace–Carleson type appear naturally in control theory, where they characterize a certain stability property for a certain class of systems. For suitable parameters, they also reflect classical results in analysis, such as the Hausdorff–Young inequality and the Carleson embedding theorem. For other choices of parameters, they form a natural substitute for the Hausdorff–Young inequality. In my talk, I will illustrate some of the mentioned connections, and discuss some results on the boundedness of these embeddings. |
Eric Ahlqvist (Thursday, 14:50–15:50)
Class field towers and Schur sigma-groups In this talk, I will present joint work in progress with Richard Pink on Schur sigma-groups and class field towers. The project originates from a classical problem in number theory: The failure of unique factorization in the ring of integers of a number field K is measured by the class group Cl(K). Fixing a prime p, class field theory gives rise to a canonical tower of unramified extensions of K — the Hilbert p-class field tower — and we consider its Galois group G. The unramified Fontaine–Mazur conjecture says that every p-adic representation of G factors through a finite quotient. When K is an imaginary quadratic field, this conjecture is known to hold except possibly when Cl(K)/p is generated by exactly two elements. In that case, the pro-p-group G can be generated by two elements and two relations, and admits an involution acting by inversion on both generators and relations; such a group is called a strong Schur sigma-group. The conjecture holds when the defining relations occur sufficiently high in the Zassenhaus filtration, so we focus on the case of Zassenhaus type (3,3), which remains open. We show if such groups are infinite, almost all of them violate the unramified Fontaine–Mazur conjecture. To gain further insight, we are currently working towards a classification of these groups. |
Anaëlle Pfister (Thursday, 16:00–17:00)
Minimal Kinematics on M_{0,n} Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. We characterize all choices of minimal kinematics on the moduli space M_{0,n} and find a bijection with 2-trees. We also compute the amplitude associated with minimal kinematics in terms of the two-trees. |
Marion Jeannin (Friday, 10:00–11:00)
Generalized exponential and integration in positive characteristic
In order to better understand a given physical system, one might often want to have a look at its group of symmetries. The Lie algebra of the latter (which is a Lie group) is nothing but the group of transformations of the system near the identity. In algebraic geometry, Lie groups are replaced by algebraic groups, while Lie algebras can still be roughly described as groups where all elements are infinitesimally close to the identity. In this context, the Lie algebra of a Lie group can be thought of as a first approximation of it. In characteristic 0, the exponential series is well defined and allows one to endow any finite-dimensional nilpotent k-Lie algebras with a group law on the vector group V(u), making it into a unipotent algebraic k-group. In other words there is an equivalence between the category of nilpotent k-Lie algebras of finite dimension and unipotent algebraic k-groups. On the other hand, the functor G → Lie(G) induces a quasi-inverse equivalence. If now k is of characteristic p > 0, such a nice conversation between (unipotent algebraic) groups and (nilpotent) Lie algebras no longer exists in general, but one can still wonder whether under suitable assumptions it is still possible to associate a unipotent algebraic group to a “nilpotent” (this notion will need to be adapted to the context) Lie algebra. Obstructions are of arithmetic, algebraic and geometric nature. |
Johan Björklund (Friday, 11:20–12:20)
Real algebraic knots and flexible (symplectic) knots in RP³ In this talk we will introduce real algebraic and flexible knots. Real algebraic knots are defined by polynomials and are more rigid objects than smooth knots. Just as smooth knots are considered up to isotopy, there is a notion of rigid isotopy for real algebraic knots. We will discuss the classification problem in this setting. Flexible knots are an attempt to capture the topological properties of real algebraic knots while retaining more flexibility leading to an easier classification problem. If time allows we will also discuss flexible symplectic knots. |
Christian Gottlieb (Friday, 13:30–14:20)
Why Artinian rings are Noetherian, and why not? I will tell you, why I don't think Artinian rings should be Noetherian. But in fact they are, so my intuition is misleading. However, even ideas which do not seem so fruitful, may have something in it. In this case the wrong intuition leads to a proof that Artinian rings ARE Noetherian. Moreover, we shall see which Artinian modules are Noetherian, and which Noetherian modules are Artinian. |
Anna Torstensson (Friday, 14:30–15:20)
Ways of describing polynomial subalgebras In this talk we will take a look at different ways to describe and work with polynomial subalgebras in both the univariate and multivariate settings. For a finitely generated subalgebra there is a method to compute a so called SAGBI basis. This is a new set of generators with a property that makes them suitable for computations such as checking if an element belongs to the subalgebra or not. The method is an algorithm in the univaraite case, but may run forever in the multivariate setting. If the subalgebra is of finite codimension in the polynomial ring we can find a finite SAGBI basis, but we can also use a finite number of conditions on the elements. For example, for A = K[x^3, x^4, x^5] the given generators are a SAGBI basis but A as also given by A = { f | f'(0) = f''(0) = f'''(0) = 0}. We will explore the nature of the latter descriptions and the possibility to generalised this method to the case of infinite codimension. |
Lionel Lang (Friday, 16:00–16:50)
Galois group of sparse polynomial systems In this talk, I want to report on recent progress in computing the Galois group of systems of polynomial equations with prescribed supports. Already in the case of two variables, we do not yet have a complete answer: for some supports, the Galois group is smaller than expected. Studying the Galois groups of simpler enumerative problems, we observed with Alex Esterov that a special attention should be paid to the product of the solutions to the system. This observation leads to a conjectural answer. If times permits, I would like to show that this conjecture is true for a simple instance of the problem. This is based on my collaboration with Alex Esterov. |
Victor Hildebrandsson (Friday, 17:00–17:30)
*-algebra structures on path algebras
A quiver is a directed graph allowing multiple arrows and loops, and its path algebra is the algebra generated by all possible paths in the quiver with multiplication defined by concatenation of paths. With motivation from noncommutative geometry, we want to know when there exists a *-algebra structure on path algebras. We consider quivers with anti-involutions, and show that a path algebra admits a *-algebra structure if and only if there exists an anti-involution on its underlying quiver. |
