Mathematical Research Seminar - Archive

The spectrum of unbounded operators, typically acting in a complex Hilbert space, plays a vital role in diverse topics such as (i) mathematical formulation of quantum mechanics, (ii) the solution of Sturm-Liouville boundary differential equation, or (iii) asymptotic behaviour of 1-parametric strongly continuous operator semigroup.

We present our recent result on classification of additive spectrum preserving bijections on the set of unbounded operators. It turns out that every such map is nothing but a change of the domain. For important classes of Banach spaces, which include Hilbert and L_p spaces the same result is obtained even withohut injectivity.

The catalogues of such maps is handy at simplifications of problems which involve unbounded operaators and were asymptotics, say, is of vital importance.

This is a joint work with G. Dolinar, J. Marovt, and E. Poon.

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A circulant nut graph is a non-trivial simple graph such that its adjacency matrix is a circulant matrix whose null space is spanned by a single vector without zero elements. Regarding these graphs, the order–degree existence problem can be thought of as the mathematical problem of determining all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n. This problem was initiated by Bašić et al. [Art Discret. Appl. Math. 5(2) (2021) #P2.01] and the first major results were obtained by Damnjanović and Stevanović [Linear Algebra Appl. 633 (2022) 127–151], who proved that for each odd t ≥ 3 such that t ̸≡10 1 and t ̸≡18 15, there exists a 4t-regular circulant nut graph of order n for each even n ≥ 4t+4. Afterwards, Damnjanović [arXiv:2210.08334 (2022)] improved these results by showing that there necessarily exists a 4t-regular circulant nut graph of order n whenever t is odd, n is even, and n ≥ 4t + 4 holds, or whenever t is even, n is such that n ≡4 2, and n ≥ 4t + 6 holds. Finally, the aforementioned results were extended once again by Damnjanović, thus yielding a complete resolution of the circulant nut graph order–degree existence problem. In other words, all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n are now determined.

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When we consider the linear action of a finite group on a polynomial ring, an invariant is a polynomial unchanged by the action. A famous result of Noether states that in characteristic zero the maximal degree of a minimal generating invariant is bounded above by the order of the group. Our work establishes that the same bound holds for invariant skew polynomials in the exterior algebra. Our approach to the problem relies on a theorem of Derksen that connects invariant theory to the study of ideals of subspace arrangements. We reduce the problem to establishing a bound on the Castelnuovo-Mumford regularity of intersections of linear ideals in the exterior algebra, which we prove using tools from representation theory. We also examine another result from classical invariant theory, Weyl’s Polarization Theorem, and show that this result does not hold in the exterior algebra but we provide an alternative bound in this context.

Everyone is welcome and encouraged to attend!