1 00:00:15,180 --> 00:00:23,940 Or if there is a my co-writer honour for me to present my inaugural lecture to you in this very special place. 2 00:00:25,260 --> 00:00:35,490 In this lecture, I would like to discuss the origins, development and loss of a partial differential equations. 3 00:00:35,960 --> 00:00:51,230 You change what? I spent my childhood in the beautiful area of Ardmore Bay where the Changing River Origin, 4 00:00:51,230 --> 00:00:56,150 a different last thing to the Promised Land in the Eastern Time. 5 00:00:56,150 --> 00:01:05,540 I see. This is actually the place to observe an effect of fantastica nature phenomenon. 6 00:01:06,030 --> 00:01:17,479 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. 7 00:01:17,480 --> 00:01:21,530 Had a ball on the middle autumn days in Chinese calendar. 8 00:01:23,060 --> 00:01:31,730 It took the lead to a height of about nine metres our speed at 40 kilometres an hour. 9 00:01:32,540 --> 00:01:35,870 I was truly impressed by this shop flung. 10 00:01:39,470 --> 00:01:48,770 Shop around for a they had to propagate in very organised fashion from east to west. 11 00:01:49,800 --> 00:01:57,110 Alonzo Change. Oliver. The father was my first encounter with this giant wave. 12 00:01:59,100 --> 00:02:04,170 The now immense. Matthew. Tom. This caused shock from shock wave. 13 00:02:07,130 --> 00:02:12,980 The Potomac. Okay. 14 00:02:12,990 --> 00:02:16,319 Anyway, so this is a celestial. 15 00:02:16,320 --> 00:02:19,830 This is a believer, young man. I was a surfer. 16 00:02:19,830 --> 00:02:23,620 And this is a giant impact. The waves are. 17 00:02:28,310 --> 00:02:37,820 So I was also there after that. I was really puzzling about the millions of petals of the form, the body water motion. 18 00:02:38,240 --> 00:02:46,820 Like Leonardo da Vinci, you know, the livers, legs and even the kitchen sink in my house. 19 00:02:47,850 --> 00:02:59,330 Okay. So I was really fascinated with all little phenomena, also little phenomena formed by the air and water along the me. 20 00:02:59,650 --> 00:03:06,200 Okay, so here's a tang form normally formed in west the Pacific Ocean. 21 00:03:06,710 --> 00:03:14,060 Travelling into the East China Sea often slam into the area found to be. 22 00:03:15,620 --> 00:03:23,390 Here's a tornado. Actually, I'll also observed a little thing there in Chicago earlier, United States. 23 00:03:26,080 --> 00:03:33,940 Then shock a wave, along with a supersonic object, as it were, struck them incorrectly. 24 00:03:34,630 --> 00:03:41,560 Here, the elaborating shockwave formed to allow the and a supersonic aircraft also 25 00:03:41,560 --> 00:03:49,390 by various explosion by TNT Sofia's explosion and the supernova explosions. 26 00:03:51,890 --> 00:04:02,959 So. Well, in many cases, it is simply very chaotic and troubling locally. 27 00:04:02,960 --> 00:04:09,860 But globally we are behaviour. So we are not from such the phenomena of the call flocking phenomena by the boats. 28 00:04:10,250 --> 00:04:17,899 You came for me. The values were beautiful patterns globally and as a same situation. 29 00:04:17,900 --> 00:04:21,710 The massive Mexican waves. The pack, the stadium. 30 00:04:23,610 --> 00:04:36,140 So such observation actually has been successfully used by impressionist artists, especially to put in place an artist painter in 19th century. 31 00:04:36,150 --> 00:04:40,590 This example. Okay, the painting made by George Peel. 32 00:04:40,890 --> 00:04:46,690 So long. Then I was also I was curious. 33 00:04:46,770 --> 00:05:00,470 It was about the various sophisticated geometry or structure created by nature for them, created by the plant leaves like here and the flowers. 34 00:05:01,710 --> 00:05:05,460 Even for protein folding structure. 35 00:05:05,720 --> 00:05:11,430 Okay. So we is also a fascinating observation of natural phenomena. 36 00:05:11,880 --> 00:05:19,320 So I was wondering what are underlying national laws behind this phenomena? 37 00:05:20,160 --> 00:05:23,950 Are there any efficient way to describe politics? 38 00:05:24,570 --> 00:05:28,440 Those little dynamical phenomena are changing. 39 00:05:28,440 --> 00:05:35,010 What? So finding an answer to this question was the wing of my child the dream? 40 00:05:36,930 --> 00:05:40,080 So first Tom went out to touch to calculus. 41 00:05:40,650 --> 00:05:46,320 I was immediately attacked by this beautiful mathematical civilian concern with tens. 42 00:05:47,650 --> 00:05:52,540 Went off of the core concepts and calculus is a derivative. 43 00:05:53,040 --> 00:06:09,239 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. 44 00:06:09,240 --> 00:06:13,890 A valuable x is equal to ten delete off the function at the z point. 45 00:06:14,490 --> 00:06:18,810 For example, we consider a moving particle in the plane. 46 00:06:19,620 --> 00:06:25,230 So trajectory of moving particles we call the position function choose our curve. 47 00:06:25,740 --> 00:06:31,020 Such curve mathematically we form as the function like those kind of functions. 48 00:06:31,680 --> 00:06:37,950 Okay, now the telemetry of this function is the velocity of this particle moving particles. 49 00:06:38,520 --> 00:06:45,780 Okay. Now, the second delivery of this position function, the first delivery velocity function is acceleration. 50 00:06:46,290 --> 00:06:55,900 So geometrically, if this is a curve, then the first derivative is the slope of the pending the line. 51 00:06:56,700 --> 00:07:02,280 Okay. The second delay repeats the the later is curvature of this curve. 52 00:07:02,470 --> 00:07:10,590 Okay. This is for that. Now, as far as I know, there are four notions about derivatives. 53 00:07:10,650 --> 00:07:13,840 Okay. So I cleared the hill as a new thing. 54 00:07:14,490 --> 00:07:19,200 Like I use the dart necessarily with you. There is the lab beneath your study. 55 00:07:20,760 --> 00:07:30,750 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. 56 00:07:31,020 --> 00:07:35,460 Okay. Now we are living in the world of three dimensional space. 57 00:07:35,850 --> 00:07:41,340 So minute quantities. They are not only dependent when malleable. 58 00:07:42,570 --> 00:07:47,160 You depend on time and allocation. So we depend on several variables. 59 00:07:47,790 --> 00:07:51,180 Okay. Then this will require the partial derivatives. 60 00:07:51,630 --> 00:07:54,360 Okay. So our project teams are functioning. 61 00:07:55,050 --> 00:08:03,780 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. 62 00:08:04,410 --> 00:08:14,490 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. 63 00:08:14,730 --> 00:08:19,200 For example, this is like a surface. You can only present the by a function for two variables. 64 00:08:19,650 --> 00:08:24,810 So you hold a constant when valuable. There will be this function to present a curve on the surface. 65 00:08:25,530 --> 00:08:29,760 So very positive. Leave it to the slope of this. Curves in the surface. 66 00:08:30,390 --> 00:08:35,370 Okay. With the same with other variable. Handle the delay with the slope of the other variables. 67 00:08:35,760 --> 00:08:46,650 Okay. So for positivity we only use noting along the de oc which first introduced by legend for answering stamp the 86. 68 00:08:47,070 --> 00:08:55,580 Okay, now we use partial derivative with respect for that time means the fixed excellent to take it that it was this valuable. 69 00:08:55,590 --> 00:09:00,600 Then we will on this form some time. Okay. With other x variables we allowed the default. 70 00:09:00,930 --> 00:09:05,540 So this is the same for high dimension case. We can do the same thing now. 71 00:09:05,880 --> 00:09:14,310 Sometime we use the reading. Okay. Now we. How many piling up later is a vector as the partial delivered all those variables. 72 00:09:14,850 --> 00:09:21,690 Okay. So that notion for that. Then when I first learned the first the course of partial differential equation, 73 00:09:22,110 --> 00:09:29,280 I find was that there was clearly I showed a study to answer my child for the question. 74 00:09:29,790 --> 00:09:30,030 Okay. 75 00:09:30,240 --> 00:09:39,930 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. 76 00:09:40,560 --> 00:09:45,690 Okay. The only dependent not only Pam also depend location, other variables. 77 00:09:46,380 --> 00:09:57,240 Okay. Then for example, like those quantity frequencies temporally choose for propagation of sound and heating velocity, density, 78 00:09:57,240 --> 00:10:06,720 pressure and momentum for motion of fluid and displacement, surface tensions covetous for moulting of elastic and materials. 79 00:10:07,650 --> 00:10:16,560 So in order to understand the those changes later, those functions will acquire positively with the project definition of creating that relation. 80 00:10:16,950 --> 00:10:27,479 Involve this, I don't know, functioning and the positive derivative with respect valuable t and x slow equal sign for example here decreasing. 81 00:10:27,480 --> 00:10:31,889 I can write of the as first of the equation releasing the present by as a 82 00:10:31,890 --> 00:10:38,010 function OC between partial delivery with you articulating upstream so forth. 83 00:10:38,880 --> 00:10:46,770 Now I'm in a physical system. Usually we want to know more than one and no functions which satisfy more than one relations. 84 00:10:47,450 --> 00:10:51,780 Okay then this kind of positive system about the differential equations. 85 00:10:52,560 --> 00:10:58,740 So clearly positive involves the sum and no function of depend more than one variables. 86 00:10:59,010 --> 00:11:04,560 So the more general automotive initial equation the way learn that in differential equation. 87 00:11:05,460 --> 00:11:15,660 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 88 00:11:15,660 --> 00:11:24,120 is a partial differential equations she look at that of me with a full of puzzlement and ask why you study different interpretation, 89 00:11:24,120 --> 00:11:33,509 partially not the holy. Now there was a little galaxy that turns out that we had a really very pleasant time together. 90 00:11:33,510 --> 00:11:34,919 Finally, agree with me, 91 00:11:34,920 --> 00:11:48,170 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 92 00:11:48,180 --> 00:12:01,470 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. 93 00:12:02,280 --> 00:12:05,880 So at the time I've got his nose in the he also to the plus. 94 00:12:06,110 --> 00:12:10,320 On the minus sign. Of course, our time equals are much longer than nowadays. 95 00:12:10,540 --> 00:12:11,500 You mean plus time? 96 00:12:12,340 --> 00:12:21,250 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. 97 00:12:23,480 --> 00:12:31,090 Now, partial differential equation allows actually mainly for two sources weighing the 98 00:12:31,090 --> 00:12:37,000 from the first one following fundamental law of and the principles in the sciences, 99 00:12:37,600 --> 00:12:47,050 including macroscopic law and principles macroscopic massless coupling law principles mach also 100 00:12:48,070 --> 00:12:55,180 macroscopic scaling and the proceed procedures from macroscopic massless approach process. 101 00:12:55,930 --> 00:13:06,280 The second source of fundamental mathematics itself as the mathematical rationales are principles include compatibility condition, 102 00:13:06,280 --> 00:13:16,510 consistent condition, construing critical points as the warehouse, the mathematical tools instrument and the generalisation, 103 00:13:16,960 --> 00:13:21,220 completeness, and also beauty and the curiosity and imagination. 104 00:13:23,120 --> 00:13:33,010 There is a study of partial differential equation as you start that 18 centuries in the work about Villa, 105 00:13:33,190 --> 00:13:43,540 the Lambert like lodging Laplace and the mini mathematicians essential tool in the description of mechanics of continuum. 106 00:13:45,400 --> 00:13:49,990 More generally as the principal mode of analysis of modes and sizes. 107 00:13:51,220 --> 00:14:00,910 This principal modes as you're still going off of the fundamental concerns of the development of PAGEAU differentiated pacing until today, 108 00:14:02,020 --> 00:14:12,250 beginning the middle of 19th centuries, particularly with the work of the iMac as a pre become essential tool and other balance of mathematics. 109 00:14:13,240 --> 00:14:18,460 This duality of viewpoint has a being an essential tool. 110 00:14:18,490 --> 00:14:21,550 Analysis of Passive Differential Equation. Until today. 111 00:14:22,090 --> 00:14:28,989 Right now for namely a posture differential equation, satisfactory ceiling has it being established. 112 00:14:28,990 --> 00:14:34,080 So no effort of several generations. May not most. 113 00:14:34,080 --> 00:14:39,180 The part of differential equation, as I mentioned, for alternative sources naturally are non-linear. 114 00:14:39,330 --> 00:14:46,200 So it's much more challenge, more difficult buying for application, more important, 115 00:14:47,130 --> 00:14:52,440 but now is really coming to mainstream research in mathematics and other sciences for that. 116 00:14:54,740 --> 00:15:05,810 Now I present the wings and pose of the colossal nonlinear partial differential equation from this category. 117 00:15:06,710 --> 00:15:12,350 Is from the macroscopic law in computer physics. 118 00:15:13,100 --> 00:15:16,640 So this is the law, the car conservation law. Okay. 119 00:15:17,360 --> 00:15:22,880 So it's a third rate of changes of the total amount of certain quantity. 120 00:15:23,630 --> 00:15:32,270 Kong pending a fixed domain for them in this domain equal to the flux of this quantity across the boundary of the e.g. 121 00:15:33,550 --> 00:15:42,460 The other war. The amount of such quantity and leaching can be measured by accounting for how much of it is a current present. 122 00:15:43,760 --> 00:15:50,300 And how much of it they are, the leave, the leading and if it's the period of time. 123 00:15:51,350 --> 00:15:56,720 Well, typical examples to suit your fundamental of nature the conservation law, 124 00:15:56,930 --> 00:16:03,080 mass conservation momentum, conservation of energy which it can formulate that this form. 125 00:16:04,790 --> 00:16:08,540 Mathematically, we can farm into this conservation as a following. 126 00:16:08,840 --> 00:16:13,370 So pull out the know that you as the benefit of this quantity value integration. 127 00:16:13,540 --> 00:16:16,550 This quantity as a whole. 128 00:16:16,640 --> 00:16:19,670 The math is the total amount of this quantity. 129 00:16:20,540 --> 00:16:24,230 Now the chanting method means you pick a delimited with a square t, right. 130 00:16:24,440 --> 00:16:31,760 So the statement that the total rate of Chanda to the equal pool, the flag of the quantity across the boundary. 131 00:16:33,220 --> 00:16:37,930 Okay. So this is a mathematical statement for this conservation law. 132 00:16:38,650 --> 00:16:46,570 Now, use, uh, uh, simple calculus manipulations and the use this law holds for any domain. 133 00:16:46,780 --> 00:16:50,709 Okay. Then we can build from here together. 134 00:16:50,710 --> 00:16:55,240 This first are the partial differential equation f of the function of view in general. 135 00:16:55,540 --> 00:16:59,980 So this is not only the partial differential equations, not in sorting system. 136 00:17:00,460 --> 00:17:07,470 Actually, they are more than two more than one quantities. 137 00:17:07,870 --> 00:17:10,960 And instead of our several laws now in the farm system. 138 00:17:11,380 --> 00:17:14,770 Okay. So then you can learn the system of conservation laws. 139 00:17:15,080 --> 00:17:26,390 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. 140 00:17:28,280 --> 00:17:39,260 One typical example of that is a like fishing for compressible flow, which consists useless formulations and constantly mass momentum and energy. 141 00:17:40,220 --> 00:17:48,140 This is a system harvest the flow flowing convective motion dominate the diffusion dissipation. 142 00:17:48,950 --> 00:17:57,560 So for those emotions specifically, while the shockwaves like gases elastica flow and the sheer lattice, so many scientists as you have studied this, 143 00:17:58,340 --> 00:18:06,959 has been studied this this equation slowed a long time ago and thus made a great contribution, 144 00:18:06,960 --> 00:18:16,400 especially many British scientist, that made the important contribution for studying the analysis of this systems. 145 00:18:16,670 --> 00:18:19,250 I would like special mention George Stokes. 146 00:18:19,730 --> 00:18:29,389 He was the probably first scientist to detect lies, discontinue jumps, now call shockwaves mathematically slow. 147 00:18:29,390 --> 00:18:39,160 This equation are now mathematical rigorous mathematical study for the constant the knowledge conservation. 148 00:18:39,170 --> 00:18:47,690 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, 149 00:18:48,050 --> 00:18:59,660 Hope, Atlantic and many others. So I merely say that the fantasy of them has been established slow the effort of several generations. 150 00:19:00,860 --> 00:19:11,179 For that now greater progress has been made. The following dimensional case a space start of flow started with GlaxoSmithKline leasing 151 00:19:11,180 --> 00:19:17,090 the by Bianchini please now for we are posting this for solution of small total variations. 152 00:19:18,220 --> 00:19:25,520 As the warehouse now existence the need for solution of large oscillations by now the modelling and function 153 00:19:25,520 --> 00:19:31,530 and that they take a technique called compensate companies must still they are many important problem. 154 00:19:31,540 --> 00:19:35,710 I open like a uniqueness stability for solution of large oscillations. 155 00:19:36,530 --> 00:19:39,100 Now formatted the magic is doing much more complicated. 156 00:19:39,450 --> 00:19:45,790 I can more challenge but scientifically more appealing since we are living the world with the full of shock waves. 157 00:19:46,300 --> 00:19:54,490 As I'm sure you're just existing those minute, those kind of high speed objects, fly objects. 158 00:19:54,820 --> 00:19:58,630 I also like various explanations for that, right? 159 00:19:59,020 --> 00:20:01,700 So when those kind of shockwave hits, 160 00:20:01,750 --> 00:20:15,860 some of the objects obstacles special are some special optical was geometry or them some flight objects metres of shockwaves then shock. 161 00:20:15,860 --> 00:20:19,450 And the flashing the flashing problem allows that. 162 00:20:21,790 --> 00:20:31,810 So one of the fundamental problems shock, shock in reflecting the phrasing by two dimensional which is set up is the following. 163 00:20:32,170 --> 00:20:37,210 Here are the two dimension weight to the plane shock moving from far field at the constant speed. 164 00:20:37,570 --> 00:20:40,840 So before. Let's do this. Wait. This is a similar plane shock. 165 00:20:40,840 --> 00:20:41,440 Just moving. 166 00:20:42,020 --> 00:20:54,730 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. 167 00:20:56,140 --> 00:21:03,220 So this setup is really simple, right? Well, however, the the wave patterns are extremely complicated. 168 00:21:03,700 --> 00:21:14,230 Right. The first two scientists who observed the complexity of those configuration was owns the Mafia in 1879. 169 00:21:14,760 --> 00:21:19,790 And he's actually experimentally frank to see if this experiment is out. 170 00:21:19,810 --> 00:21:27,250 He found a tool. Uh, two different types of configuration knowledge, unification, model fractions. 171 00:21:27,790 --> 00:21:38,050 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. 172 00:21:38,740 --> 00:21:45,700 But especially the experimental scientists, they do various experiments in the labs around the world. 173 00:21:46,420 --> 00:21:49,690 What did they find? A situation much more complicated, Ma observed. 174 00:21:50,620 --> 00:21:54,230 Okay, so one case is like this. 175 00:21:54,280 --> 00:21:59,800 The shock wang. This angle large enough for if we fixed the stress of shock, 176 00:22:00,220 --> 00:22:07,660 they after later power what they find that this new shock wave formed this caused a leaf rather shockwaves. 177 00:22:08,170 --> 00:22:13,690 So Wang, this information will move all the left arm of the backwards for that other direction. 178 00:22:14,080 --> 00:22:16,780 Right. So this is a picture of Lily. Beautiful. 179 00:22:16,780 --> 00:22:23,950 And then you can see this Wellesley of similar and similar flowers most here nobody living very nicely. 180 00:22:24,550 --> 00:22:33,290 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. 181 00:22:33,310 --> 00:22:37,880 You can form the full wave configuration SUNY Sharks and the wing vortex sheets. 182 00:22:38,380 --> 00:22:40,930 So now because it's simple modification. 183 00:22:41,500 --> 00:22:49,510 Now, if you could decrease the angle a little more that evening, now you observe the this four wave configuration, 184 00:22:49,530 --> 00:23:02,799 the vorticity waves form that even very recently and new padding has been discovered by the modelling as and your computation and 185 00:23:02,800 --> 00:23:13,210 experiments experiment through results for them the behind the mask them the form the supersonic bubbles from the generalised generator, 186 00:23:13,220 --> 00:23:25,720 the theories of shockwaves about that. So the scientific issue we really want to understand the first the structure of those patterns. 187 00:23:26,170 --> 00:23:29,530 The second transition collectively between the patterns. 188 00:23:30,930 --> 00:23:35,850 Man. We want to live in a hopefully we want to understand the dependence of those things. 189 00:23:36,240 --> 00:23:39,250 Are those parameters very important? 190 00:23:39,270 --> 00:23:43,290 We can model which angle. Now I show you a picture that is a dependent. 191 00:23:43,290 --> 00:23:50,370 We can also depend how strong string as the the the stress of the shock wave, the tonality, 192 00:23:50,820 --> 00:23:56,100 the present, the fruit America my number and it's all dependent with a kind of flow. 193 00:23:56,250 --> 00:24:03,870 Okay, we're talking about. So those issues have been studied slow in the earliest interdisciplinary approach. 194 00:24:04,590 --> 00:24:09,090 Okay. If I show you the first two approaches, then there are also a lot of work. 195 00:24:09,900 --> 00:24:13,980 Uh, slow. This is a larger and small scale in scientific computing. 196 00:24:14,340 --> 00:24:19,350 I just released the oldest contributed now from more than 400 papers in this collection. 197 00:24:19,860 --> 00:24:24,059 Also asymptotic analysis early now 1942 for the lab. 198 00:24:24,060 --> 00:24:31,590 [INAUDIBLE] talk to the many people that trying to think, uh, and analyse those patterns. 199 00:24:32,490 --> 00:24:43,440 So what we are thinking, whether possible, we develop a rigorous mathematical analysis to construct the those patterns globally for existence, 200 00:24:43,500 --> 00:24:52,469 stability, regularity in the bifurcation, those this legal analysis is very important somehow in specially for them we want to 201 00:24:52,470 --> 00:24:57,990 understand the transition criteria so we really want really sharp up to certain parameter. 202 00:24:58,110 --> 00:25:02,200 Jim Wang Other project, you always have alums. 203 00:25:02,280 --> 00:25:09,030 Okay, so this is very important to really understand those issues, the regular mathematical analysis. 204 00:25:11,130 --> 00:25:18,000 Another layer of anger behind of that is those kind of way of getting actually the call patterns in 205 00:25:18,000 --> 00:25:25,330 general solution of two dimensional conservation laws listing 20 years that are largely suitable. 206 00:25:25,350 --> 00:25:30,120 Image problem, two dimensional image problem for this hybrid a body conservation laws. 207 00:25:30,660 --> 00:25:33,230 Now this solution came from an analysis of similar solution. 208 00:25:33,690 --> 00:25:39,910 When you're through the analysis and in your macro simulation, you would find by you choose appropriate email. 209 00:25:39,930 --> 00:25:43,470 Problem means is a special classification of data. 210 00:25:43,650 --> 00:25:53,580 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. 211 00:25:54,810 --> 00:25:59,820 Now this is a recently so it's true that you can just choose those appropriate 212 00:25:59,820 --> 00:26:05,160 data you will form all those pet terms assure you in the previous classes. 213 00:26:05,470 --> 00:26:15,690 Okay. So but also the solution actually is the asymptotic state attractors and the building blocks and look for general solutions. 214 00:26:15,960 --> 00:26:19,290 So we have somehow we have to understand those configurations. 215 00:26:20,300 --> 00:26:27,560 Now, first we started for legal reflection. So Bomani, he had idea he had some conjecture in parliament. 216 00:26:27,920 --> 00:26:31,430 Well, he proposed casali conjecture. 217 00:26:32,420 --> 00:26:38,330 So he first posted this as a necessary condition in order to get the legality of this kind of form. 218 00:26:38,570 --> 00:26:46,610 Well, this is necessary. This information will touch those of the weight and the form, the two wave configurations. 219 00:26:47,000 --> 00:26:56,090 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. 220 00:26:57,160 --> 00:27:02,350 It turns out this plot, this question actually can translate as Anglo problem. 221 00:27:02,710 --> 00:27:06,910 You can find a critical angle. Where is this angle with angle bigger than that? 222 00:27:06,910 --> 00:27:11,780 A critical angle is always the case. Okay. Less than impossible. 223 00:27:12,280 --> 00:27:20,290 Okay. So this is the first argument here. Now, when this angle the now you are usually this still to you now unique. 224 00:27:20,300 --> 00:27:27,880 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. 225 00:27:28,240 --> 00:27:31,780 Then we choose to show this shock to the weak shock. 226 00:27:31,990 --> 00:27:36,820 In this podcast, supersonic means hot mess metal, hyperbolic leaching. 227 00:27:37,060 --> 00:27:42,490 Okay, now, when this angle becomes smaller, now this region becomes smaller, smaller than after. 228 00:27:42,490 --> 00:27:48,070 So the angle, this curve will beat that. So this cause the angle cause sonic angle. 229 00:27:49,180 --> 00:27:51,250 So the nightmare conjecture. Well, 230 00:27:51,820 --> 00:28:01,809 maybe they are there because it's a reflection configuration when we which angle bigger than that sonic angle leading because after 231 00:28:01,810 --> 00:28:08,020 that it was this will be after hopefully this would be elliptical then we'll ten he's saying we should change some structure. 232 00:28:09,090 --> 00:28:16,860 So last eight years I joined with the mission to fill the mag as we think about this problem. 233 00:28:17,250 --> 00:28:26,430 So we in two papers and the last papers we first the process issue this is we can really come out of 234 00:28:26,430 --> 00:28:32,550 this kind of configurations and which is stable you mean as Anglo's pi over two for larger angle keys. 235 00:28:33,180 --> 00:28:42,780 So in the wing months ago we just finished our research monograph and the way to solve that is conjecture for potential floor value. 236 00:28:42,780 --> 00:28:48,400 Listen to. Are now formally flashing companies to be much more competitive. 237 00:28:48,910 --> 00:28:57,010 So unlike a incomprehensible case, the competitive option, we still haven't understood the well enough. 238 00:28:57,400 --> 00:29:05,490 Okay. For the apple specially to ask what are the lightest basis of of vorticity allow this 3G and is it possible. 239 00:29:05,490 --> 00:29:09,700 Well, this vortex fruit is called dark, for example. 240 00:29:09,970 --> 00:29:15,590 So this awareness of passion with requires of further understanding with those kinds of things of possibility. 241 00:29:15,590 --> 00:29:20,380 And we can further understand those more complicated, complicated configurations. 242 00:29:20,710 --> 00:29:32,290 So I hope to show you some ideas for trying to solve those kind of multidimensional problem, require overcome some core difficulties. 243 00:29:32,290 --> 00:29:38,050 We we are also facing other mathematics especially are the part of differential equations, 244 00:29:38,710 --> 00:29:43,920 a mixed type composite equation and the fully bounded Antigone. 245 00:29:44,470 --> 00:29:50,290 And then we show the new understanding of complex of our texture and the vorticity and are 246 00:29:51,970 --> 00:29:59,230 bounded up and hammering inequalities are our values and I mean the numerical analysis that. 247 00:30:01,780 --> 00:30:08,780 The conservation law is also. Those glasses have many connections. 248 00:30:08,840 --> 00:30:13,310 For example, I was instead preaching, mostly preaching youth. 249 00:30:14,650 --> 00:30:19,590 Write it down. This is the actual students energy pencil. This is the Einstein pencil here. 250 00:30:19,990 --> 00:30:23,730 That equals four pi because it's a four dimensional case we're talking about. 251 00:30:24,240 --> 00:30:31,710 So those are the equations with the geodesic equation form the core of a mathematical formulation of general relativity. 252 00:30:32,520 --> 00:30:40,680 So if you like nicely achieved this equation, you can really form a system of pairing coupled, nonlinear, hyperbolic, elliptic, positive equation. 253 00:30:41,520 --> 00:30:51,090 So the analysis of X exact solutions of Einstein equations, a wing of the activity of the cosmological cosmology. 254 00:30:51,600 --> 00:30:58,770 So actually last maybe half century that this leads to the predicting our black holes are different. 255 00:30:58,770 --> 00:31:01,770 More of the evolution of the universe. 256 00:31:01,770 --> 00:31:12,080 For example, parallel hawking singularity clearly. Now, the beauty of this equation is that this is a type of conservation of energy and momentum. 257 00:31:13,550 --> 00:31:21,710 By design, just geometrical studies. You knew how to automatically inherently do this conservation of energy and the momentum. 258 00:31:22,020 --> 00:31:25,820 Okay. So this is really the beauty of this equation. 259 00:31:26,000 --> 00:31:31,990 I want to mention that another connection of the a little bit was the Texas variation. 260 00:31:32,300 --> 00:31:36,440 Okay. This is a field of mathematics that that deal with. 261 00:31:36,800 --> 00:31:41,810 It's to me, the function knows it's opposite to the ordinal calculus. 262 00:31:41,840 --> 00:31:45,820 We did weighted like for functions deal with a as you do a functional. 263 00:31:46,430 --> 00:31:51,500 Okay. Those are the function of the a lot applications of problem. 264 00:31:51,530 --> 00:31:56,240 Right that energy of actually functional things physics engineering industry. 265 00:31:56,660 --> 00:32:07,460 Although this turns metric function both the optics and the geometry uh, like in the geometry and also cost functioning optimum optimisation. 266 00:32:08,180 --> 00:32:13,610 So ideas we want to seek to minimise the are critical part of this. 267 00:32:13,610 --> 00:32:20,290 The functional right for greater progress has been made in the last listen to 268 00:32:20,350 --> 00:32:24,919 read the case as I should point Adam as you also fact the man who made a great 269 00:32:24,920 --> 00:32:32,450 contribution in this field especially John Paul he especially the judgement called 270 00:32:32,450 --> 00:32:38,000 not partial him of party convexity and his insights on the weak convergence method. 271 00:32:38,420 --> 00:32:44,030 And as you are responsible for listen at the major advances in this earlier. 272 00:32:44,840 --> 00:32:47,150 Okay. Now what are the connection with conservation law. 273 00:32:47,570 --> 00:32:57,049 Well actually you check the minimise are the critical point that includes after this the critical part as you said, 274 00:32:57,050 --> 00:33:05,390 the fastest system of all on the ecological creation opposite by all of the closing, which is the form as a conservation of form. 275 00:33:05,930 --> 00:33:12,830 Okay. But I like is a significant also seen in see as you find observe the following. 276 00:33:13,340 --> 00:33:16,070 If you give out a like ology, you give a domain. 277 00:33:16,310 --> 00:33:26,540 If like a large you don't have soothing symmetrical properties, then such a symmetric symmetry actually has a corresponding conservation loss. 278 00:33:27,140 --> 00:33:36,850 In other words, an imbalance of variational integrals into ago leads to outcomes a corresponding conservation of a critical point. 279 00:33:37,380 --> 00:33:47,570 Right. The part of which you find a systematic approach to generate all those conservation levels, of which very useful in many areas that. 280 00:33:48,890 --> 00:33:50,540 So also the PD. 281 00:33:50,570 --> 00:34:00,110 Yeah, I would say partial differential equations, uh, came from the, my macroscopic massive scorpion law and the principles right for them. 282 00:34:00,110 --> 00:34:08,690 A Connecticut Theorem The Bosma equation with the max, the wavelength and the last of a Poisson Landau equations, 283 00:34:09,110 --> 00:34:12,770 and then the quantum mechanics on the fuel cell in the tilak equations. 284 00:34:13,010 --> 00:34:19,820 Schrödinger equations are the many thanks to function theory and the local conservation law for probabilities, so forth. 285 00:34:21,890 --> 00:34:30,470 Now, as I point out earlier, that in many cases it's simply chaotic and turbulent locally, but globally well-behaved. 286 00:34:31,010 --> 00:34:35,390 So should you think about something big, a system with buildings, particles? 287 00:34:36,110 --> 00:34:40,550 So it's very difficult to use those kind of macros, copula, to describe it individually. 288 00:34:40,850 --> 00:34:45,110 And put together using some system if you want to understand some behaviour. 289 00:34:45,440 --> 00:34:50,600 You mean the the well the faster the computation the the figure 100 years for that. 290 00:34:51,170 --> 00:34:56,270 Okay. So it's now. So how to how to deal with this one? 291 00:34:56,300 --> 00:35:02,520 This kind of situations. Why this? Well, we use we have a mathematical approach, okay. 292 00:35:03,050 --> 00:35:14,570 To do that. So ideally, you use the macroscopic scaling and the procedures flaws as a microscopic macroscopic process for larger particles. 293 00:35:15,200 --> 00:35:20,060 So our plots include the like averaging expectation closure process. 294 00:35:20,540 --> 00:35:25,160 Now you throw macroscopic scaling, you get a high dynamic limit. 295 00:35:25,550 --> 00:35:29,540 Now, how much innovation, the mass of the flow fields clean and so forth. 296 00:35:29,930 --> 00:35:36,080 So that's the idea. You help us understand the midfield upon your new laws and the principles. 297 00:35:36,350 --> 00:35:41,900 So no macroscopic level. And it turns out, again, a new approach of differential equations and models. 298 00:35:42,230 --> 00:35:51,080 They allow the application, use this idea to model this kind of idea, putting in essential mathematical modelling of various area. 299 00:35:53,220 --> 00:36:02,370 Now I come to the second source for part of differential equations which the as you fly mathematical lesson s and the principles. 300 00:36:03,360 --> 00:36:06,810 Included a compatibility condition, consistent condition, 301 00:36:07,590 --> 00:36:15,270 constraining critical point s whereas mathematical tools instrument and the generalisation completeness, 302 00:36:15,510 --> 00:36:24,330 beauty, chaos, imaginations so with just the example belong to the as if from compatibility condition and the constraints. 303 00:36:25,450 --> 00:36:32,380 So this is a classical problem. Isometric embedding problems which form a following. 304 00:36:33,070 --> 00:36:39,370 If we are given a curve, then we can describe a length on the curve. 305 00:36:39,880 --> 00:36:43,290 Okay, so this is called metrics. 306 00:36:43,810 --> 00:36:46,150 So in that term of definition, 307 00:36:46,150 --> 00:36:54,820 geometry is a fundamental forms that if we want to know curvature of those curves bearing the various curves only later with the, 308 00:36:55,510 --> 00:37:00,820 uh, we call each idea this in mathematical thinking, the fundamental forms. 309 00:37:01,210 --> 00:37:04,630 Okay, so the idea, each idea. 310 00:37:05,200 --> 00:37:09,110 So giving surface, of course we can come to judge idea. 311 00:37:09,670 --> 00:37:13,780 Right now the question for the other way along, we give a metric, 312 00:37:14,710 --> 00:37:23,560 maybe certain curvatures can we find a surfacing earlier water with this matrix and the corresponding curvature? 313 00:37:24,610 --> 00:37:28,150 Okay. So this is a really legalisation problem, right? 314 00:37:28,390 --> 00:37:31,840 You're giving the idea whether I can find a surface for that. 315 00:37:32,830 --> 00:37:40,330 Okay. Now, as we observe nature, as you give it a lot of very sophisticated geometry, 316 00:37:40,480 --> 00:37:47,710 as I show you those of the plant leaves, flowers, even things as the plotting for these. 317 00:37:48,700 --> 00:37:54,560 Right. So the question now, can we produce even more sophisticated therapies? 318 00:37:55,270 --> 00:38:02,660 Are things you can't even reproduce humanely? Right, by having studios as the nature companies structure. 319 00:38:03,570 --> 00:38:10,530 So this question of fundamentally differential in mathematics, very different for geometry and topology, 320 00:38:10,990 --> 00:38:22,640 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 321 00:38:22,650 --> 00:38:33,150 a large elasticity the mature size also in biology especially I want to mention as a U.S. Defence Advanced Research Projects Agency, 322 00:38:33,900 --> 00:38:37,710 the form the 20 silly challenged problem in science. 323 00:38:38,160 --> 00:38:42,210 So this problem actually formed in the 10th in the question. 324 00:38:42,780 --> 00:38:49,620 The statement following beauty are strong mathematical clearly as in chop peak and the digital embedding 325 00:38:50,460 --> 00:38:56,280 can give insight into plotting [INAUDIBLE] here where the possible okay so this off so that many 326 00:38:56,280 --> 00:39:05,549 mathematicians I study this is a problem started actually in 1873 I just released to all those contributors 327 00:39:05,550 --> 00:39:14,070 and in positive list I have people and I want the lack of single out of one column is the by Nash. 328 00:39:14,400 --> 00:39:17,610 Okay so Nash put all the following thing in. 329 00:39:18,720 --> 00:39:30,150 Early in the morning in early May. Manafort and Latakia are the key tempo camps seeking asymmetric and Badi into Ukraine's 330 00:39:30,170 --> 00:39:37,290 basing so his claims this being said to be bigger in capital each the larger enough. 331 00:39:37,790 --> 00:39:46,580 Okay so this leaves Syria. Some of you, if you have watched the Hollywood movie A Beautiful Man. 332 00:39:46,850 --> 00:39:50,350 He actually was the subject of that movie of that. 333 00:39:50,550 --> 00:39:59,690 Okay. So for his surname, as you relate how the cumulative column of original history makers is diminishing 334 00:39:59,690 --> 00:40:05,540 numbers smaller as you might be 89 by also as you further improve the dimension. 335 00:40:06,080 --> 00:40:12,440 Now for application, we really want to question whether we can legitimately any dimension, 336 00:40:12,710 --> 00:40:17,000 the lowest dimension but arestill in the space dimension much higher. 337 00:40:17,010 --> 00:40:21,500 So you can visualise this the surface in this high dimension space. 338 00:40:22,190 --> 00:40:27,390 Okay. So open problem now really whether we can reach the optimal dimension. 339 00:40:27,780 --> 00:40:32,880 The first question. The second question. Can we get lazy when we ask a matching painting? 340 00:40:33,540 --> 00:40:42,389 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, 341 00:40:42,390 --> 00:40:48,990 so this possible right now, if we will give example, actually the optimal dimension is to see total in impossible. 342 00:40:49,380 --> 00:40:53,340 You do a comp example if without stretching out by curvature. 343 00:40:54,480 --> 00:41:04,000 So this is actually the current very active research activity to try to understand this problem so difficult. 344 00:41:04,020 --> 00:41:05,760 Why is this problem so difficult? 345 00:41:05,820 --> 00:41:14,010 I want to point out somehow this involves not only a partial differential equation, often mix the hyperbolic types in mathematical terms. 346 00:41:14,370 --> 00:41:25,890 Okay. So here's a fundamental C, I mean, differential geometry that's exist as surface actually with the idea, which I mentioned earlier. 347 00:41:26,610 --> 00:41:29,610 Okay, J.J. Positive, definite answer matches. Okay. 348 00:41:31,320 --> 00:41:40,830 This is possible. You can construct the surface provided that the coefficients each year should satisfy costs for the AC system. 349 00:41:41,640 --> 00:41:46,050 Okay. So a gospel that is just a following. 350 00:41:46,410 --> 00:41:54,090 You give it idea. The second fundamental form each idea of by you through this aiming self by this two equation this 351 00:41:54,090 --> 00:42:00,600 actually compatibly condition from okay you want to find the existence of the surface of this system, 352 00:42:01,350 --> 00:42:05,940 then you need a compatible condition. This came from all quadratic equations. 353 00:42:06,390 --> 00:42:09,780 Then this is a constraint. By the cost equation you give cost curve it. 354 00:42:09,780 --> 00:42:13,530 You may want to find that your surface keep the cost curvature for that. 355 00:42:14,250 --> 00:42:20,690 Okay, it provides. This system is not only a passive differential equation of mixed type in mathematic. 356 00:42:21,170 --> 00:42:24,270 Okay, you have a body equation for that. 357 00:42:26,060 --> 00:42:31,040 So recently we asked we we thought, well, this is the feature of this system. 358 00:42:31,040 --> 00:42:34,310 So very like the equation, we observed the flow dynamics. 359 00:42:34,850 --> 00:42:40,250 So the question whether we can live form of this kind of equation into the flow dynamic formulation. 360 00:42:40,820 --> 00:42:50,420 So we found the following observation we can make have introduced artificially velocity u v and the density low IMU on this form. 361 00:42:51,140 --> 00:43:00,710 Now if we choose constitutively constitutively the Chaplin guess means pleasure and thanks to the from minus one over lo. 362 00:43:01,700 --> 00:43:05,060 Then that decreasing can be formed at the moment. 363 00:43:05,270 --> 00:43:16,179 Conservation of momentum. And the cost equation is really binary law binary releasing it like other basically like equipment, 364 00:43:16,180 --> 00:43:24,760 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. 365 00:43:25,280 --> 00:43:29,180 The action is really exactly like a flow dynamics in the transonic flow. 366 00:43:29,810 --> 00:43:36,170 If it's supersonic then relative to car hyperbolic equations subsonic elliptic equation which mix type. 367 00:43:37,060 --> 00:43:42,300 The the mixing slows is a gospel curvature where the gospel is positive. 368 00:43:42,310 --> 00:43:47,049 I'll go to the negative for that. Okay. So with this observation, actually, 369 00:43:47,050 --> 00:43:57,280 we found a solution that we found out one way to establish existing stability of isometric embedding through the weak convergence method. 370 00:43:58,200 --> 00:44:03,099 Now those conflicts tend to sum up curvature. It's very normal for that pole. 371 00:44:03,100 --> 00:44:07,820 You hear the very simple this is the geometry ended up with almost every day. 372 00:44:08,460 --> 00:44:14,920 Not so much. Okay. So one side of the court coverage of politics, the other side, the gospel, the negative. 373 00:44:15,150 --> 00:44:24,090 So this law changes side on the surface. So we cannot avoid padgett differential equation mixed by elliptic hyperbolic equations for that. 374 00:44:25,150 --> 00:44:27,940 Now for high demand in situation much more complicated. Okay. 375 00:44:28,240 --> 00:44:34,780 So besides the gossip equation, quadratic equations very with the compatibility condition for normal butter. 376 00:44:35,500 --> 00:44:39,310 So this is the cognitive equation which forms the bigger systems. 377 00:44:39,340 --> 00:44:46,920 Looks really awful by. I want to point out they have very beautiful geometric structure for their systems. 378 00:44:47,700 --> 00:44:57,509 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 379 00:44:57,510 --> 00:45:02,460 to look that up with continuity and the stability of asymmetric bed for the problem. 380 00:45:04,290 --> 00:45:10,230 Okay. So now I come to the PDF with the mathematical tools and the instruments. 381 00:45:10,760 --> 00:45:15,750 Okay. So literally closely following many mathematical problems. 382 00:45:17,550 --> 00:45:21,480 Listen to example the Bengali conjecture by poor man. 383 00:45:22,380 --> 00:45:26,700 He began his work in 2002 and 2003. 384 00:45:27,360 --> 00:45:37,260 He's a pullover really based. I'll let you fly equation to study dynamical behaviour of solution of equation to solve this conjecture. 385 00:45:38,570 --> 00:45:47,520 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. 386 00:45:47,830 --> 00:45:57,330 What you see in which the lead leader the lack of value in policy or implement largest union at the single index ceilings. 387 00:45:57,910 --> 00:46:02,570 There's a lot of those kind of mathematical look towards a ladder. 388 00:46:03,670 --> 00:46:08,730 If the really essential are those problems, you mean an adequate relation between algebra, 389 00:46:08,740 --> 00:46:13,600 geometry and the solid, solid concealing slow collective equations. 390 00:46:14,350 --> 00:46:22,870 And now for stochastic project, stochastic PD and also the scientific computing the as actual one of the main 391 00:46:23,530 --> 00:46:28,180 sources for project the for scientific motivation for time through computing. 392 00:46:30,410 --> 00:46:37,460 Now I've got P.D. also, as a mathematician, you really usually will say All I want to do generalisation. 393 00:46:37,760 --> 00:46:41,569 I'll make theatre more beautiful than completeness, right? 394 00:46:41,570 --> 00:46:53,000 I'll curiosity and imaginations. So those kinda ideas actually do actually mean, uh, lead me to generalise general or mathematical. 395 00:46:53,000 --> 00:46:59,090 C earlier and further development and the more wide applications are tend toward the for that. 396 00:47:01,850 --> 00:47:04,940 So I hope I have convinced you. 397 00:47:05,510 --> 00:47:11,630 The calculus there is clearly a mathematical element conveying concern with changes. 398 00:47:12,440 --> 00:47:18,860 Okay. The power differential equations of a mathematician foremost aim to describe a change. 399 00:47:20,020 --> 00:47:21,610 So behave yourself heavily. 400 00:47:21,640 --> 00:47:36,700 Material objects in nature with timescales ranging from picoseconds to millennia and with a lengthy scale ranging from subatomic to astronomical, 401 00:47:37,570 --> 00:47:44,340 can be maddening by nonlinear parts of differential equations equations with similar features, right? 402 00:47:46,360 --> 00:47:55,910 The law of partial differential equation have become increasingly significant within mathematics as a science since. 403 00:47:57,200 --> 00:48:02,570 So the mathematically you only have a partial differential and then it has a little history. 404 00:48:03,200 --> 00:48:08,840 And in listing the case this is related to somebody has to experience a vigilance clause. 405 00:48:09,080 --> 00:48:12,920 The research is a much more at a brisk pace. 406 00:48:14,290 --> 00:48:19,570 So if you want maybe a more kids as of tomorrow, reschedule another event. 407 00:48:22,210 --> 00:48:26,140 Five lectures by five distinguished mathematicians. 408 00:48:27,040 --> 00:48:36,430 The 915 in the morning for talk from them to our morning and the mathematical oracle by professor lawyer carefully from Texas Austin. 409 00:48:37,120 --> 00:48:47,530 So this event was organised by the Oxford Centre for Nonlinear Parts of Differential Equations, which open the October 1st, 2007. 410 00:48:48,640 --> 00:48:54,790 Uh, followed by the actual 2003 International Review of Mathematics in the UK, 411 00:48:55,930 --> 00:49:05,110 which indicated the critical attention is needed to both analysis a part of the future equation in the UK. 412 00:49:05,770 --> 00:49:08,770 So finding intellect is a professor. Sir John Paul. 413 00:49:09,100 --> 00:49:22,190 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. 414 00:49:22,200 --> 00:49:25,580 Also, PD is a part of the Mathematical Institute. 415 00:49:26,540 --> 00:49:33,030 There are four centres massive at the centre and the event 11 research groups. 416 00:49:33,860 --> 00:49:40,700 Okay, so this is the picture four new mathematical things that the new beautiful mathematics institute. 417 00:49:41,300 --> 00:49:54,350 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. 418 00:49:56,900 --> 00:50:07,100 Well, as a cable fellow, I would like to acknowledge the cables plan to establish at the Vantage Study Centre. 419 00:50:07,850 --> 00:50:18,080 Okay. Excellent side, which will hopefully be this building the other side on UMass, the Matthew Institute building, colossal Woodstock. 420 00:50:18,890 --> 00:50:27,890 Right. So for this that yeah, we're very excited about that. 421 00:50:28,250 --> 00:50:37,400 And of course, the more support is required to make this plan to ensure that they are additive for that from community that. 422 00:50:40,530 --> 00:50:50,520 So finally, I would like to thank the University of Oxford, set this post of a professorship for analysis of differential equations. 423 00:50:53,070 --> 00:51:03,270 There I find the Oxford is really unique, a place with a stimulating academic atmosphere. 424 00:51:03,510 --> 00:51:07,200 Firstly, the faculty and the excellent body of students. 425 00:51:08,010 --> 00:51:19,080 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 426 00:51:19,590 --> 00:51:28,380 and the Oxford community mobilised to make the UK community and the European Community for challenging scientific, 427 00:51:29,280 --> 00:51:36,870 socio educational issues, especially those issues where partial differential equations can play alone. 428 00:51:37,710 --> 00:51:51,420 Thank you very much. Please don't.