• Abigail Croker Poster (pdf)

• Jarmo Kikstra Poster (pdf)

• Jingwei Xie Poster (pdf)

Fabrication and kinetic characterisation of metal oxide heterojunctions for photoelectrochemical water splitting

Photoelectrochemical water splitting utilises solar energy directly to split water into hydrogen and oxygen via two half-reactions. This method of green hydrogen generation could play a crucial role in the global transition to zero-carbon energy. The water oxidation half-reaction at the photoanode has received much focus in recent years, owing to its challenging kinetics.

Metal oxides have many properties that make them suitable candidates for photoanode materials, and by combining materials in the form of a heterojunction it is possible to benefit from complementary properties. Transient spectroscopic techniques can be used to probe the charge carriers within heterojunctions, and provide insight into the function of each material. Such understanding is critical for device optimisation.

Bismuth vanadate (BVO) is a promising photoanode material. My research thus far has involved investigating the effect of electrolyte pH on BVO stability, and the role of an interlayer in enhancing BVO/co-catalyst heterojunction photoelectrochemical performance.

• View Louise's research poster (PDF)
• F‌ind out more about the Durrant's group research
• Find out more about Louise's experience as a Schrödinger scholar

This project will develop a new approach to using renewable carbon sources in chemical manufacture and is of direct relevance to carbon capture and utilisation. We aim to explore cooperative action between both earth-abundant and transition metals to construct carbon chains directly from CO and CO₂. These chains can be used as building blocks to prepare complex organic molecules, useful building blocks in pharmaceutical and chemical synthesis. This work is of direct importance to a future sustainable energy economy and will provide understanding toward long-term CO2 remediation strategies. Further, this approach will compliment industrially relevant processes that use CO to make low-value fuels (Fischer-Tropsch process) or CO2 to make polymers (lactide or CO₂/epoxide polymerization).

Recent publications

Read a recent article from Joe and the Crimmin group: 

Magnesium-Stabilised Transition Metal Formyl Complexes: Structures, Bonding, and Ethenediolate Formation

Abstract: Herein we report the first comprehensive series of crystallographically characterised transition metal formyl complexes. In these complexes, the formyl ligand is trapped as part of a chelating structure between a transition metal (Cr, Mn, Fe, Co, Rh, W, and Ir) and a magnesium (Mg) cation. Calculations suggest that this bonding mode results in significant oxycarbene-character of the formyl ligand. Electron-rich late-transition metal complexes have the highest oxycarbene-character to the bonding and are the most stable in solution. Further reaction of a heterometallic Cr---Mg formyl complex results in a rare example of C–C coupling and formation of an ethenediolate complex. These results show that well-defined transition metal formyl complexes are potential intermediates in the homologation of carbon monoxide.

Find out more

• Find out more about Joe's experience as a Schrödinger scholar
• Visit the Crimmin Group website
• Follow Joe on Twitter

Beetle Evolution and Climate Change

Studying the past can give us an insight into the current environmental crisis. I will reconstruct the evolutionary history of beetles and compare this with climate records since the Mesozoic Era. I will explore which lineages of beetles had higher extinction rates during periods of climate change, and if characteristics such as body size, habitat or diet affected survival. In addition, geographic variables such as latitude, range size and dispersal ability will be interesting considerations. I aim to use this data to predict the impact of future climate change. We may be able to identify potentially vulnerable lineages and regions without carrying out population surveys. This a new and exciting area of research which has the potential both to inform future conservation work and to strengthen the argument for urgent climate action.

• View Aileen's research poster (PDF)
• Find out more about Aileen's experience as a Schrödinger scholar

Why are we not just blobs of cells? Legendary mathematician Alan Turing thought he might know the answer: Our cells are governed by simple rulesthat lead to complex patterning. These ‘Turing patterns’ determine the shape of our brains, the spacing between our fingers, and the intricate patterns on fish skin. Engineering thesepatterns from the ground up couldlead to many medical advances –including self-healing tissuesand engineering of organoids, as well as complex biomaterials.Mathematical theory can facilitate this engineering effort by guiding biological designs into more favourable directions. In my PhD project, I thereforetake an interdisciplinary approach, tightly integrating mathematical theory with synthetic biology to engineer synthetic Turing patterns.

• View Georg's research poster (PDF)
• Find out more about Georg's experience as a Schrödinger scholar

Coercive inequalities in noncommutative analysis

Noncommutativity is a phenomenon that has a significant impact on mathematics and physics. It was realised during the development of quantum mechanics that the noncommutative framework is necessary to understand the events at a quantum scale. The Hörmander theorem, which proves the hypoelliptic generator associated with the system of partial differential equations under a rank condition. In this project, we want to extend the existing classical theory for the quantum setting. The generator that satisfies this rank condition, Hypoellipticity often occurs with Hypocoercivity. Hypocoercivity was introduced by Cédric Villani in 2009. The theorems and estimates of exponential convergence to equilibrium associated with such systems have been studied. We want to prove these hypocoercive estimates in a noncommutative framework.

Furthermore, we would prove the associated coercive inequalities like the Poincare and Logarithmic Sobolev inequalities.

• View Shreya's research poster (PDF)
• Find out more about Shreya's experience as a Schrödinger scholar

A Leland Model for Delta-Hedging in Central Risk Books

In an effort to consolidate related trading activities, banks and hedge funds promote the usage of central risk books, in which trading exposures can be managed as a whole. This brings along two benefits: reduction in the amount of delta-hedging needed and reduction in the cost of each delta-hedge with limit orders. Both lead to more competitive prices being quoted on the option. This work revisits the Leland'85 delta-hedging strategy in the context of the second benefit. We propose an actionable option hedging strategy using both market and limit orders and derive a nonlinear partial differential equation describing the corresponding option price.

• View Zexin's reasearch poster (PDF)
• Find out more about Zexin's experience as a Schrödinger scholar

Formalising the Gaga theorem

This project is to formalise the famous GAGA theorem by Jean-Pierre Serre in his Géométrie Algébrique et Géométrie Analytique [8]. The process of formalising mathematics is to use an interactive theorem prover to verify mathematical proofs. In this project, we will be using Lean.

Why formalising?

The first reason is to validate mathematical proofs. Undoubtedly, mathematics is hard, so hard that sometimes even top experts cannot agree what the correct answer should be. For example, the two articles in Annals of mathematics [4, 7] give two different answers to the same question. Because we couldn't perform experiments to validate mathematical proofs and mathematicians make mistakes, so we need other tools to help us check mathematics. The second reason is that, by formalising more and more mathematics, we will build a larger and larger database of mathematical proofs which could potentially be utilised by modern machine learning algorithms so that machine not only can verify mathematical proofs, it can prove theorems on its own, see this paper [6] and the IMO grand challenge where AI will attempt to solve IMO problem on its own [1].

Why GAGA?

Very roughly speaking, algebraic geometry "uses only polynomials to cut shapes" while analytic geometry "uses holomorphic functions to cut shapes" and the two geometries have very different topology, one has the Zariski topology while the other has the analytic topology. Intuitively, a statement in analytic geometry might not be true in algebraic geometry. However, the two geometries are still closely related, prior to GAGA theorem, Chow's theorem states that analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety, i.e. closed in Zariski topology as well [3]. The GAGA theorem is a stronger theorem from which Chow's theorem immediately follows. The GAGA theorem very roughly states that under some conditions, the two geometries are "equivalent". Besides the interesting contents of GAGA, its importance lies in the fact that it is used in the proof of Fermat Last Theorem. So if one wants to formalise Fermat Last theorem, one need to unavoidably formalise the GAGA theorem.

• View Jujian's research poster (PDF)
• Find out more about Jujian's experience as a Schrödinger scholar

References

1. Imo grand challenge, 2020.
2. Jeremy Avigad, Leonardo de Moura, and Soonho Kong. Theorem proving in lean, 2015.
3. Wei-Liang Chow. On compact complex analytic varieties. American Journal of Mathematics, 71(4):893-914, 1949.
4. János Kollár. Non-quasi-projective moduli spaces. Annals of mathematics, pages 1077-1096, 2006.
5. Amnon Neeman. Algebraic and analytic geometry. Number 345. Cambridge University Press, 2007.
6. Stanislas Polu and Ilya Sutskever. Generative language modeling for automated theorem proving. arXiv preprint arXiv:2009.03393, 2020.
7. Georg Schumacher and Hajime Tsuji. Quasi-projectivity of moduli spaces of polarized varieties. Annals of mathematics, pages 597-639, 2004.
8. Jean-Pierre Serre. Géométrie Algébrique et Géométrie Analytique. In Annales de l'institut Fourier, volume 6, pages 1-42, 1956.

Gravitational wave polarisations measurements using detector networks

Gravitational waves are perturbations of the spacetime due to several astrophysical and cosmological events, such as the close encounter of two black holes or neutron stars. Through gravitational wave observations we can learn several important features of the physics of stars, compact objects and the history of the universe. My PhD program is aimed to understand in detail which are the properties of some astrophysical and cosmological models which can be probed with next generation gravitational wave experiments, producing forecasts on the observative power of present and planned missions, with a focus on ground based interferometers, very powerful instruments which can make very accurate measurements of gravitational wave properties.

• View Giorgio's research poster (PDF)
• Find out more about Giorgio's experience as a Schrödinger scholar

The accelerating expansion of our universe has seen a few evidences for decades, however, its identity is still mysterious. One of the candidates is some unknown scalar fields. However, this model predicts an additional force that has not been observed in precise test with macroscopic objects (e.g. monitoring lunar orbits). Thus, chameleon screening mechanism has been proposed to hide this dark energy force in macroscopic objects.

Recently, quantum technology provides a precise tool to measure acceleration using the wave nature of atoms. Given an atom has such a small microscopic size, it won’t be able to hide any chameleon screened forces. Thus, atom interferometry becomes a powerful probe for this dark energy model. My work is focusing on improving the equipment to obtain more atoms and reduce background noises. Hopefully, this upgraded atomic physics experiment of better signal-to-noise ratio can cast some light about this cosmological mystery.

• View Guancheng's research poster (PDF)
• Find out more about Guanchen's experience as a Schrödinger scholar