Raj Dahya

Role: Doktorand
Advisor: Tanja Eisner
Affiliation: Universität Leipzig, Fakultät für Mathematik u. Informatik
Orchid: 0000-0002-4446-497X

🇩🇪 Kurzfassung auf deutsch: Wissenschaftlicher Mitarbeiter für die Fakultät für Mathematik und Informatik zw. 2020–3. Während dieser Jahre unterrichtete ich in Mathematik (lineare Algebra, Analysis [I] & [II], Mathematik für Physikstudierende [I] & [II]), Informatik (Logik), und Bioinformatik (Algorithmen & Datenstrukturen [I], [II], [I]). Dazu forschte ich in Operatorentheorie, v.a. über globale Eigenschaften von Halbgruppen sowie »Umkehrbarkeit«. Diese Forschung resultierte in diverse Veröffentlichungen (manche davon schon erfolgreich publiziert, siehe unten). Aus dieser Forschung verfasse ich gerade meine Doktorarbeit.

🇬🇧 Summary in English: Researcher at the Fakultät für Mathematik und Informatik, Universität Leipzig between 2020–3. During this period I taught in mathematics (linear algebra, analysis [I] & [II], mathematics for physicists [I] & [II]), computer science (logic), und bioinformatics (algorithms & data-structures [I], [II], [I]). Beyond this, the subject matter of my research is primarily from operator theory, viz. global properties of semigroups and notions of "reversibility". This research has resulted in various papers (some of which have been successfully published, see below). Based on this research, I am currently writing my phd thesis.

Papers

	Title	Rough summary	Status
1	The space of contractive $\mathcal{C}_{0}$-semigroups is a Baire space	We prove that the space of $1$-parameter contractive semigroups under the $\mathcal{k}_{\text{wot}}$-topology is a Choquet (and thus Baire) space. This relies on transfering properties of the dense subspace of unitary semigroups to the larger space. To achieve this transfer result, we use the Banach-Mazur infinite two-player game. This result is generalisable to other settings with the afore mentioned density property.	Published by J. Math. Analysis Appl. (Preprint available on arxiv.)
2	On the complete metrisability of spaces of contractive semigroups	We prove that the spaces of $d$-parameter contractive semigroups under the $\mathcal{k}_{\text{wot}}$-topology is Polish (in particular completely metrisable) for each $d \geq 1$. This relies on the automatic strong-continuity of weakly continuous semigroups, and this result is generalised to a large class of examples, including non-commutative non-discrete cases. The result is achieved via a Borel-complexity calculation.	Published by Archiv der Mathematik (Basel). (Preprint available on arxiv.)
3	Dilations of commuting $\mathcal{C}_{0}$-semigroups with bounded generators and the von Neumann polynomial inequality		Published by J. Math. Analysis Appl. (Preprint available on arxiv.)
4	Characterisations of dilations via approximants, expectations, and functional calculi		Published by J. Math. Analysis Appl. (Preprint available on arxiv.)
5	Interpolation and non-dilatable families of $\mathcal{C}_{0}$-semigroups		Published by Banach J. Math. Analysis (Preprint available on arxiv.)

Interests (restricted to current research)

Put short, my research revolves around semigroup theory. However, I am not a classical semigroup-theorist. I am interested in sufficiently natural generalisations of semigroups. The goal of my research is to understand what happens for all (or "almost all" under suitable topological frameworks) semigroups of a particular kind.

Whilst my research does not revolve around studying properties of generators or convergence rates of SPDEs, etc. my research exploits classical results (e.g. the Hille-Yosida theorem, the Trotter-Kato theorem, etc.) and generalises others (e.g. the strong continuity of weakly continuous semigroups).

The key components of my research are as follows:

"classical" dynamical systems: contractive $\mathcal{C}_{0}$-semigroups on Hilbert spaces over time domains which we can desribe via (locally compact Polish) topological monoids, $M$, instead of just $[0,\:\infty)$.
Mult-parameter semigroups: A concrete subclass, which is nonetheless more general than $M=[0,\:\infty)$, is $M=\mathbb{R}^{d}$ for $d \in \{1,2,3,\ldots\}$. This setting corresponds (both algebraically and topologically) to the context of $d$-parameter contractive semigroups (or equivalently: commuting families of $d$ contractive semigroups). We can refer to this as the continuous setting. There is of course the discrete analogue, viz. under the choice of $M=\mathbb{N}_{0}^{d}$. (This corresponds to commuting families of $d$ contractions.)
Dilations: There are a plethora of diverse notions of dilation. They all share a common idea: embedding the system into a larger one which enjoys "nicer" properties. In our classical setting I deal with two notions of dilation 1) the notion of unitary dilations à la Sz.-Nagy and 2) the notion of regular unitary dilations attributed to Brehmer.
Interpolation: In our latest results, we utilise an interpolation-technique due to Bhat and Skeide (developed for isometric systems on $C^{\ast}$-modules) and generalise this to the contractive multi-parameter setting. Roughly put, given a(n arbitrarily large!) family of commuting contractions, there is a (tensor-product) enlargement of the system and a continuous family of contractive semigroups, which "goes through" the contractions at discrete time points. This is related to (but does not solve) the embedding problem.
Approximations/residuality results (in the Baire-category sense): We work with a certain "weak" topology for operator-valued function spaces (more precisely, this topology is equivalent to a certain compact-open topology). Under this topology a raft of residuality results was developed by Eisner, et al. for $1$-parameter unitary/isometric semigroups. I show that these can be extended to the contractive case. In the $d$-parameter case, my research shows that this continues to be possible for $d=2$, but not for $d \geq 3$. The results occur in a very strong 0-1 dichotomy: either all contractive semigroups are reversible (holds for $d\in\{1,2\}$) or "almost none" (in a Baire-category sense) are (holds for all $d \geq 3$).

Interests (beyond current research)

My passion can be summarisd in 3 words: (non-conventional) computational paradigms. There are various algorithms/models based on or inspired by natural systems, e.g. Monte Carlo Tree Search, or image segmenatation inspired by diffusion, or Generative Adversarial (neural) Networks.

Currently I am particularly gripped by quantum computing: from the physical theories (the ontological descriptions), to the mathematics (of course!) incl. the presentation due to Kraus et al. of quantum processes via completely positive operators on "states" (dually: on self-adjoint observables), to the various discrete-mathematical algorithms, through to implementation e.g. via IBM's qiskit (see e.g. an implementation on one of my public git repositories).

What is of particular interest for me is that there is a simple axiomatic definition of the processes underlying quantum computations, dubbed quantum operations, which we can ascribe to Kraus. These can described in a very "building-block"-like fashion as is common in mathematical logic. The quantum operations are given by the largest class $$ Q \subseteq \{ (\mathcal{H}, \Phi) \mid \mathcal{H}~\text{Hilbert sp.}, \:\Phi \colon L^{1}(\mathcal{H}) \to L^{1}(\mathcal{H}) ~\text{linear} \} $$ satisfying:

(restriction₁: positivity) for each $(\mathcal{H},\Phi) \in Q$ it must hold that $\Phi\geq\mathbf{0}$;
(restriction₂: tracial-contractivity) for each $(\mathcal{H},\Phi) \in Q$ it must hold that $\mathrm{tr}(\Phi(\rho))\leq\mathrm{tr}(\rho)$ for $\rho\in L^{1}(\mathcal{H})^{+}$.
(basic elements) $(\mathcal{H},\mathrm{id}_{\mathfrak{L}(\mathcal{H})}) \in Q$ for all Hilbert spaces $\mathcal{H}$;
(closure under tensor-products) $ (\mathcal{H}_{1},\Phi_{1}), (\mathcal{H}_{2},\Phi_{2}) \in Q \Longrightarrow (\mathcal{H}_{1} \otimes \mathcal{H}_{2}, \Phi_{1} \otimes \Phi_{2}) \in Q $;

One can show that this is equivalent to Kraus’s simpler definition, viz. $\Phi$ is a quantum operation if and only if it is tracially-contractive and completely positive, i.e. $$ \forall{n\in\mathbb{N}:~} \Phi\otimes\mathrm{id}_{M_{n}(\mathbb{C})} \geq \mathbf{0}, $$ which is a slight weakening of the above axioms. From Kraus’s (or equivalently: the above) axiomatisation one obtains one of the most beautiful representation theorems in mathematics (or rather mathematical physics):

Theorem (Kraus). Let $\mathcal{H}$ be a Hilbert space, and $\Phi \colon L^{1}(\mathcal{H}) \to L^{1}(\mathcal{H})$ be linear. Then $\Phi$ is a quantum operation if and only if it can be expressed as $$ \Phi(s) = \mathrm{tr}_{2}( \mathrm{ad}_{(I \otimes P)} \:\mathrm{ad}_{U} \:(s \otimes \omega) ) $$ for each $s\in L^{1}(\mathcal{H})$, where, for some auxiliary space $\tilde{\mathcal{H}}$, $U \in \mathfrak{L}(\mathcal{H} \otimes \tilde{\mathcal{H}})$ is a unitary operator, $P \in \mathfrak{L}(\tilde{\mathcal{H}})$ a projection, and $\omega$ can be taken to be any fixed pure state on $\tilde{\mathcal{H}}$. Note, the operation $\mathrm{ad}_{W}$ denotes the adjoint action $\mathrm{ad}_{W}S = WSW^{\ast}$.

This representation theorem allows for physical interpretation of the process underlying an arbitrary quantum operation $\Phi$: Firstly, the system can be enlarged to a closed system, whereby the state is coupled to a pure state on an auxiliary system in an at first separable (thus non-entangled) fashion. Next, scattering effects are applied, as described by the unitary evolution $U$. This may leads to an overall entangled state of the combined system. Finally, the auxiliary system is measured out, whereby this measurement has the effect (as Kraus aptly refers to the role observables) described by the projection $I \otimes P$.

There are generalisations of Kraus’s result to parameterised operations, e.g. by Davies. What is common in all of these results is a notion of dilation to a closed system (i.e. describable via a unitary evolution) on an enlargement of the given system via an auxiliary space. These themes occur in the classical setting, esp. when considering the interpolation result of Bhat and Skeide (in the non-classical context) and our generalisation thereof (in the classical setting). There are thus some overlaps between my general interests, and current research project.