Lie theory, deformations and the life sciences | 91TV
Transcript
- Okay. Thank you very much for this nice introduction. I'm very pleased to welcome all of
- you coming for this lecture today. First of all, this is a great honour that I have been awarded
- by the Royal Society, and deep thanks to all the staffs here for organising this prestigious
- ceremony. I would like to thank especially His Excellency, our Ambassador of Tunisia in London,
- Mr Yassine El Oued, and all the staffs of the Embassy. So it is a big honour that they are
- here today. So they represent our country, they represent Tunisia, and this is big moment actually
- to share together. I would like also to thank many people here coming from far away. Professor Kamel
- Barkaoui from Paris. Great honour also. Thank you very much. He's a member of the Tunisian
- Academy of Sciences, Letters and Arts. This is a pleasure. Dr [unclear name 0:01:21.9] from
- Brussels. Thank you very much for coming. So he's also the President of the Association of
- Sustainable Development, for which I am a member right now, and Professor Dominique Marchand
- from Clermont-Ferrand. Also, great thanks to Professor [unclear name 0:01:41.6] first of all,
- for constant help and for steady endeavours for mathematics in Tunisia, and thank you
- very much, [unclear name 0:01:54.9], for your nomination. I would like also to thank all people
- attending online, my friends, my colleagues. Especially our colleague from the Tunisian
- Academy of Science, Letters and Arts. So they are watching online, this ceremony. I thank them very,
- very, very much. Also, I would like to extend some moments first to the memory of my parents. I think
- they would have been very, very happy to see this moment, and to my family, starting with my wife,
- Professor Akila Baklouti Sellami, who is a professor of linguistics, present here in the
- room, and also my daughter Zainab Baklouti, and my son, [unclear name 0:02:57.4] from Lyon, France,
- and also my son, [unclear name 0:03:01.9] from Poland. So I thank them very much. I dedicate this
- award to my family, to my country, to the African community and also to the mathematical community.
- So I would say, for the Royal Society, this is the first time that the mathematician get this prize.
- So it is very honourable for mathematics and for the development of language around the globe. So
- now let me start a little bit, this lecture. So as you see, this is about Lie theory and some
- developments and some impact about what can be done with Lie theory, about some life sciences
- and some natural sciences. So I would like to say some word about the research profile of
- Ali Baklouti. So Ali Baklouti a professor of mathematics. So my primary research
- interest Lies in the intersection of algebra and analysis. I'm working on representation theory of
- solvable groups and non-commutative harmonic analysis. So actually, the Herbart method,
- introduced by Kirillov in 1962, was a very powerful tool to, first of all, consider this
- harmonic analysis around the nilpotent Lie groups and exponential solvable group, and also for
- solvable Lie groups. My second area of research is geometry, with emphasis on deformation theory
- of discontinuous groups, with a focus on some pioneering contributions by Professor Toshiyuki
- Kobayashi from Tokyo University and with deep connections with topology rich framework rigidity
- and stability of geometric structure. Of course, for a lot of you, this is a little bit strange,
- something which is not understandable but don't worry, I will go through some applications.
- Fundamentally, this is the main question, how to make this mathematical accessible to a lot
- of people here, because it's quite complicated to explain mathematics to a general audience.
- So the challenge Lies in the intersection of analysis, geometry and algebra, and they continue
- to inspire a wide array of mathematical investigations. For nearly three decades,
- my research has been devoted to exploring and resolving a number of longstanding and
- fundamental problems within this context. So first of all, this question of Pukánszky polarisation
- is a big and longstanding problem. So during my thesis, I constructed some intertwining operators
- between two representations irreducible in the setting of nilpotent Lie groups,
- but when you go up to more complicated contexts, the problem becomes very, very complicated. Also,
- Duflo's polynomial conjecture. Michel Duflo is a French mathematician. I'm very happy
- that he's watching us online now and by the end of this talk, I'm very happy to say that we are
- capable to solve this conjecture called the Duflo polynomial conjecture and also the Benson-Ratcliff
- conjecture, this concerns actually some deep quantisation problems. The Corwin-Greenleaf
- conjecture I will explain a little bit later and the two polynomial conjecture for restrictions.
- So followed some work I did during my thesis. Finally, the Zariski-Closure conjecture. I'm
- working with Professor Dominique Marchand here. So all these problems actually are based
- on a theory which is called Lie theory. I think very much my advisor and my first collaborator,
- Professor Jean Ludwig from Metz University. So up to now he's an emeritus professor, and he's
- actually my first co-author until 2025 but by the end of this year, Professor Hidenori Fujiwara,
- he's my first collaborator, so he's the referee of my PhD thesis. In majority, all these problems
- and conjectures, we pursued a long collaboration to try to have these solutions to these difficult
- problems. So I thank them very much because they are watching us online. For the problem
- about geometry and the discontinuous problems around these actions on discontinuous spaces,
- etc., I would like to thank all my students in Sfax University and collaborators because they
- did a very, very nice job about this difficult problem. Also, I will explain a little bit
- what is the originality of this problem and what are the outcomes and what are the impacts of these
- problems on life sciences and so on. So the plan of the talk today is something like that. What
- is the Lie theory, I explain in, let's say a clear way for people who are not originated, who are not
- based in mathematics. So this is very important. Then I explain what is the impact of Lie theory
- in life sciences. There is a big necessity, how to go from groups through homogeneous spaces,
- and this is also very important. I will explain our contributions about the deformation theory
- and some outcomes. Finally, I will cite some of these solutions to this longstanding problem
- in the last slides. So what is Lie theory? So first, a Norwegian mathematician named Sophus Lie
- was looking at a differential equation, something like that, and he wanted to solve them in a single
- unified way using continuous symmetries. Of course, I think almost everybody in this room
- knows about what is a differential equation. So an equation, when you have something which moves…
- So behind, you have some equation which is called a differential equation. Sophus Lie wanted to
- solve this equation using a uniform theory called continuous symmetries, but why continuous
- symmetries? So he was inspired by an earlier mathematician, Galois, a French mathematician,
- who was studying polynomial equations of this type, and he used discrete symmetries. Why
- discrete? Because in the setting of polynomial equations, you have only finite number of roots
- of these polynomials. For instance, if you look at the setting when N=2, you have only
- two roots and it is very easy to have some groups which acts on these roots. So Lie, Sophus Lie,
- wanted to replicate the success of Galois theory and made the analogy between differential and
- polynomial equations. And why this? Because when it concerns this differential equation, there is
- in general a continuous family of solutions, and it is only when you specify the initial conditions
- that you have a single one. If you look here, you have some axis as a continuous symmetry. So the
- idea comes, how to build some group of symmetries around, in order to have some solution to this
- general differential equations. So the idea is to construct or to go through an easy, a very easy
- concept which is called the group. What is the group? So a group is a set where we can multiply.
- So just to take two elements of the group, and the multiplication of these two elements has meaning,
- and the product is still another element of G, for instance, the product is associative,
- like the real numbers, and you have a unity element which is actually inactive element,
- adjust the element one, in the case of the real line. When you multiply one by any element, Y
- point X=X. So this is very easy to understand. In this group, we can invert. So if you have any X,
- so one over X exists. For instance, the easiest example is you take the circle, just to take the
- complex number on the circle, and this satisfies all these axioms, but groups actually manifest
- themselves through symmetries. I will explain in a moment, what does it mean. So they must often
- appear as symmetry of object and multiplication is the composition of symmetries. So let me give
- an example. So if you look at, for instance, the rotation of the sphere around an axis with
- a determined angle. For instance, you have this X of symmetry. So you have this rotation around this
- axis and you can have another rotation. So the multiplication, the composition in this setting
- has a meaning. So this is our multiplication. You get another rotation around the globe. So the
- identity element, this is the inactive rotation. So you take any rotation around G=R theta and you
- take the rotation which does not do anything. So it is inactive. Then the result that the product,
- which is the composition, gives you the first rotation. What is the inverse element? So if
- you take any rotation, R theta, and you look at this inverse R minus theta. So the composition,
- the multiplication, gives you the inactive element, the identity element. This is an
- example of a group. The set of all rotations of this sphere is this group which is noted
- S02 of R. This is just a mathematical notation, and if we allow reflections across subspace,
- we get a bigger group noted O2 of R. I will explain that, what does it mean, reflections.
- A Lie group, actually, so it is a beautiful, structured puzzle composed of two perfectly
- interlocked pieces. First of all, it is a group and then it is a smooth manifold. What does it
- mean? It is something like, when you can draw some coordinates. So we badly need coordinates on
- a given group, and the structure of the manifold makes the group look like an Euclidean space, and
- this is used for instance when you do integration, differentiation, complex calculus, et cetera. So
- this is very useful, but these coordinates would be local, not global, as the setting of R or R2,
- et cetera, but it will be local coordinates. These Lie groups are among the most basic and
- fundamental building blocks of the modern geometry. So you will see that Lie theory
- is the mathematical study of symmetry groups and their transformation. So it appears whenever such
- symmetries are found in nature. It provides a powerful framework for analysing the behaviour
- of these symmetries. So let me explain more the fundamental aspect of this theory. For instance,
- if you look at the Lie group, so and you take any point in the Lie group. So the coordinates
- are drawn on a Cartesian space, said the Lie algebra. So the Lie algebra is just, for instance,
- if you look at the setting of the circle. So the element ,the unit element is E and the
- Lie algebra is just the real line, and of course, locally it is some interval when you can do a lot
- of things. So the circle is a curved space, and in any point, you can trace the tangent space and you
- end up with something like a real line. We can think about higher dimensional examples and the
- meaning is similar. It simply means that at the neighbourhood of given point, does not look line.
- So for instance if you look at these examples. So this is a higher dimensional torus. In any point,
- you can trace a plane, and in a plane, you can do a lot of things. For instance for anything
- of computations or integration and so on. So this is the idea behind a Lie group and Lie algebra.
- So if I can illustrate this example, if you look at the surface of the Earth, you can imagine that
- these coordinates can be something like that. So in any point in the Earth, you have something like
- an Euclidean space, which is the tangent space, and it is equivalent to an illustration of an
- example of a Lie group and Lie algebra. So if you go now for higher dimensional subspaces, so
- instead of looking at SO2 of R, you can look at S1 of R. The fact of these groups are manifold means
- two things actually. Instead of considering them simply as a set of matrices, we can approach them
- geometrically. S1 is a group in itself because it consists as rotation, as I explained before,
- but we can approach this set of matrices geometrically, and they are among the most
- fundamental family of Lie groups. For instance, if you complexify instead of looking at matrices
- with real entries, you go through complex entries. You can consider, for example, SU of N. This is
- the Hermitian matrices with some equality of the determinant, something like that. Also, a bigger
- group is SLN. This is of all matrices with the determinant. This is very useful actually when you
- look at, in terms of applications in many, many sciences. So for instance, what is the impact of
- all this Lie theory? For instance, in mathematics, it is almost perfectly well connected. If you
- look at this kind of domain. Algebraic topology. Algebraic geometry. Combinatorics. Differential
- geometry. Number theory. Low dimensional topology. Riemannian geometry, et cetera. So these are very
- related to Lie theory. Also, Lie theory is ubiquitous in physical sciences. So it is
- associated, for instance, to classical mechanics, electromagnetism, particle physics, quantum theory
- and just an application, about what I have said about this classical group, for instance,
- U1. U1 is just the circle. It describes the phase symmetry crucial in electromagnetism and SU2,
- it appears in weak interactions and in the theory of angular momentum, and SU3 describes the strong
- force, and the SO3 governs rotational symmetries. SL2C is related to the Lorentz group and special
- relativity. This is just an example how these Lie groups and this theory intervene, for instance,
- in the physical sciences. In chemistry, the same happens, maybe deeply, for instance,
- if you look at the relationship with rotational symmetries of molecules and quantum chemistry.
- SO3, for instance, and SO2, U1, SO2, et cetera, have a lot of impact about something related to
- this domain of research. In biology too, so if you look at the symmetries of living systems and
- molecular biology, the morphogenesis, et cetera, these groups have their impact and their role
- within this context. But also for medicine, if you look at the problem of disease progression,
- the drug development, et cetera. So these Lie groups have a lot of interaction with this kind of
- research. The second group, SE3, which I have not defined yet. I will define that in two moments.
- So this is also very important and this is all related to the Euclidian motion groups, positive
- equilibrium. I will explain that in a moment how it is important to build the geometrical vision
- of all this law we are dealing with actually, and also in Earth sciences, a lot of domains are
- related to Lie theory. So it has a deep impact on that. So for instance, seismology, dynamics
- of geophysical fluids, crystallography and mineralogy, plate tectonics, et cetera. So
- now I explain that groups or Lie groups in general are not very sufficient. So we need
- to move from Lie groups to something which is very complicated, more complicated let's say,
- but more efficient which I call the homogeneous spaces. So in general, group G acts on its space.
- It has an action. For instance, in the case of the permutations, polynomial roots. Galois established
- the profound notion of Galois group. So these are permutation groups acting on roots of polynomial,
- in order to have solutions of complicated polynomial equations. A group can be understood,
- and it is often based on the tools through the way it acts on or transforms other mathematical
- objects. I will explain that by some mathematical animation, let's say. The most important thing,
- there is a notion of transitivity of the action of a group on a set, which means physically, simply,
- that if you take two elements of the group, of the space, let's say, you can find a mirror which
- transforms the first one to the second one. This is the meaning of transitive action of a group
- in a space. So for the audience, please all the time have in mind that when I speak about group,
- just a group of symmetry. If you look what happens in this room, there are a lot of
- groups of symmetry which transform a lot of parts to other parts symmetrically. So please
- have in mind that when I talk about group, this is a group of symmetry, and this transit of equations
- defines the geometry of the space. So through this space, any transitive equation, of a group on X,
- allows us to write down the space X as homogeneous spaces, G/H like that. H is a closed subgroup of
- G which defines the model geometry of X. For instance, Élie Cartan was elected as
- a foreign member of the Royal Society here in 1947, so he published, between the Two Wars,
- the following very, very nice paragraph. So he said that elementary geometry is essentially the
- theory of invariants of a certain group, the group of Euclidean motions. So its purpose is
- indeed to study the properties of figures that are preserved by an arbitrary motion,
- stating that all motions from a group which expresses in precise language the axiom
- according to which two figures equal to a third are equal to each other. I will explain that with
- some mathematical animation. So imagine that you have some geometric forms like that. I'd say that
- two objects are identical in form, if we can map each one onto the other by motion. For instance,
- these objects are identical in forms. Why indeed? Because if you look at this example. So you have a
- group action. So this is translation, translation, translation, translation. This is group actions,
- translations, translations, translations, translations. Now rotations. So I rotate
- the object, I rotate it again, I rotate it again, and then I use the reflections I spoke in the very
- beginning. So with this action of this group, I can identify which objects are identical in
- form and if you have many objects like that, by a first group action, you can get this form,
- but in the second one, you can get this word mathematics. So this group is called the Euclidean
- motion groups actually. So the O of 2 are the rotation groups and reflection groups, and R2
- represent the translation. So the semidirect product consists of the Euclidean motion group
- which acts transitively on the Euclidean space E2. It is the group of all transformations preserving
- the geometrical structure of E2 of the Euclidean space. So E2 results in geometry. It is defined
- by the symmetry, determined by the action of E2. When you have this transitive action,
- so you have a homogeneous space, as I explained before, and these homogeneous spaces Lie in the
- heart of my research, just as they did nearly two centuries ago for Felix Klein when he formulated
- his modern and unified definition of geometry. Felix Klein published this so-called, the Erlangen
- Program in 1872, which classified geometry through the lens of transformation group. So the idea is
- a homogeneous space for a group is equivalent to the notion of geometry. For instance, what I have
- explained now about the Euclidean motion group by Élie Cartan, this led to the Euclidean geometry,
- but of course, you can have another different type of actions. For instance, you can have
- the affine geometry. So this is the action of the affine group. You can speak about the projective
- geometry, the action of the projective group, and you can also consider the Riemannian geometry,
- this is the action of local isometries. So the idea of this Erlangen program is to
- stimulate the relationship between homogeneous spaces and the geometry. After that, Albert
- Einstein took the idea of the Erlangen program and interpreted it physically. So he's asserted that
- the laws of nature are invariant under general coordinate transformations. Einstein modelled
- the spacetime as a pseudo-Riemannian manifold whose curvature reflects the distribution of
- mass and energy, but the idea comes from this notion, or this program elaborated by Felix
- Klein. Then Élie Cartan adds saying that this is more philosophical questions, nature is not
- absolutely homogeneous. If you think about nature, the universe, he adds that it is not homogeneous,
- which means that you cannot find a group acting transitively on this. Then Charles Hirschman
- reflects the realistic complexity of geometric and physical system, by saying no absolute
- homogeneity, but rather local homogeneity. So we move from the homogeneity to the local
- homogeneity, but what does it mean? What is the local? It is something not very complicated. You
- take your space X and you take any point. So this means that around this point, you
- have some neighbourhood which is homoeomorphic, which is isomorphic to some homogeneous spaces.
- So this notion is very important. So we move from the homogeneity to local homogeneity. In terms
- of let's say, let's formulate it mathematically. So the local geometry of spaces is determined by
- geometry first. So we regroup acting locally, continuously, and transitively on some specific
- X. So X is something like that, and then you have a discrete subgroup acting… So you have
- a discrete subgroup here acting on this space X to produce another cushion space, which is
- called a Clifford-Klein form. So discontinuous means that the action is proper and free. I won't
- go through details, but just to say that you have a smaller group acting transitively with proper
- and free actions. So let me give you an example. So if we generalise the Euclidean motion group
- by Élie Cartan for N=2, so you get the Euclidean group for general N, and this acts transitively
- on the Euclidean space, and we have the geometry of En produced by this transitive action. In this
- case, the discrete subgroup would be something like the integers. I think everybody in this room
- knows about what are integers. So just… Sorry. I think we have to move slowly in order to…
- Yes, so you have a discrete subgroup acting on the Euclidean space, and the result is a
- quotient space, which is a torus here, because the action of Zn on this Euclidean space gives
- you these stories, because the Euclidean space already is a homogeneous space. So the easiest
- way to say that is the torus as a mathematical set is something which is compact. So these spaces are
- locally the same but different globally. This is the idea. So to move from local to global. So any
- geometric structure in space X is inherited by the Clifford-Klein forms. So for instance, I generated
- this image with artificial intelligence to show you how complicated and how sophisticated one can
- get some Clifford-Klein forms, which are very, very important. First of all, in nature and in
- mathematics. Now, building on the foundational notions of groups, manifold geometry, and the
- profound interplay between them, my contributions focus on advancing the deformation theory
- of Clifford-Klein forms within the setting of solvable groups and their compact extensions. This
- framework offers a concrete approach to tackling the local rigidity conjecture. So for instance,
- the Erlangen program states that geometry is, if you have space and you have transitive action,
- so group of transformations. So the deformation is something equivalent to a geometry which is
- equivalent to a homogeneous space and then a discontinuous group actions. My framework is
- lying within Lie groups. The classification of Lie groups gives this kind of abelian,
- nilpotent, exponential solvable, reductive, semi-simple, and simple. I'm working in this
- area of solvable group, but also I consider some compact extension from some reductive group acting
- on solvable groups. For instance, the setting of the Euclidean motion groups. You have the
- compact group of the rotation group acting on R2, which is compact extension in some sense of R2. So
- the local rigidity conjecture says that whenever you have a connected, simply connected Lie group
- and you have gamma, which is not trivial, so this discrete subgroup, which is a discontinuous
- group built on the homogeneous space, then I beLieve that, in this case, the Clifford-Klein
- form deforms continuously. Let me explain a little bit, in let's say, a more explicit way. So imagine
- that you have some homogeneous spaces and you have some actions. This conjecture would say
- that there is an infinite continuous way to deform the discrete subgroup gamma, with the assumption
- that the geometric structure of the deformed Clifford-Klein forms are preserved. So this
- is for instance, the first one is deformation, the first deformation, but you can get a new
- one and then you can get another one, and the conjecture says in some circumstances there is
- an infinite way to deform this Clifford-Klein forms. This is conjecture, and unfortunately,
- the proof resists so far. So I will give you some examples, which is very simple. So if you look at
- the situation of the plane, G=R2, and you have the X axis is R/H, and the Y axis is G/H, and you look
- at this group. Gamma is the integers. Then in this case, you have two orbits, the positive orbits. So
- the parameter space is formed by two orbits, the positive orbit and the negative orbit. In this
- case, there are only two deformation of the torus. So you cannot deform the Clifford-Klein form in
- this case with infinite way, and the reason is that this group is not nilpotent. Nilpotent in
- some explicit way. It is, let's say if I illustrate this example, this means that it is
- characterised by some polynomial coordinates, something like that. So I published in 2022, the
- first book to explain the dynamics of all these deformations in the setting of compact extension
- of solvable group. How this works, how this conjecture can solve several problems, and this
- year, last month, I published the second version of this book where I explained the geometry of
- Clifford-Klein forms and also the local rigidity and the stability concepts. So in this book,
- you'll find, for instance, a lot of things about the structure theory of solvable Lie groups,
- various geometric and topological concepts related to the deformation and moduli spaces of
- discontinuous group actions. The newest approaches and models in deformation theory in the setting
- of compact extension of solvable Lie groups. The locally rigid conjecture in the setting of
- compact extensions. So I extend a little bit the setting of this conjecture from nilpotent to the
- extension, compact extension of flat potent with some other conditions. Also, I explain the notion,
- the geometric notion of stability, Hausdorffness, and the Calabi-Markus phenomena. Just to say that
- this deformation theory has a lot of applications, for instance in physics, for general relativity
- and liquid crystals, in chemistry, for crystallography and molecular reactions,
- in biology, proteins and biomolecular structures in medicine, et cetera. So these are just some
- examples, but if you go through the literature, so the application of this deformation theory really
- has a wide spectrum. I end this presentation to say some words about these problems I have
- mentioned in the very beginning. Just to say that the Representation Theory is a way to understand
- abstract mathematical objects by expressing them as transformations, so that we can work
- with them more easily and mathematically. This means that unity representation of a group is a
- pair by H, H is a Hilbert space, and you have a group homomorphisms. When you move from the
- classical harmonic analysis, for instance, from the Fourier analysis, when you have this Fourier
- transform to the infinite dimensional Fourier transform, you need these representations. So
- the noncommutative analogue of the Fourier transform in the setting of noncommutative
- groups is this one. I would like to thank all my students working on this area. They did a very,
- very nice job in how to prove some noncommutative versions of uncertainty principles that govern the
- trade-off between localisation in group and representation space. So really did a very,
- very nice job. The group here acts on the dual vector space of the Lie algebra, and they produce
- some quadrant orbits. So these quadrant orbits are smooth manifolds. So this image shows you
- layer of quadrate orbits which are smooth manifolds, and they have all odd dimensions.
- They have related symplectic forms. The question now is about quantisation and quantisation.
- So in order to qualify, so in the Kirillov method, I mentioned it in the first slide, characterise
- unitary representation as corresponding to a unique coadjoint orbit and the problem is there,
- how to make this problem of quantisation and dequantisation. For induced representations and
- restrictions, they are the most typical kind of representations, and we published with Professor
- Jean Ludwig and Hidenori Fujiwara, this book in 2022, and we explain all the harmonic analysis
- around these representations. All these positions are related to algebra of differential operators,
- which are quite a bit complicated, but they allow, at least in a physical way, to get some solution
- to many, many differential equations and so on. Finally, I am very pleased to announce that,
- for instance, for the first problem of closedness of the product of Pukánszky position. So in my
- thesis, I solved the problem of how to find an operation with two representations, and now this
- problem is completed for the exponential setting case this year. The Duflo polynomial conjecture,
- I'm very pleased to announce that it is solved, at least for nilpotent situations, and
- two papers are underway with Professor Fujiwara to complete the exponential setting. I gave a
- counterexample to the Benson-Ratcliff conjecture, saying that the dequantisation process using the
- cohomological invariant are not very efficient in many, many cases. Also the parallel conjecture for
- restrictions also, it is solved. So this is how to go from induced to restricted representations,
- and also finally, the Zariski-Closure conjecture for exponential, I am very pleased to say that it
- is completely solved this year. I think Dominique Marchand, a revised paper already and a second in
- progress, looking at some general situations. Let me thank you very much for your listening.
- Thank you so much. Congratulations. Thank you for this wonderful lecture.
- So next is the Q&A session. Before we start with your questions, we have a couple of
- colleagues from Tunisia who have been watching I think live, the lecture. The first one is
- a speech, actually commentary from Professor Hafid Abdul-Malik, who is the President of
- the Department of Mathematical and Natural Sciences in the Tunisian Academy of Sciences,
- Letters and Arts, and this is known as Bayt al-Hikmah, the House of Wisdom in Tunisia. I
- don't know if we will have the… Oh, we have to do something here. There we go. So unfortunately, we
- don't have the time to go through the full speech. Can we move to the next slide? Here we go. Okay,
- so this is the speech. So again, not possible to read through all of it given time but essentially,
- he's recognising the award and the prize to Professor Baklouti and also he's highlighting
- the connections between the different communities and different scientific academies, and hopefully
- this will also be a stage for further interactions between the Royal Society and the Tunisian
- Academy, which I think we're all very passionate about, especially here, the Tunisian community in
- the UK. We've been working really hard to create a stronger link and reinforce the links between the
- two communities. So thank you, thank you so much, Professor Abdul-Malik, for these comments. We have
- a second guest from Tunisia and I think, is that on the slide? Are you going to do the questions?
- We have questions by Professor Amel Benammar Elgaaied,
- who is a permanent member of the Department of Mathematical and Natural Sciences at the
- Tunisian Academy of Sciences, Letters and Arts. So his first question is,
- how can deformation theory within Lie groups be used to model adaptive biological systems,
- such as immune response or neuroplasticity, in a mathematically rigorous way?
- Thank you very much. So I have explained, first of all, the impact of Lie theory on
- a lot of domains in biology, in medicine, and the deformation theory, as explained
- in this lecture also can be… So the actions of these discontinuous groups
- comes basically from symmetry groups which acts around a lot a lot of domains in biology and in
- medicine. So we cannot predict precisely how we can use this theory unless we can modelise
- correctly the problems. So once we have modelised correctly the problems and we built the necessity
- framework, the necessity arsenal about all this problem, we can produce a lot of solutions,
- as I explain in this lecture, to a lot of problems around the questions.
- Thank you. His second question is how might advances in Lie theory contribute
- to the development of personalised medicine,
- particularly in modelling the dynamic transformations of biological systems?
- So same kind of question. So we have to actually modelise. So once we have
- modelised the problem and looked at the circumstances and the difficulties,
- we can precisely look how to predict these transformation groups and these symmetry
- groups to solve a lot of problem around these questions. Thank you very much.
- Thank you. That was wonderful. Any questions here from the audience?
- Please raise your hand if you have any questions. Please wait for the microphone.
- Thank you, Ali, for this. I'm working on AI actually, and I can several times, I think
- about all these symmetries that we have and we apply through this Lie theory. So, I'm thinking
- if you are looking at it to see how you can apply that in modelling, but in data driven modelling.
- In a precise way, I cannot answer directly this question because
- this is not my scope for the moment, but what we can say, whenever you look at some local
- asymmetries on sharp spaces and sharp problems, you can get a lot of solutions. For instance, the
- problem of solving these differential operators problems in some complicated space give you some
- solutions to a lot of differential equations. I guess as I answered the first question,
- we have to modelise the problem. We have to look at the nature of the action.
- What are the groups which have the capability to act transitively,
- freely, properly on a lot of components, and then we can predict the nature of answers.
- Yes, exactly. It's exactly that. We need to get some presentation of this,
- all this data in a way we can get this natural network, natural presentation.
- No problem. Thank you.
- Question here.
- Thank you, Professor Baklouti, for this interesting presentation and for nicely
- showing the application of Lie groups to other sciences, physics, chemistry,
- medicine. I was just wondering if you are aware of any applications to human sciences,
- of the theory of groups to human sciences, sociology, psychology,
- if there are any applications and if you are aware of them? Thank you very much.
- Yes, especially for the theory, I don't know, but the impact of mathematics is very large on
- how to develop the human sciences, for instance, because mathematics now is the science of logic,
- and mathematics can mobilise a lot of things. If you think about how to make some statistics or
- to make some… So mathematics is everywhere. I made a presentation a couple of days ago
- about the role of mathematics in general on how the human sciences can be touched by this science.
- So for sure, when you look about how mathematics can affect a lot of sciences, for instance,
- because it is related directly to, for instance, to cryptography, it is related to data science,
- probability, statistics. So it has a lot of application, including the human sciences.
- Thank you so much for this talk. So you earLier mentioned that SO2 describes the
- symmetries of an electromagnetic system. I want to know if these nilpotent groups that
- you mentioned earLier sort of model symmetries in a certain kind of like physical system as well.
- SO2 is very well known to act about a lot of Euclidean geometry because SO2
- is the set of all rotation, positive rotation, let's say, not reflections,
- acting in Euclidean spaces. So R2 in our spaces. So it can model a lot of phenomena
- about physical systems and about something like some local phenomena acting in
- chemistry and biology and so on. So it has a deep impact. That's why ÉLie Cartan specified
- his saying about this group, because it has many, many, many, many things.
- What about these nilpotent groups that you spoke about?
- Yes, nilpotent group means that the group is diffeomorphic to something like Rd but with
- the polynomial coordinates. For instance, the group I mentioned, the group R2, it's
- exponential. It's not nilpotent. So this means that when… So when I define a group that there is
- a multiplication law when you take two elements. So X point Y, this is an element of the group,
- but if this element depends polynomially on the first one, it's called a nilpotent Lie group.
- So the application. So the multiplication is polynomial. So first the group is isomorphic
- to something like Rd or R2 or R, and then the multiplication group is defined by polynomial.
- Thank you.
- Hello. I have a high level question. What is the secret to be a great mathematician, and what is
- your advice for the next generation in Tunisia that want to follow your path, for example?
- Thank you very much. So the advice I can tell to all young researchers is be prominent,
- be perseverant. Please pay a long silent moment in research and don't be in a hurry to publish
- your papers everywhere. So these moments of silence in thinking deeply about the
- outcomes of your publications, the outcome of your research, the application gives you
- a lot of advantages, and when you publish your papers in a very well-known journals,
- this counts a lot for you. So for young people, especially people who are now
- taking away for researcher experience, be patient, be prominent, and be perseverant. This is what
- happened to me when I started my thesis. So the first result I gained took a lot of time. I didn't
- get some result from the very beginning, but it took a long time. I think Professor Jean Ludwig
- who is following us online now. We took a lot of time to experience a lot of examples and so on,
- but my first publication, my good publication, took a lot of time to be to be done.
- Thank you. One last question, because I think we're running out of time.
- Yes, sorry, I'm not really an expert in this, but protein folding. So predicting
- protein structure from the amino acid sequence. AlphaFold made some great progress in there but
- they did it in this very messy computational way where it's really kind of try and error
- wiggling things around. Do you think, like with your theory of Lie groups,
- you could develop more of a theoretical way of predicting protein structures?
- Very nice question, yes. So the problem is also when you deal with Lie groups and when you look at
- first examples, sometimes it is very complicated to pursue a long computation and so on, and
- the idea is to have some groups of permutations, acting on something fascinates a lot of things.
- For instance, in the case of Galois. So the solution of all these polynomial equations,
- so the actions of these groups facilitate a lot, the finding of solutions, but as I said before,
- all the questions, you have first to modelise the problem, to look what are the action,
- necessary action and sufficient action and prominent action, which could give you
- a lot of solution and a lot of ways to find solutions and to find the results.
- Okay, thank you.
- Excellent. Thank you so much. Let's thank Professor Baklouti again.
Join us for the Royal Society Africa Prize Lecture delivered by Professor Ali Baklouti.
91TV Africa Prize is awarded to Professor Ali Baklouti for his work on non-commutative harmonic analysis and geometry on homogeneous spaces.
This talk explores the fundamental role of Lie theory in advancing diverse scientific fields, particularly those related to living systems. A key aspect of this contribution lies in deformation theory, which enhances our understanding of symmetries and their transformations across various disciplines.
By examining Lie groups and their deformations, this presentation illustrates their far-reaching impact in fields such as medicine, physics, chemistry, and biology. These mathematical structures offer new perspectives and innovative solutions to complex scientific challenges, fostering interdisciplinary advancements.
Beyond its applications in the life sciences, Lie theory remains central to pure mathematics, particularly in harmonic analysis, differential geometry, and algebra, while also influencing modern developments in theoretical physics and computer science. This talk underscores the unifying power of Lie groups and deformations, positioning them as essential tools for both theoretical exploration and practical innovation across multiple scientific domains.
This talk will also introduce new conjectures in deformation theory and present key breakthroughs in representation theory, solving long-standing open problems. These contributions provide deeper insights into the structure of Lie groups and their applications, paving the way for further discoveries in mathematics and beyond.
About the Royal Society
91TV is a Fellowship of many of the world's most eminent scientists and is the oldest scientific academy in continuous existence.
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