Invariant Differential Forms on Complexes of Graphs and Feynman Integrals

We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential forms in question generate the stable real cohomology of the general linear group, as shown by Borel. By integrating such invariant forms over the space of metrics on a graph, we define canonical period integrals associated to graphs, which we prove are always finite and take the form of generalised Feynman integrals. Furthermore, canonical integrals can be used to detect the non-vanishing of homology classes in the commutative graph complex. This theory leads to insights about the structure of the cohomology of the commutative graph complex, and new connections between graph complexes, motivic Galois groups and quantum field theory.

Differential operators and Higher Specht polynomials

In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

Differential operators and reection group of type Dn

In this note, we study the actions of rational quantum Olshanetsky-Perelomov systems for finite reflections groups of type D n . we endowed the polynomial ring C[x 1,..., xn ] with a differential structure by using directly the action of the Weyl algebra associated with the ring C[x 1,..., xn ] W of invariant polynomials under the reflections groups W after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the reflections groups. We use the higher Specht polynomials associated with the representation of the reflections group W to exhibit the generators of its simple components.

Differential operators and reection group of type Bn

In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.

Differential-algebraic systems are generically controllable and stabilizable

We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. 10.1007/978-3-642-34928-7_1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.

Holonomic modules for rings of invariant differential operators

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals [Formula: see text].

Quantum gravity and Riemannian geometry on the fuzzy sphere

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $$[x_i,x_j]=2\imath \lambda _p \epsilon _{ijk}x_k$$ [ x i , x j ] = 2 ı λ p ϵ ijk x k modulo setting $$\sum _i x_i^2$$ ∑ i x i 2 to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric $$3 \times 3$$ 3 × 3 matrices g and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature $$ \frac{1}{2}(\mathrm{Tr}(g^2)-\frac{1}{2}\mathrm{Tr}(g)^2)/\det (g)$$ 1 2 ( Tr ( g 2 ) - 1 2 Tr ( g ) 2 ) / det ( g ) . As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure.

Invariant Differential Expression Analysis Reveals Mechanism of Cancer Resistance to Cell Cycle Inhibitors

Retinoblastoma (RB) is a good model to study drug resistance to cell-cycle inhibitors because it is driven by mutations in the core components of cell-cycle, i.e, Rb gene. However, there is limited gene expression dataset in RB which has major reproducibility issues. We have developed invariant differential expression analysis (iDEA) that improves the state of the art in differential expression analysis (DEA). iDEA uses strong Boolean implication relationships in a large diverse human dataset GSE119087 (n = 25,955) to filter the noisy differentially expressed genes (DEGs). iDEA was applied to RB datasets and a gene signature was computed that led to prediction and mechanism of drug sensitivity. The prediction was confirmed using drugs-sensitive/resistant RB cell-lines and mouse xenograft models using CDC25 inhibitor NSC663284. iDEA improved reproducibility of differential expression across diverse retina/RB cohorts and RB cell-lines with different drug sensitivity (Y79/Weri vs NCC). Pathway analysis revealed WNT/β-catenin involved in distinguishing drug sensitivity to CDC25 inhibitor NSC663284. NSC663284 inhibited tumour cell proliferation in mouse xenograft model containing Y79 cells indicating novel therapeutic option in RB. Invariant differentially expressed genes (iDEGs) are robustly associated with outcome in diverse cancer datasets and supports for a fundamental mechanism of drug resistance.