# Dr Cesare Tronci

### Biography

I obtained a Laurea in Nuclear Engineering in May 2004 at the Politecnico di Torino (Italy). Then, after spending two years (06/2003 - 05/2005) at CERN (Switzerland) working on microwave electronics under the direction of Ugo Amaldi (also at TU München and TERA Foundation), I moved to the Theoretical Division (in the former Plasma Theory Group) of the Los Alamos National Lab (LANL, USA), where I visited Giovanni Lapenta (now at KU Leuven, Belgium) for several months.

In 10/2005, I entered a PhD programme in Applied Mathematics at Imperial College, under the direction of Darryl Holm. Between 2006 and 2007, I also spent two summers at LANL, working with Bruce Carlsten in the former International, Space & Response Division (High Power Electrodynamics Group). In 09/2008, I joined the Mathematics Section of EPF Lausanne (Switzerland) as a research assistant in Tudor Ratiu's group (Geometric Analysis). In 01/2012, I was appointed as a Lecturer (Assistant Professor) at Surrey, where I became Reader (Associate Professor) in 08/2018. In 06/2020, I was also appointed Adjunct Professor at Tulane University (USA) within the Department of Physics & Engineering Physics.

Over the last 8 years, I was an invited participant fellow at several research programs at the Fields Institute (Toronto, CA), the Newton Institute (Cambridge, UK), the Bernoulli Center (Lausanne, CH), and the MSRI (Berkeley, USA). In 2019, I was awarded an Alexander von Humboldt fellowship and spent a year working at the Max Planck Institute for Plasma Physics in Garching bei München (Germany).

### Research

### Research interests

Over the years, I have worked with engineers, physicists and mathematicians and shared with them the excitement of doing research in each of the corresponding fields. I have developed an understanding of the needs and the challenges in these disciplines and this experience is for me a continuous source of inspiration. This diversified path has led me to the discovery of mathematical research and its interconnections with other pure and applied sciences. Then, applied mathematics emerges not only as a discipline that applies mathematical concepts to other fields, but also as an area in which other fields of science serve as a continuous inspiration to develop new exciting directions in mathematical research - this two-way vision is precisely where my research stands. Over the last ten years, I have been working on geometric approaches to the formulation of nonlinear multi-physics models systems with multiple scales. Applications include complex fluids, plasma physics and quantum molecular dynamics.

*Specific interests include:*Momentum maps and reduction by symmetry; applications of symplectic and Poisson geometry; Hamiltonian and Lagrangian techniques in nonlinear dynamics; multi-physics modelling of nonlinear multiscale dynamics; nonlinear modelling of magnetized plasmas; liquid crystals and fluids with internal structure; nonlocal dynamics of aggregation and self-assembly; phase-space methods in classical and quantum mechanics; mixed classical-quantum dynamics and applications in chemistry.

### Research collaborations

*International collaboration network:*Within a 4-year Leverhulme Research Project, I am leading an international collaboration composed of Emanuele Tassi at the Centre of Theoretical Physics in Marseille (France), Giovanni Lapenta at the Centre for Plasma Astrophysics of KU Leuven (Belgium), and Philip J. Morrison at the Institute of Fusion Studies at UT Austin (USA). Within the same project, I also collaborate with Alain J. Brizard (St. Michael College of Vermont, USA), Enrico Camporeale (CWI Amsterdam, The Netherlands), and Maria Elena Innocenti (KU Leuven, Belgium).

*National collaboration network:*Within a London Mathematical Society Scheme 3 award, I am the coordinator of a national Network in Applied Geometric Mechanics involving three groups working on core topics in geometric mechanics: geometric quantum dynamics (Brunel University), geometric imaging science (Imperial College), and geometric fluid dynamics (University of Surrey).

*Other collaborations:*Over the last 10 years, I have worked with several (mainly international) collaborators on different mathematical problems, with special emphasis on applications. Recent collaborators include: Joshua W. Burby (Courant Institute - NYU, USA), Bin Cheng (U. Surrey, UK), François Gay-Balmaz (ENS-CNRS Paris, France), Darryl D. Holm (Imperial College, UK), Tomoki Ohsawa (UT Dallas, USA), Tudor S. Ratiu (EPF Lausanne, Switzerland), Vakhtang Putkaradze (U. Alberta, Canada), Endre Süli (Oxford, UK), Cornelia Vizman (U. West Timisoara, Romania).

### Research grants

Leverhulme Research Project Grant: In 2014, I was awarded the Leverhulme Research Grant RPG2014-112 “From geometry to kinetic-fluid systems (and back)”. Total budget: £252K. Duration: 4 years (started 7/1/15). This grant involves two 2-years research assistants and three international collaborations in Europe and USA.

Small awards: Over the last five years, I have been awarded a series of small grants (<£3K) by the London Mathematical Society, the Institute of Mathematics and Applications and the internal Faculty Research Support Fund at the University of Surrey.

### Selected invited talks

*Multi-physics models for hybrid kinetic-fluid and classical-quantum systems*, Hamiltonian Systems, from Topology to Applications through Analysis I, 8-12 October 2018, Mathematical Sciences Research Institute, Berkeley CA, USA

*Symmetry methods for quantum variational principles and expectation value dynamics*, PACM Colloquium, Princeton University, November 6 2017, Princeton NJ, USA

*Variational formulations of low frequency kinetic-MHD in the current-coupling scheme*, Mini-conference on "New Developments in Algorithms and Verification of Gyro Kinetic Simulations", 58th Annual Meeting of the APS Division of Plasma Physics, October 31 - November 4, 2016, San Jose CA, USA. (Solicited talk)

*Modeling efforts in hybrid kinetic-MHD and fully kinetic theories*, Princeton Plasma Physics Laboratory, October 27 2016, Princeton NJ, USA

*Multiphysics models for hybrid kinetic-fluid systems*, Courant Institute of Mathematical Sciences, New York University, April 7 2016, New York NY, USA

*Classical-quantum variational principles*, Classic and Stochastic Geometric Mechanics Workshop, June 8-11, 2015, École Polytechnique Fédérale de Lausanne, Switzerland

*Relabeling symmetry in fluid dynamics*, Geometry and Fluids, April 7-11 2014, Clay Mathematics Institute, Oxford, UK

*Hydrodynamic vorticity and helicity of conservative liquid crystal flows*, Isaac Newton Institute for Mathematical Sciences, June 11 2103, Cambridge, UK

*Geometry and symmetry in multi-physics models for magnetized plasmas*, Fields Institute for Research in Mathematical Sciences, July 9 2012, Toronto, Canada

### My teaching

Teaching portfolio: *Linear Algebra & Vector Calculus (MAT1037), Geometric Mechanics* (MATM032), *Geometric Mechanics* (MAGIC080)

Summary: Since I joined the University of Surrey, I introduced a new 4th year module (thought 6 times) in Geometric Mechanics. The module introduces symmetry methods in mechanical problems for different applications, from rigid body dynamics to quantum mechanics. I introduced a new partial assessment method that requires students to give presentations to the rest of the class. Also, I have thought (5 times) an online mini-course (10 hours) in Geometric Mechanics (with more advanced content) to PhD students across the UK, within the Mathematics Access Grid group of universities. Since 2016, I have also been teaching a first-year module in Linear Algebra & Vector Calculus (including a small part on MATLAB).

### My publications

### Publications

magnetohydrodynamics (MHD) part that describes a bulk fluid component and a

Vlasov kinetic theory part that describes an energetic plasma component. While

most hybrid models in the plasma literature are non-Hamiltonian, this paper

investigates a recent Hamiltonian variant in its two-dimensional configuration.

The corresponding Hamiltonian structure is described along with its Casimir

invariants. Then, the energy-Casimir method is used to derive explicit

sufficient stability conditions, which imply a stable spectrum and suggest

nonlinear stability.

describing the interaction of a bulk fluid plasma obeying MHD and an energetic

component obeying a kinetic theory. Upon using the Vlasov kinetic theory for

energetic particles, two planar Vlasov-MHD models are compared in terms of

their stability properties. This is made possible by the Hamiltonian structures

underlying the considered hybrid systems, whose infinite number of invariants

makes the energy-Casimir method effective for determining stability.

Equilibrium equations for the models are obtained from a variational principle

and in particular a generalized hybrid Grad-Shafranov equation follows for one

of the considered models. The stability conditions are then derived and

discussed with particular emphasis on kinetic particle effects on classical MHD

stability.

(governed by a kinetic theory) interacts with a fluid bulk (governed by MHD).

Different nonlinear coupling schemes are reviewed, including the

pressure-coupling scheme (PCS) used in modern hybrid simulations. This latter

scheme suffers from being non-Hamiltonian and to not exactly conserve total

energy. Upon adopting the Vlasov description for the hot component, the

non-Hamiltonian PCS and a Hamiltonian variant are compared. Special emphasis is

given to the linear stability of Alfv\'en waves, for which it is shown that a

spurious instability appears at high frequency in the non-Hamiltonian version.

This instability is removed in the Hamiltonian version.

oriented particles, for example, moving magnetized particles. This is achieved

by introducing a double bracket dissipation in kinetic equations using an

oriented Poisson bracket, and employing the moment method to derive continuum

equations for magnetization and density evolution. We show how our continuum

equations generalize the Debye-Hueckel equations for attracting round

particles, and Landau-Lifshitz-Gilbert equations for spin waves in magnetized

media. We also show formation of singular solutions that are clumps of aligned

particles (orientons) starting from random initial conditions. Finally, we

extend our theory to the dissipative motion of self-interacting curves.

Ericksen-Leslie director theory and the Eringen micropolar approach. Up to this

day, these two descriptions have remained distinct in spite of several attempts

to show that the micropolar theory comprises the director theory. In this paper

we show that this is the case by using symmetry reduction techniques and

introducing a new system that is equivalent to the Ericksen-Leslie equations

and includes disclination dynamics. The resulting equations of motion are

verified to be completely equivalent, although one of the two different

reductions offers the possibility of accounting for orientational defects.

After applying these two approaches to the ordered micropolar theory of

Lhuiller and Rey, all the results are eventually extended to flowing complex

fluids, such as nematic liquid crystals.

reconnection. The formulation relies on fundamental principles in Lagrangian

dynamics, in which the inertia of the electron mean flow is neglected in the

expression of the Lagrangian, rather then enforcing a zero electron mass in the

equations of motion. This is done upon splitting the electron velocity into its

mean and fluctuating parts, so that the latter naturally produce the

corresponding pressure tensor. The model exhibits a new Coriolis force term,

which emerges from a change of frame in the electron dynamics. Then, if the

electron heat flux is neglected, the strong electron magnetization limit yields

a hybrid model, in which the electron pressure tensor is frozen into the

electron mean velocity.

Maxwell-Vlasov system. This limit produces a neutral Vlasov system that

captures essential features of plasma dynamics, while neglecting radiation

effects. Euler-Poincar\'e reduction theory is used to show that the neutral

Vlasov kinetic theory possesses a variational formulation in both Lagrangian

and Eulerian coordinates. By construction, the model recovers all collisionless

neutral models employed in plasma simulations. Then, comparisons between the

neutral Vlasov system and hybrid kinetic-fluid models are presented in the

linear regime.

Q-tensor dynamics is compared to the Volovik & Kats (VK) theory of biaxial

nematics by using Hamilton's variational principle. Under the assumption of

rotational dynamics for the Q-tensor, the variational principles underling the

two theories are equivalent and the conservative VK theory emerges as a

specialization of the QS model. Also, after presenting a micropolar variant of

the VK model, Rayleigh dissipation is included in the treatment. Finally, the

treatment is extended to account for nontrivial eigenvalue dynamics in the VK

model and this is done by considering the effect of scaling factors in the

evolution of the Q-tensor.

and extends it to comprise the interaction between classical and quantum

degrees of freedom. Euler-Poincar\'e reduction theory is applied to the

Schr\"odinger, Heisenberg and Wigner-Moyal dynamics of pure states. This

construction leads to new variational principles for the description of mixed

quantum states. The corresponding momentum map properties are presented as they

arise from the underlying unitary symmetries. Finally, certain

semidirect-product group structures are shown to produce new variational

principles for Dirac's interaction picture and the equations of hybrid

classical-quantum dynamics.

dynamics of the corresponding Gaussian Wigner function from the Hamiltonian and symplecticgeometric

point of view. The main result states that the momentum map corresponding to the

natural action of the symplectic group on the Siegel upper half space yields the covariance matrix

of the corresponding Gaussian Wigner function. This fact, combined with Kostant?s coadjoint

orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two

dynamics.

particle effects in the pressure-coupling scheme,Journal of Plasma Physics 84 (4) 905840401 Cambridge University Press

ices are commonly

studied by hybrid kinetic-fluid simulation codes whose unde

rlying continuum evolu-

tion equations often lack the correct energy balance. While

two different kinetic-fluid

coupling options are available (current-coupling and pres

sure-coupling), this paper

applies the Euler-Poincar ?e variational approach to formu

late a new conservative

hybrid model in the pressure-coupling scheme. In our case th

e kinetics of the en-

ergetic particles are described by guiding center theory. T

he interplay between the

Lagrangian fluid paths with phase space particle trajectori

es reflects an intricate

variational structure which can be approached by letting th

e 4-dimensional guiding

center trajectories evolve in the full 6-dimensional phase

space. Then, the redundant

perpendicular velocity is integrated out to recover a four-

dimensional description.

A second equivalent variational approach is also reported,

which involves the use

of phase space Lagrangians. Not only do these variational st

ructures confer on the

new model a correct energy balance, but also they produce a cr

oss-helicity invariant

which is lost in the other pressure-coupling schemes report

ed in the literature.

incompressible rotational fluid flow with a free

surface in a moving vessel when both the vessel

and fluid motion are to be determined. The fluid

is represented by a stream function and the vessel

motion is represented by a path in the planar

Euclidean group. Novelties in the formulation include

how the pressure boundary condition is treated, the

introduction of a stream function into the Euler-

Poincaré variations, the derivation of free surface

variations, and how the equations for the vessel path

in the Euclidean group, coupled to the fluid motion,

are generated automatically.

of mixed states. Both quantum and classical mechanics are shown to possess analogous

momentum map pairs associated to left and right group actions. In the quantum setting,

the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to

different realizations of the density operator, which are of interest in quantum molecular

dynamics. Finally, the paper shows how alternative representations of both the density

matrix and the classical density are equivariant momentum maps generating new Clebsch

representations for both quantum and classical dynamics. Uhlmann?s density matrix and Koopman wavefunctions are shown to be special cases of this construction.

in the standard Vlasov theory, the same method is applied to the drift-kinetic equation

from Littlejohn?s theory of guiding-center motion. This process sheds a new light on driftordered

fluid (drift-fluid) models, whose anisotropic pressure tensor is then considered in

detail. In addition, current drift-fluid models are completed by the insertion of magnetization

terms ensuring momentum conservation. Magnetization currents are also shown to lead

to challenging aspects when drift-fluid models are coupled to Maxwell?s equations for the

evolution of the electromagnetic field. In order to overcome these difficulties, a simplified

guiding-center theory is proposed along with its possible applications to hybrid kinetic-fluid

models.

nonadiabatic approaches, the Born-Oppenheimer adiabatic approximation is still adopted

as a valid alternative in certain circumstances. In the context of Mead-Truhlar minimal

coupling, this paper presents a new closure of the nuclear Born-Oppenheimer equation,

thereby leading to a molecular dynamics scheme capturing geometric phase effects. Specifically, a semiclassical closure of the nuclear Ehrenfest dynamics is obtained through a convenient prescription for the nuclear Bohmian trajectories. The conical intersections

are suitably regularized in the resulting nuclear particle motion and the associated

Lorentz force involves a smoothened Berry curvature identifying a loop-dependent geometric

phase. In turn, this geometric phase rapidly reaches the usual topological index

as the loop expands away from the original singularity. This feature reproduces the phenomenology

appearing in recent exact nonadiabatic studies, as shown explicitly in the Jahn-Teller problem for linear vibronic coupling. Likewise, a newly proposed regularization

of the diagonal correction term is also shown to reproduce quite faithfully the energy

surface presented in recent nonadiabatic studies.