Or if there is a my co-writer honour for me to present my inaugural lecture to you in this very special place.
In this lecture, I would like to discuss the origins, development and loss of a partial differential equations.
You change what? I spent my childhood in the beautiful area of Ardmore Bay where the Changing River Origin,
a different last thing to the Promised Land in the Eastern Time.
I see. This is actually the place to observe an effect of fantastica nature phenomenon.
Okay, so I remember when I was three years old, my parents took me to watch the meg and this magnificent change I live with.
Had a ball on the middle autumn days in Chinese calendar.
It took the lead to a height of about nine metres our speed at 40 kilometres an hour.
I was truly impressed by this shop flung.
Shop around for a they had to propagate in very organised fashion from east to west.
Alonzo Change. Oliver. The father was my first encounter with this giant wave.
The now immense. Matthew. Tom. This caused shock from shock wave.
The Potomac. Okay.
Anyway, so this is a celestial.
This is a believer, young man. I was a surfer.
And this is a giant impact. The waves are.
So I was also there after that. I was really puzzling about the millions of petals of the form, the body water motion.
Like Leonardo da Vinci, you know, the livers, legs and even the kitchen sink in my house.
Okay. So I was really fascinated with all little phenomena, also little phenomena formed by the air and water along the me.
Okay, so here's a tang form normally formed in west the Pacific Ocean.
Travelling into the East China Sea often slam into the area found to be.
Here's a tornado. Actually, I'll also observed a little thing there in Chicago earlier, United States.
Then shock a wave, along with a supersonic object, as it were, struck them incorrectly.
Here, the elaborating shockwave formed to allow the and a supersonic aircraft also
by various explosion by TNT Sofia's explosion and the supernova explosions.
So. Well, in many cases, it is simply very chaotic and troubling locally.
But globally we are behaviour. So we are not from such the phenomena of the call flocking phenomena by the boats.
You came for me. The values were beautiful patterns globally and as a same situation.
The massive Mexican waves. The pack, the stadium.
So such observation actually has been successfully used by impressionist artists, especially to put in place an artist painter in 19th century.
This example. Okay, the painting made by George Peel.
So long. Then I was also I was curious.
It was about the various sophisticated geometry or structure created by nature for them, created by the plant leaves like here and the flowers.
Even for protein folding structure.
Okay. So we is also a fascinating observation of natural phenomena.
So I was wondering what are underlying national laws behind this phenomena?
Are there any efficient way to describe politics?
Those little dynamical phenomena are changing.
What? So finding an answer to this question was the wing of my child the dream?
So first Tom went out to touch to calculus.
I was immediately attacked by this beautiful mathematical civilian concern with tens.
Went off of the core concepts and calculus is a derivative.
Okay. So I believe it actually is a measure of how a function changes in the changes, more precisely, the ability to malfunction if x was it is bad.
A valuable x is equal to ten delete off the function at the z point.
For example, we consider a moving particle in the plane.
So trajectory of moving particles we call the position function choose our curve.
Such curve mathematically we form as the function like those kind of functions.
Okay, now the telemetry of this function is the velocity of this particle moving particles.
Okay. Now, the second delivery of this position function, the first delivery velocity function is acceleration.
So geometrically, if this is a curve, then the first derivative is the slope of the pending the line.
Okay. The second delay repeats the the later is curvature of this curve.
Okay. This is for that. Now, as far as I know, there are four notions about derivatives.
Okay. So I cleared the hill as a new thing.
Like I use the dart necessarily with you. There is the lab beneath your study.
And like a larger use applied. So I use of study for so many different tests for the use of different machines for going for that.
Okay. Now we are living in the world of three dimensional space.
So minute quantities. They are not only dependent when malleable.
You depend on time and allocation. So we depend on several variables.
Okay. Then this will require the partial derivatives.
Okay. So our project teams are functioning.
For example, if I put a two dimension case is a delivery team with a let's get a one off is also variables with others hold a constant.
Okay. In other words, the project delivered to me. The change of the quantity will expand as the other variable with the other constant for the.
For example, this is like a surface. You can only present the by a function for two variables.
So you hold a constant when valuable. There will be this function to present a curve on the surface.
So very positive. Leave it to the slope of this. Curves in the surface.
Okay. With the same with other variable. Handle the delay with the slope of the other variables.
Okay. So for positivity we only use noting along the de oc which first introduced by legend for answering stamp the 86.
Okay, now we use partial derivative with respect for that time means the fixed excellent to take it that it was this valuable.
Then we will on this form some time. Okay. With other x variables we allowed the default.
So this is the same for high dimension case. We can do the same thing now.
Sometime we use the reading. Okay. Now we. How many piling up later is a vector as the partial delivered all those variables.
Okay. So that notion for that. Then when I first learned the first the course of partial differential equation,
I find was that there was clearly I showed a study to answer my child for the question.
Okay.
So this is a positive delivered partial differential equation in the later with a function, as I mention, some animal function we really want to know.
Okay. The only dependent not only Pam also depend location, other variables.
Okay. Then for example, like those quantity frequencies temporally choose for propagation of sound and heating velocity, density,
pressure and momentum for motion of fluid and displacement, surface tensions covetous for moulting of elastic and materials.
So in order to understand the those changes later, those functions will acquire positively with the project definition of creating that relation.
Involve this, I don't know, functioning and the positive derivative with respect valuable t and x slow equal sign for example here decreasing.
I can write of the as first of the equation releasing the present by as a
function OC between partial delivery with you articulating upstream so forth.
Now I'm in a physical system. Usually we want to know more than one and no functions which satisfy more than one relations.
Okay then this kind of positive system about the differential equations.
So clearly positive involves the sum and no function of depend more than one variables.
So the more general automotive initial equation the way learn that in differential equation.
Well this element of me as choose starting actually once I meet a novelist American literature after she got to know me my list which
is a partial differential equations she look at that of me with a full of puzzlement and ask why you study different interpretation,
partially not the holy. Now there was a little galaxy that turns out that we had a really very pleasant time together.
Finally, agree with me,
the term partial differential equation is really I got a term and you mean literally are now also so we cannot avoid equal sign and she
I would like product to mention that actually equal sign was invented by Oxford the okay labour the recall and in in his book in 1557.
So at the time I've got his nose in the he also to the plus.
On the minus sign. Of course, our time equals are much longer than nowadays.
You mean plus time?
Well, I hadn't really had a time to read this the longest paragraph because it wasn't something that early modern English before Shakespeare.
Now, partial differential equation allows actually mainly for two sources weighing the
from the first one following fundamental law of and the principles in the sciences,
including macroscopic law and principles macroscopic massless coupling law principles mach also
macroscopic scaling and the proceed procedures from macroscopic massless approach process.
The second source of fundamental mathematics itself as the mathematical rationales are principles include compatibility condition,
consistent condition, construing critical points as the warehouse, the mathematical tools instrument and the generalisation,
completeness, and also beauty and the curiosity and imagination.
There is a study of partial differential equation as you start that 18 centuries in the work about Villa,
the Lambert like lodging Laplace and the mini mathematicians essential tool in the description of mechanics of continuum.
More generally as the principal mode of analysis of modes and sizes.
This principal modes as you're still going off of the fundamental concerns of the development of PAGEAU differentiated pacing until today,
beginning the middle of 19th centuries, particularly with the work of the iMac as a pre become essential tool and other balance of mathematics.
This duality of viewpoint has a being an essential tool.
Analysis of Passive Differential Equation. Until today.
Right now for namely a posture differential equation, satisfactory ceiling has it being established.
So no effort of several generations. May not most.
The part of differential equation, as I mentioned, for alternative sources naturally are non-linear.
So it's much more challenge, more difficult buying for application, more important,
but now is really coming to mainstream research in mathematics and other sciences for that.
Now I present the wings and pose of the colossal nonlinear partial differential equation from this category.
Is from the macroscopic law in computer physics.
So this is the law, the car conservation law. Okay.
So it's a third rate of changes of the total amount of certain quantity.
Kong pending a fixed domain for them in this domain equal to the flux of this quantity across the boundary of the e.g.
The other war. The amount of such quantity and leaching can be measured by accounting for how much of it is a current present.
And how much of it they are, the leave, the leading and if it's the period of time.
Well, typical examples to suit your fundamental of nature the conservation law,
mass conservation momentum, conservation of energy which it can formulate that this form.
Mathematically, we can farm into this conservation as a following.
So pull out the know that you as the benefit of this quantity value integration.
This quantity as a whole.
The math is the total amount of this quantity.
Now the chanting method means you pick a delimited with a square t, right.
So the statement that the total rate of Chanda to the equal pool, the flag of the quantity across the boundary.
Okay. So this is a mathematical statement for this conservation law.
Now, use, uh, uh, simple calculus manipulations and the use this law holds for any domain.
Okay. Then we can build from here together.
This first are the partial differential equation f of the function of view in general.
So this is not only the partial differential equations, not in sorting system.
Actually, they are more than two more than one quantities.
And instead of our several laws now in the farm system.
Okay. So then you can learn the system of conservation laws.
That is the form like that. Now the notion is a simple to describe but assured massive weather treatment present difficulties for that this company.
One typical example of that is a like fishing for compressible flow, which consists useless formulations and constantly mass momentum and energy.
This is a system harvest the flow flowing convective motion dominate the diffusion dissipation.
So for those emotions specifically, while the shockwaves like gases elastica flow and the sheer lattice, so many scientists as you have studied this,
has been studied this this equation slowed a long time ago and thus made a great contribution,
especially many British scientist, that made the important contribution for studying the analysis of this systems.
I would like special mention George Stokes.
He was the probably first scientist to detect lies, discontinue jumps, now call shockwaves mathematically slow.
This equation are now mathematical rigorous mathematical study for the constant the knowledge conservation.
I'll start with the actual scalar equations in the work by Allah the last century for the United Forties in the work by LAX,
Hope, Atlantic and many others. So I merely say that the fantasy of them has been established slow the effort of several generations.
For that now greater progress has been made. The following dimensional case a space start of flow started with GlaxoSmithKline leasing
the by Bianchini please now for we are posting this for solution of small total variations.
As the warehouse now existence the need for solution of large oscillations by now the modelling and function
and that they take a technique called compensate companies must still they are many important problem.
I open like a uniqueness stability for solution of large oscillations.
Now formatted the magic is doing much more complicated.
I can more challenge but scientifically more appealing since we are living the world with the full of shock waves.
As I'm sure you're just existing those minute, those kind of high speed objects, fly objects.
I also like various explanations for that, right?
So when those kind of shockwave hits,
some of the objects obstacles special are some special optical was geometry or them some flight objects metres of shockwaves then shock.
And the flashing the flashing problem allows that.
So one of the fundamental problems shock, shock in reflecting the phrasing by two dimensional which is set up is the following.
Here are the two dimension weight to the plane shock moving from far field at the constant speed.
So before. Let's do this. Wait. This is a similar plane shock.
Just moving.
Then the question is, what happens when this shock leads to the vortex of which and the what the kind of wave padding to form along this which.
So this setup is really simple, right? Well, however, the the wave patterns are extremely complicated.
Right. The first two scientists who observed the complexity of those configuration was owns the Mafia in 1879.
And he's actually experimentally frank to see if this experiment is out.
He found a tool. Uh, two different types of configuration knowledge, unification, model fractions.
Now, since the importance of this is the fundamental importance of this problem, I checked as many scientists along 9040s to visit to this problem.
But especially the experimental scientists, they do various experiments in the labs around the world.
What did they find? A situation much more complicated, Ma observed.
Okay, so one case is like this.
The shock wang. This angle large enough for if we fixed the stress of shock,
they after later power what they find that this new shock wave formed this caused a leaf rather shockwaves.
So Wang, this information will move all the left arm of the backwards for that other direction.
Right. So this is a picture of Lily. Beautiful.
And then you can see this Wellesley of similar and similar flowers most here nobody living very nicely.
Now the same problem if you change which angle and it knows more than situation, much more complicated, you see a form instead two way of conflict.
You can form the full wave configuration SUNY Sharks and the wing vortex sheets.
So now because it's simple modification.
Now, if you could decrease the angle a little more that evening, now you observe the this four wave configuration,
the vorticity waves form that even very recently and new padding has been discovered by the modelling as and your computation and
experiments experiment through results for them the behind the mask them the form the supersonic bubbles from the generalised generator,
the theories of shockwaves about that. So the scientific issue we really want to understand the first the structure of those patterns.
The second transition collectively between the patterns.
Man. We want to live in a hopefully we want to understand the dependence of those things.
Are those parameters very important?
We can model which angle. Now I show you a picture that is a dependent.
We can also depend how strong string as the the the stress of the shock wave, the tonality,
the present, the fruit America my number and it's all dependent with a kind of flow.
Okay, we're talking about. So those issues have been studied slow in the earliest interdisciplinary approach.
Okay. If I show you the first two approaches, then there are also a lot of work.
Uh, slow. This is a larger and small scale in scientific computing.
I just released the oldest contributed now from more than 400 papers in this collection.
Also asymptotic analysis early now 1942 for the lab.
[INAUDIBLE] talk to the many people that trying to think, uh, and analyse those patterns.
So what we are thinking, whether possible, we develop a rigorous mathematical analysis to construct the those patterns globally for existence,
stability, regularity in the bifurcation, those this legal analysis is very important somehow in specially for them we want to
understand the transition criteria so we really want really sharp up to certain parameter.
Jim Wang Other project, you always have alums.
Okay, so this is very important to really understand those issues, the regular mathematical analysis.
Another layer of anger behind of that is those kind of way of getting actually the call patterns in
general solution of two dimensional conservation laws listing 20 years that are largely suitable.
Image problem, two dimensional image problem for this hybrid a body conservation laws.
Now this solution came from an analysis of similar solution.
When you're through the analysis and in your macro simulation, you would find by you choose appropriate email.
Problem means is a special classification of data.
It means that you initially not only depend the piece with constant you depend and go okay so with well the by agreement that the 19th century.
Now this is a recently so it's true that you can just choose those appropriate
data you will form all those pet terms assure you in the previous classes.
Okay. So but also the solution actually is the asymptotic state attractors and the building blocks and look for general solutions.
So we have somehow we have to understand those configurations.
Now, first we started for legal reflection. So Bomani, he had idea he had some conjecture in parliament.
Well, he proposed casali conjecture.
So he first posted this as a necessary condition in order to get the legality of this kind of form.
Well, this is necessary. This information will touch those of the weight and the form, the two wave configurations.
So we asked whether Huang's it's a possible you're given insulin shock given then can we find the sleuthing angles such that this is possible.
It turns out this plot, this question actually can translate as Anglo problem.
You can find a critical angle. Where is this angle with angle bigger than that?
A critical angle is always the case. Okay. Less than impossible.
Okay. So this is the first argument here. Now, when this angle the now you are usually this still to you now unique.
But in order to get your neck in the liquid, this one to the strike is stable and this angle goes to pi over two with the other one.
Then we choose to show this shock to the weak shock.
In this podcast, supersonic means hot mess metal, hyperbolic leaching.
Okay, now, when this angle becomes smaller, now this region becomes smaller, smaller than after.
So the angle, this curve will beat that. So this cause the angle cause sonic angle.
So the nightmare conjecture. Well,
maybe they are there because it's a reflection configuration when we which angle bigger than that sonic angle leading because after
that it was this will be after hopefully this would be elliptical then we'll ten he's saying we should change some structure.
So last eight years I joined with the mission to fill the mag as we think about this problem.
So we in two papers and the last papers we first the process issue this is we can really come out of
this kind of configurations and which is stable you mean as Anglo's pi over two for larger angle keys.
So in the wing months ago we just finished our research monograph and the way to solve that is conjecture for potential floor value.
Listen to. Are now formally flashing companies to be much more competitive.
So unlike a incomprehensible case, the competitive option, we still haven't understood the well enough.
Okay. For the apple specially to ask what are the lightest basis of of vorticity allow this 3G and is it possible.
Well, this vortex fruit is called dark, for example.
So this awareness of passion with requires of further understanding with those kinds of things of possibility.
And we can further understand those more complicated, complicated configurations.
So I hope to show you some ideas for trying to solve those kind of multidimensional problem, require overcome some core difficulties.
We we are also facing other mathematics especially are the part of differential equations,
a mixed type composite equation and the fully bounded Antigone.
And then we show the new understanding of complex of our texture and the vorticity and are
bounded up and hammering inequalities are our values and I mean the numerical analysis that.
The conservation law is also. Those glasses have many connections.
For example, I was instead preaching, mostly preaching youth.
Write it down. This is the actual students energy pencil. This is the Einstein pencil here.
That equals four pi because it's a four dimensional case we're talking about.
So those are the equations with the geodesic equation form the core of a mathematical formulation of general relativity.
So if you like nicely achieved this equation, you can really form a system of pairing coupled, nonlinear, hyperbolic, elliptic, positive equation.
So the analysis of X exact solutions of Einstein equations, a wing of the activity of the cosmological cosmology.
So actually last maybe half century that this leads to the predicting our black holes are different.
More of the evolution of the universe.
For example, parallel hawking singularity clearly. Now, the beauty of this equation is that this is a type of conservation of energy and momentum.
By design, just geometrical studies. You knew how to automatically inherently do this conservation of energy and the momentum.
Okay. So this is really the beauty of this equation.
I want to mention that another connection of the a little bit was the Texas variation.
Okay. This is a field of mathematics that that deal with.
It's to me, the function knows it's opposite to the ordinal calculus.
We did weighted like for functions deal with a as you do a functional.
Okay. Those are the function of the a lot applications of problem.
Right that energy of actually functional things physics engineering industry.
Although this turns metric function both the optics and the geometry uh, like in the geometry and also cost functioning optimum optimisation.
So ideas we want to seek to minimise the are critical part of this.
The functional right for greater progress has been made in the last listen to
read the case as I should point Adam as you also fact the man who made a great
contribution in this field especially John Paul he especially the judgement called
not partial him of party convexity and his insights on the weak convergence method.
And as you are responsible for listen at the major advances in this earlier.
Okay. Now what are the connection with conservation law.
Well actually you check the minimise are the critical point that includes after this the critical part as you said,
the fastest system of all on the ecological creation opposite by all of the closing, which is the form as a conservation of form.
Okay. But I like is a significant also seen in see as you find observe the following.
If you give out a like ology, you give a domain.
If like a large you don't have soothing symmetrical properties, then such a symmetric symmetry actually has a corresponding conservation loss.
In other words, an imbalance of variational integrals into ago leads to outcomes a corresponding conservation of a critical point.
Right. The part of which you find a systematic approach to generate all those conservation levels, of which very useful in many areas that.
So also the PD.
Yeah, I would say partial differential equations, uh, came from the, my macroscopic massive scorpion law and the principles right for them.
A Connecticut Theorem The Bosma equation with the max, the wavelength and the last of a Poisson Landau equations,
and then the quantum mechanics on the fuel cell in the tilak equations.
Schrödinger equations are the many thanks to function theory and the local conservation law for probabilities, so forth.
Now, as I point out earlier, that in many cases it's simply chaotic and turbulent locally, but globally well-behaved.
So should you think about something big, a system with buildings, particles?
So it's very difficult to use those kind of macros, copula, to describe it individually.
And put together using some system if you want to understand some behaviour.
You mean the the well the faster the computation the the figure 100 years for that.
Okay. So it's now. So how to how to deal with this one?
This kind of situations. Why this? Well, we use we have a mathematical approach, okay.
To do that. So ideally, you use the macroscopic scaling and the procedures flaws as a microscopic macroscopic process for larger particles.
So our plots include the like averaging expectation closure process.
Now you throw macroscopic scaling, you get a high dynamic limit.
Now, how much innovation, the mass of the flow fields clean and so forth.
So that's the idea. You help us understand the midfield upon your new laws and the principles.
So no macroscopic level. And it turns out, again, a new approach of differential equations and models.
They allow the application, use this idea to model this kind of idea, putting in essential mathematical modelling of various area.
Now I come to the second source for part of differential equations which the as you fly mathematical lesson s and the principles.
Included a compatibility condition, consistent condition,
constraining critical point s whereas mathematical tools instrument and the generalisation completeness,
beauty, chaos, imaginations so with just the example belong to the as if from compatibility condition and the constraints.
So this is a classical problem. Isometric embedding problems which form a following.
If we are given a curve, then we can describe a length on the curve.
Okay, so this is called metrics.
So in that term of definition,
geometry is a fundamental forms that if we want to know curvature of those curves bearing the various curves only later with the,
uh, we call each idea this in mathematical thinking, the fundamental forms.
Okay, so the idea, each idea.
So giving surface, of course we can come to judge idea.
Right now the question for the other way along, we give a metric,
maybe certain curvatures can we find a surfacing earlier water with this matrix and the corresponding curvature?
Okay. So this is a really legalisation problem, right?
You're giving the idea whether I can find a surface for that.
Okay. Now, as we observe nature, as you give it a lot of very sophisticated geometry,
as I show you those of the plant leaves, flowers, even things as the plotting for these.
Right. So the question now, can we produce even more sophisticated therapies?
Are things you can't even reproduce humanely? Right, by having studios as the nature companies structure.
So this question of fundamentally differential in mathematics, very different for geometry and topology,
and now it's essential to why off a sense path to understand evolution of a sophisticated ship of surface and things ship in it you include
a large elasticity the mature size also in biology especially I want to mention as a U.S. Defence Advanced Research Projects Agency,
the form the 20 silly challenged problem in science.
So this problem actually formed in the 10th in the question.
The statement following beauty are strong mathematical clearly as in chop peak and the digital embedding
can give insight into plotting [INAUDIBLE] here where the possible okay so this off so that many
mathematicians I study this is a problem started actually in 1873 I just released to all those contributors
and in positive list I have people and I want the lack of single out of one column is the by Nash.
Okay so Nash put all the following thing in.
Early in the morning in early May. Manafort and Latakia are the key tempo camps seeking asymmetric and Badi into Ukraine's
basing so his claims this being said to be bigger in capital each the larger enough.
Okay so this leaves Syria. Some of you, if you have watched the Hollywood movie A Beautiful Man.
He actually was the subject of that movie of that.
Okay. So for his surname, as you relate how the cumulative column of original history makers is diminishing
numbers smaller as you might be 89 by also as you further improve the dimension.
Now for application, we really want to question whether we can legitimately any dimension,
the lowest dimension but arestill in the space dimension much higher.
So you can visualise this the surface in this high dimension space.
Okay. So open problem now really whether we can reach the optimal dimension.
The first question. The second question. Can we get lazy when we ask a matching painting?
What do you mean, a little weak? Okay. See, when Bebe, for example, thinks that if if we move, give a comp example, Najibullah was a C,
so this possible right now, if we will give example, actually the optimal dimension is to see total in impossible.
You do a comp example if without stretching out by curvature.
So this is actually the current very active research activity to try to understand this problem so difficult.
Why is this problem so difficult?
I want to point out somehow this involves not only a partial differential equation, often mix the hyperbolic types in mathematical terms.
Okay. So here's a fundamental C, I mean, differential geometry that's exist as surface actually with the idea, which I mentioned earlier.
Okay, J.J. Positive, definite answer matches. Okay.
This is possible. You can construct the surface provided that the coefficients each year should satisfy costs for the AC system.
Okay. So a gospel that is just a following.
You give it idea. The second fundamental form each idea of by you through this aiming self by this two equation this
actually compatibly condition from okay you want to find the existence of the surface of this system,
then you need a compatible condition. This came from all quadratic equations.
Then this is a constraint. By the cost equation you give cost curve it.
You may want to find that your surface keep the cost curvature for that.
Okay, it provides. This system is not only a passive differential equation of mixed type in mathematic.
Okay, you have a body equation for that.
So recently we asked we we thought, well, this is the feature of this system.
So very like the equation, we observed the flow dynamics.
So the question whether we can live form of this kind of equation into the flow dynamic formulation.
So we found the following observation we can make have introduced artificially velocity u v and the density low IMU on this form.
Now if we choose constitutively constitutively the Chaplin guess means pleasure and thanks to the from minus one over lo.
Then that decreasing can be formed at the moment.
Conservation of momentum. And the cost equation is really binary law binary releasing it like other basically like equipment,
the conservation in energy for this case, those of it. So then we can park this later from this relation we can also define song speed.
The action is really exactly like a flow dynamics in the transonic flow.
If it's supersonic then relative to car hyperbolic equations subsonic elliptic equation which mix type.
The the mixing slows is a gospel curvature where the gospel is positive.
I'll go to the negative for that. Okay. So with this observation, actually,
we found a solution that we found out one way to establish existing stability of isometric embedding through the weak convergence method.
Now those conflicts tend to sum up curvature. It's very normal for that pole.
You hear the very simple this is the geometry ended up with almost every day.
Not so much. Okay. So one side of the court coverage of politics, the other side, the gospel, the negative.
So this law changes side on the surface. So we cannot avoid padgett differential equation mixed by elliptic hyperbolic equations for that.
Now for high demand in situation much more complicated. Okay.
So besides the gossip equation, quadratic equations very with the compatibility condition for normal butter.
So this is the cognitive equation which forms the bigger systems.
Looks really awful by. I want to point out they have very beautiful geometric structure for their systems.
So this thing that we observe when structure for that, we observe that it is this geometric start to enable us that give us actually get us
to look that up with continuity and the stability of asymmetric bed for the problem.
Okay. So now I come to the PDF with the mathematical tools and the instruments.
Okay. So literally closely following many mathematical problems.
Listen to example the Bengali conjecture by poor man.
He began his work in 2002 and 2003.
He's a pullover really based. I'll let you fly equation to study dynamical behaviour of solution of equation to solve this conjecture.
The other example I don I sincerely enforce most for the emerging manifolds and the like is it's a very important and complex analysis surfaces.
What you see in which the lead leader the lack of value in policy or implement largest union at the single index ceilings.
There's a lot of those kind of mathematical look towards a ladder.
If the really essential are those problems, you mean an adequate relation between algebra,
geometry and the solid, solid concealing slow collective equations.
And now for stochastic project, stochastic PD and also the scientific computing the as actual one of the main
sources for project the for scientific motivation for time through computing.
Now I've got P.D. also, as a mathematician, you really usually will say All I want to do generalisation.
I'll make theatre more beautiful than completeness, right?
I'll curiosity and imaginations. So those kinda ideas actually do actually mean, uh, lead me to generalise general or mathematical.
C earlier and further development and the more wide applications are tend toward the for that.
So I hope I have convinced you.
The calculus there is clearly a mathematical element conveying concern with changes.
Okay. The power differential equations of a mathematician foremost aim to describe a change.
So behave yourself heavily.
Material objects in nature with timescales ranging from picoseconds to millennia and with a lengthy scale ranging from subatomic to astronomical,
can be maddening by nonlinear parts of differential equations equations with similar features, right?
The law of partial differential equation have become increasingly significant within mathematics as a science since.
So the mathematically you only have a partial differential and then it has a little history.
And in listing the case this is related to somebody has to experience a vigilance clause.
The research is a much more at a brisk pace.
So if you want maybe a more kids as of tomorrow, reschedule another event.
Five lectures by five distinguished mathematicians.
The 915 in the morning for talk from them to our morning and the mathematical oracle by professor lawyer carefully from Texas Austin.
So this event was organised by the Oxford Centre for Nonlinear Parts of Differential Equations, which open the October 1st, 2007.
Uh, followed by the actual 2003 International Review of Mathematics in the UK,
which indicated the critical attention is needed to both analysis a part of the future equation in the UK.
So finding intellect is a professor. Sir John Paul.
Thank. Now there are as you as the 11 faculty members in the centre, they are experts in partial differential equation as early as now.
Also, PD is a part of the Mathematical Institute.
There are four centres massive at the centre and the event 11 research groups.
Okay, so this is the picture four new mathematical things that the new beautiful mathematics institute.
And as all of us actually are looking forward to moving in to solving 30 sona of that with support of university staff of a university.
Well, as a cable fellow, I would like to acknowledge the cables plan to establish at the Vantage Study Centre.
Okay. Excellent side, which will hopefully be this building the other side on UMass, the Matthew Institute building, colossal Woodstock.
Right. So for this that yeah, we're very excited about that.
And of course, the more support is required to make this plan to ensure that they are additive for that from community that.
So finally, I would like to thank the University of Oxford, set this post of a professorship for analysis of differential equations.
There I find the Oxford is really unique, a place with a stimulating academic atmosphere.
Firstly, the faculty and the excellent body of students.
I feel very honoured, a great honour to be a part of the Oxford community and looking forward in the coming year to work with all of you
and the Oxford community mobilised to make the UK community and the European Community for challenging scientific,
socio educational issues, especially those issues where partial differential equations can play alone.
Thank you very much. Please don't.