# Posters

(preliminary list)

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This poster is based on joint work with A. Sitarz [1]. For a complex parameter \( q \), the algebra \( \mathcal{A}(SU_q(2)) \) was shown to be a braided Hopf algebra in some braided tensor category [2]. Our original motivation was to construct braided differential calculi on this algebra. We show that there is a class of Hopf algebras for which, using the general framework of twisting, one can construct their braided versions. In particular, the Hopf algebras of \( SU_q(n) \) and the quantum double torus belong to this class. Furthermore, the twisting can also be applied to the (co-,bi-)modules over the braidable Hopf algebras, and permits us to consider braided differential calculi.

[1] A. Bochniak, A. Sitarz. *Braided Hopf algebras from twisting*. arXiv:1701.01106.

[2] P. Kasprzak, R. Meyer, S. Roy, S.L. Woronowicz. *Braided quantum \( SU(2) \) groups*, J. Noncommut. Geom. 10, p. 1161-1125 (2016)

Consider two symbols \( V,U \) which satisfy the commutation relation \( UV-qVU=hV^s \). These relations appear in quantum mechanics, combinatorics and non-commutative geometry. For instance, when \( q=h=1,s=0 \), the symbols generate the Heisenberg-Weyl algebra of boson operators and when \( h=1,s=2 \), the symbols generate the so-called Meromorphic Weyl algebra or Jordan plane. In this research, the coefficients that arise from the normal ordering of strings in the algebra generated by \( V,U \) (modulo the ideal defined by the commutation relation) are studied using two approaches. First, we continue the study by Varvak [1] where the coefficients were interpreted as \( q \)-rook numbers under the row creation rook model introduced by Goldman and Haglund [2]. Second, we express the coefficients in terms of a kind of generalization of some symmetric functions.

[1] A. Varvak, Rook numbers and the normal ordering problem, *J. Combin. Theory Ser. A* **112** (2005), 292–307.

[2] J. Goldman and J. Haglund, Generalized rook polynomials, *J. Combin. Theory Ser. A* **91** (2000), 509–530.

Poincaré duality is a fundamental result in differential geometry that allows one to understand the cohomology of orientable compact manifolds more easily. An analogue of it exists in \( K \)-theory, where instead of requiring orientability, one requires that the manifold possess a Spin\( ^c \) structure. We first review these notions and illustrate the importance of Poincaré duality, before explaining some recent work [1], joint with Prof. Mathai Varghese and Dr. Hang Wang, that extends the \( K \)-theoretic Poincaré duality to the equivariant setting for certain types of group actions, building on an earlier result of Kasparov [2].

[1] H. Guo, V. Mathai and H. Wang, *Spin\( ^c \), \( K \)-homology and proper actions.* Submitted. arxiv.org/abs/1609.01404

[2] G. G. Kasparov, *Equivariant \( KK \)-theory and the Novikov conjecture*. Invent. Math. **91** (1988), no. 1, 147-201.

In this poster we shall present a detailed account on the pseudodifferential calculus on noncommutative tori [1]. This calculus turns out to be very useful (see, e.g., [2]), but its analytic aspect hasn't been studied well in any literature. The aim of this poster is to fulfill this deficiency. This is joint work with Hyun-su Ha and Raphaël Ponge.

[1] A. Connes, *C*-algèbres et géométrie différentielle*. C.R. Acad. Sc. Paris, t. 290, Série A, 599-604, 1980.

[2] A. Connes, P. Tretkoff, *The Gauss-Bonnet theorem for the noncommutative two torus. Noncommutative geometry, arithmetic, and related topics*, pp. 141–158, Johns Hopkins Univ. Press, Baltimore, MD, 2011.

Einstein's causality is one of the fundamental principles underlying modern physical theories. Whereas it is readily implemented in classical physics founded on Lorentzian geometry, its status in quantum theory has long been controversial. On the mathematical side, the classical causal structure is a certain binary relation on the set events, i.e. points of spacetime. But in the quantum world the very notion of a point-like event becomes blurred and it is widely believed that the causal structure would also be afflicted with uncertainty.

However, together with my colleagues M. Eckstein and N. Franco, we have developed a rigorous framework, in which the causal structure remains rigid, although the events themselves become nonlocal. The idea, founded on the operational viewpoint on physics compelling us to exchange the classical spacetime for the space of states of some abstract algebra of observables, is incarnated with a Lorentzian version of noncommutative geometry à la Connes. The explanation of the basic concepts of our formalism will be followed by an illustrative example of an almost-commutative spacetime, which, moreover, sheds a new light on the peculiar properties of massive fermions.

[1] N. Franco and M. Eckstein, An algebraic formulation of causality for noncommutative geometry. *Classical Quant Grav*, 30(13):135007, 2013.

[2] M. Eckstein and T. Miller, Causality for nonlocal phenomena. *Ann. Henri Poincaré*. Published online 13 March 2017.

[3] M. Eckstein, N. Franco, and T. Miller, Noncommutative geometry of Zitterbewegung. *Phys Rev D*, 95:061701, 2017.

In this work, join and meet algebraic structure which exists in non-near-linear finite geometry is discussed. Lines in non-near-linear finite geometry were decomposed as products of lines in near-linear finite geometry and the existence of lattice between any pair of near-linear finite geometry \( \mathcal{Z}_{\mathfrak{p}_i} \) for \( \mathfrak{q} \) a divisor of \( \mathbf{d} \) and between a subsystems of a finite quantum systems with variables in \( \mathcal{Z}_\mathbf{d} \) and its subsystems with variables in \( \mathcal{Z}_{\mathfrak{p}_i} \) was confirmed. A duality between floor and roof in a phase space of a finite quantum system is shown using examples. A bijection between the set of subgeometry and finite geometry from the subsets of the set of divisors of \( \mathbf{d} \) and set of subsystems of a finite quantum system and a finite system from the subsets of the set of divisors of \( \mathbf{d} \) is established.

[1] A. Vourdas, Rep. Prog. Phys. 67, 1 (2004)

[2] M. Shalaby, A. Vourdas, J. Phys. A45, 052001 (2012)

[3] S.O. Oladejo,C.Lei, and A. Vourdas, J. Phys. A: Math. Theor. 47 485204 (2014)

Vilenkin groups are a natural generalisation of the Walsh group, and have been extensively studied in the context of classical harmonic analysis. We consider the Vilenkin-Fourier transformation acting upon noncommutative \( L^p \) spaces associated to the hyperfinite type \( II_1 \) factor \( R \), and study weak type inequalities for the partial summation operators.

The structure of the Vilenkin groups induces a natural filtration on the factor \( R \), and so we also investigate noncommutative martingale spaces corresponding to this filtration.