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There isn’t a single “next big thing” mathematicians are trying to solve. There are many next big things, and sometimes the next big thing that actually takes place isn’t any of the next big things anyone expected. But we can try, cautiously, to portray some of the most prominent challenges ahead, which are likely to attract attention and effort.How can we identify these challenges? There are a few reasonable sources:Hilbert’s famous list of 23 problems[1] is the closest thing we’ve ever had to an explicit portrayal of the greatest mathematical problems facing us. This was in 1900, but several of the problems are still only partially solved, or not at all.The Clay Mathematics Institute’s seven Millennium Problems[2] addressed a similar need in the year 2000. Only one of the seven has been solved so far.Books, articles and other publications addressing themes for the future of mathematics, such as the tome Mathematics Unlimited – 2001 and beyond[3] by Engquist and Schmid (eds.)Anyone attempting to write down such a list is bound to be at least somewhat biased, and likely often wrong. I will likely be more than somewhat biased and more than often wrong, but I’ll do my best, for whatever that’s worth.Complexity, Computability and LogicThe foundational questions of what can be computed and how efficiently are intimately linked to questions of mathematical logic: proofs, models and the boundaries of the mathematical endeavor. Indeed, one of the meta-questions here is, in my mind: what is the future of mathematics, now that computers have begun to augment our own abilities? This question will get answered, one way or another, in the coming 100 years.There are, certainly, more concrete problems we are facing. The most famous one is, of course, P vs NP,[4]which is one of the Millennium problems (and, in my humble opinion, the most profound one). Apart from this mother-of-all-problems, making any significant dent in the polynomial hierarchy and the complexity zoo is “the next big thing” for an entire industry of thinkers, working on a dazzling array of complexity classes.[5]Specifically, understanding the theoretical and practical significance of quantum computation is a major endeavor for the coming decades.One of the most intriguing (if speculative) directions in the area of computability, provability and foundations are Homotopy Type Theory[6]and Univalent Foundations[7] whose development was profoundly and tragically set back by the untimely death, in September 2017, of Fields medalist Vladimir Voevodsky.[8]Photo by Andrea KaneVoevodsky was, among many other things, a cautious intuitionist, or at least he was intrigued by intuitionism and finitism as a possible foundation for aspects of mathematics. My personal belief is that finitist methods and ideas will play an important role in the future of mathematics, not because they are somehow philosophically or morally superior, but because they are interesting.We’ve learned a lot about provability, but I think we’ve only skimmed the surface of proof complexity. What if a number-theoretic conjecture is false, but holds until some vast, uncomputable number? What if a statement is provable in ZFC, but the shortest proof is TREE(999) characters long? I believe that elucidating the fascinating frontiers of feasible thought will continue to define central pieces of mathematics for decades, or centuries.The Langlands ProgramRobert Langlands[9] is another person whose vision is deep and far-reaching enough to create an entire program, one that has guided the lives of many people since the late 1960s and is far from running out of steam. The Langlands program has been called “A Grand Unified Theory of Mathematics” by Edward Frenkel,[10] and I think this is only a slight exaggeration. It is an incredibly daring program, and it is peculiar that it’s absent from some of the sources I had mentioned.Describing the program is a daunting task. In An Elementary Introduction to the Langlands Program,[11]Stephen Gelbart writes:Herein lies the agony as well as the ecstasy of Langlands' program. To merely state the conjectures correctly requires much of the machinery of class field theory, the structure theory of algebraic groups, the representation theory of real and [math]p[/math]-adic groups, and (at least) the language of algebraic geometry. In other words, though the promised rewards are great, the initiation process is forbidding.Very, very roughly, the Langlands Program proposes that all [math]L[/math]-series, traditional objects of central importance in number theory, “come from” representations of certain groups. It connects together automorphic forms, the theory of adeles, representation theory and many other threads of geometry, algebra, analysis and number theory.The Langlands Program is not a single question or conjecture. It is an entire web of ideas. Ngô Bảo Châu’s[12] proof of the “Fundamental Lemma”[13]and Laurent Lafforgue’s[14] proof of the Langlands Correspondence over function fields were tremendous achievements that landed them both a Fields medal, but they are only steps in the grand program. Much is yet to be discovered and done, and I’m pretty sure that progress in the program will continue to happen over the coming decades.Children’s DrawingsI know this seems a little tongue-in-cheek, but it is only slightly so. The idea of “dessins d’enfants”[15]is one of Alexander Grothendieck’s[16] legacies, and I submit that elucidating, clarifying and building upon Grothendieck’s ideas is a significant outstanding challenge for mathematics.Specifically, dessins d’enfants offer a perspective on an object which (in my personal view) is one of the most mysterious and magnificent things in the deep, true universe (the one that is indifferent to the shackles of our actual, random physical world). It is perhaps the single most mysterious and magnificent of all: the absolute Galois group [math]\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})[/math]. Whatever progress is being made on understanding this profound group – and progress will, I am sure, be made – will shed light on the deepest questions in Number Theory and beyond. Grothendieck’s idea is not the only direction, but it is a promising and exciting one, like most everything this singular man came up with.Another thread of Grothendieck’s ideas, that of higher categories and topos theory, is pursued by many incredible and dedicated people, perhaps most vigorously by Jacob Lurie.[17]I don’t think we can expect him to finish it all on his own, wildly ingenious as he may be. It is hard to predict, but it is possible that higher category theory will grow in significance to play an absolutely central role in future math. It is also, by the way, closely connected to some of the ideas I mentioned in the first section (univalent foundations and so on).The Mathematics of PhysicsOne of the most ambitious, and most vague, of Hilbert’s 23 problems is the sixth.[18]It is phrased:6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.Remember: this is 1900. Before Einstein’s 1905,[19]before General Relativity, before Quantum Mechanics, before the Standard Model and Quantum Field theory and Superstring Theory. Hilbert couldn’t have known how far his physics colleagues were from understanding physics, let alone axiomatizing it. And the challenge remains: to form a coherent, unified mathematical framework that describes our physical world.From this massive endeavor, the Clay Mathematics Institute picked a much more concrete, yet still profound, challenge for the Millennium Problems: the problem[20] of establishing a Yang-Mills theory corresponding to any gauge group, and showing that it has a minimal mass. A different way to phrase this challenge in a mathematical context is, perhaps: organize, axiomatize and understand topological quantum field theories.In a wonderful survey describing the Yang-Mills problem, Jaffe and Witten wrote:[21]…one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so!To be sure, physics poses many other additional challenges for mathematics. Another Millennium Problem[22] seeks a proof (or refutation) that the Navier-Stokes[23] equations admit a smooth solution given any smooth boundary conditions.These are equations which describe fluid motion; they are not quantum or relativistic or anything, but they are fundamental for our understanding of the macroscopic world. Also, they are a key example of nonlinear partial differential equations (PDEs), and understanding nonlinear PDEs is a huge and multifaceted mathematical challenge. We really, really want to understand if, despite their chaotic nature, such equations always have smooth solutions or if they can “break down” given the right initial conditions.Specifically for Navier-Stokes, I think many people are just waiting for Terry Tao to dispose of the damn thing already, despite the fact that he’s so far made progress on showing why the problem is harder than it may seem. Incidentally, the way he does that hints at establishing “sufficiently strong” computational models within the framework of fluid dynamics, which brings us back to the domain of computability and proofs. If solutions to nonlinear PDEs are rich enough to accommodate non-trivial logic, it may be that certain problems in PDE are as non-solvable as the halting problem.Whether it’s Tao or someone else, making progress here would be huge.The (Generalized) Riemann HypothesisAnother major candidate for “the next big thing” is the Riemann Hypothesis, the only Millennium Problem which has survived intact since Hilbert’s list in 1900. It may not be “the next big thing” in terms of being solved – heaven knows how long this might take – but it will surely continue to consume people’s minds, hopes and dreams.The Riemann Hypothesis seems kind of exotic: it deals with a specific function, the Riemann zeta function [math]\zeta(s)[/math], and asks about the location of its roots (other than a series of “trivial” ones). They are all expected to lie on the line [math]\Re(s)=\frac{1}{2}[/math].Specific functions and their roots aren’t usually a matter of universal interest, but this one is a huge exception. The Riemann zeta function has central importance in several fields of mathematics (most obviously, Number Theory). Furthermore, there’s a natural generalization of the Riemann Hypothesis which deals with other zeta functions, and has applications even more far-reaching than RH alone.This problem has now stood for over 150 years, and I don’t think it’s going away any time soon. Whenever anyone proves it, or even significantly enhances our understanding of the difficulty, that’s a Next Big Thing indeed.The ABC ConjectureSince its introduction in the mid-1980s, the ABC conjecture[24] was found to offer a unified way of understanding many old and new problems in Number Theory. In fact its proof would imply[25] many of the Fields-winning problems of the past 50 years, and papers regularly show up showing that it implies this or that open problem.The problem is unsolved. I’m pointing this out since there’s currently a pretty confusing state of affairs in which a proposed proof by Shinichi Mochizuki[26] has stood unconfirmed since 2012, and so far only a very small number of mathematicians believe that it does indeed prove the ABC Conjecture. Many questions about this have[27] been[28] asked[29] on[30] Quora.[31](This situation saddens me. The ABC Conjecture is a thing of beauty, and the deep work done around it is truly exciting and intriguing. But the general public is attracted to drama, and the drama obscures the real essence of the theory. There’s no reason to be excited by Mochizuki’s proof. Not yet, at any rate.)Developing Mochizuki’s “Inter-Universal Teichmüller Theory”, if at all possible, will be a monumental progress. Alternatively, finding other ways to attack the ABC Conjecure and the closely related Szpiro’s Conjecture[32] would certainly qualify as a Big Thing.There is no shortage of huge open problems in many other areas of mathematics, any one of which could be the Next Big Thing. We don’t know where things will lead us, or which dramatic new innovations would transform some or all of math. The last century was transformative; it’s entirely reasonable that this one will be, too.Footnotes[1] Hilbert's problems - Wikipedia[2] Clay Mathematics Institute[3] Mathematics Unlimited - 2001 and Beyond | Björn Engquist | Springer[4] Clay Mathematics Institute[5] Complexity Zoo[6] Homotopy type theory - Wikipedia[7] Univalent foundations - Wikipedia[8] Vladimir Voevodsky 1966–2017[9] Robert Langlands - Wikipedia[10] Edward Frenkel - Wikipedia[11] Bulletin of the American Mathematical Society[12] Ngô Bảo Châu - Wikipedia[13] Fundamental lemma (Langlands program) - Wikipedia[14] Laurent Lafforgue - Wikipedia[15] Dessin d'enfant - Wikipedia[16] Alexander Grothendieck - Wikipedia[17] Jacob Lurie - Wikipedia[18] Hilbert's sixth problem - Wikipedia[19] Annus Mirabilis papers - Wikipedia[20] Yang–Mills existence and mass gap - Wikipedia[21] http://www.claymath.org/sites/default/files/yangmills.pdf[22] http://www.claymath.org/sites/default/files/navierstokes.pdf[23] Navier–Stokes equations - Wikipedia[24] abc conjecture - Wikipedia[25] http://www.ams.org/notices/200210/fea-granville.pdf[26] Shinichi Mochizuki - Wikipedia[27] What is the reason why the proof for ABC conjecture of Shinichi Mochizuki has still not been generally considered as a proof passing the peer review stage as usual though it had already been examined by more than 10 mathematicians?[28] Did Shinichi Mochizuki solve the ABC Conjecture?[29] How difficult is it to penetrate the Mochizuki’s unverified proof of abc conjecture?[30] What does Terence Tao think of Mochizuki´s proof of the abc conjecture?[31] Has proof of the ABC conjecture, written by Shinichi Mochizuki, already been checked?[32] Szpiro's conjecture - Wikipedia

IFOS Interview 201912feb 2020 afternoon 5th to goRN Choubey sir board.Chirag Jain, From Bharatpur RajasthanMechanical engineering, worked in tata motors PuneMarks obtained : 172/300 ( not selected)Chairman:1. Why did you leave tata motors job?2. how can you apply mechanical engineering knowledge in forest conservation?3. What are various acts for environmental conservation?4. What is the Purpose of biodiversity act 2002?5. What steps have govt taken to conserve biodiversity?6. What is the Difference in NP and WS?7. What is per capita carbon footprint of India8. What is Per capita emissions of India9. What about USA ?10. Is it fair for developing country to reduce there GHG emissions when in developed countries they are so high?Member1:1. Do you know about Kuldhara village?2. Why brahmins people migrated from there in single night..?3. what is Ethnobotonist ?4. What is the IUCN Status of Great Indian busturd5. What are there cause of extinction?6. What is ecological succession?7. What are its different types?8. How are different polluting industry are classified in India..9. Name some red and white category industries.10. What can be done to reduce pollution from paper and pulp industry.11. What do you know about eco task force? Where it has been formed? What is it purpose?Member2:1. Why govt is doing bank merger?2. Why NPA problem in banks?3. When banks have so much NPA, from where they can get fund to give loans?4. What are different ways for investment in economy?5. How FDI come into India?6. What were the provisions in budget to increase FPI?7. Any provision related to Deposit insurance?Member3:1. What was your role as alumni coordinator?2. What are different role of alumni in college?3. Do you think focus on cultural activities in technical and engineering institutes is good ?4. What is the current Status of ozone layer depletion?5. Who is the Animal trials approving authority in India?Chairman:1. Do you want as to ask anything?(I told about Bharatpur kevaladeo national park)2. Tell me about it?3. Why Siberian crane are not coming there now?4. Where are they stoping now?Thank you..your interview is over.(Interview went for around 25 minutes..Board was very cordial)IFoS interview 2018Date: 30 Jan, 2019 afternoon, 4th to goBS Bassi sir boardMarks obtained in interview : 180/300, though not recommended( missed by 8 marks).Chairman:1. Is this your photo?2. You have worked in ongc, so Tell me about digboi?3. You have worked with tata motors so Why did Nano fail?4. Have you seen minicooper ad ?5. Suppose you are head of my marketing team, What should be the marketing strategy to launch new car?6. What you mean by small is beautiful?M1:1. What is Climate change?2. what is Forest role in it?3. what is Phytoremediation?4. What happens to plant when they absorb heavy metals?5. As a mechanical engineer what would you do to treat a site affected by pollutants?6. Whats canola oil?7. What special about golden rice?8. In which vitamin it is fortified?9. What is carbon footprint?10. How can we calculate our carbon footprint?M2:1. What is famous about Deeg fountain ( hometown)2. How did it operate ?3. How every fountain give unique color?4. Tell me about history of Bharatpur?( home district)5. Why do you think that British not able to conquer Bharatpur?6. Tell me some Famous work of raja surajmal?7. What is Skill India mission?8. What is the latest changes proposed by the govt in this mission(pmkvy 2.0)?M3:1. How to stop oil and gas release from rigs??2. What was your work in mentorship program you mentioned?3. What do you mean by fitness?( hobby)4. How do you if anyone is fit or not?5. What is your BMI?6. How to measure BMI?7. Is BMI range varies with age?8. Why alwar tiger reserve was in news sometimes back?( home state)9. Have you visited Bharatpur NP?10. What Migratory birds you saw there?11. Have you visited ranthambore tiger sReserve?12. What is CSR ???13. What is tata doing about it?Chairman:1. what is trade war?2. Latest development in usa china Trade war?3. Is there any leader scheduled to visit usa for this trade war( Chinese vice premier)?4. Quantum of import and export between usa and china?5.What is Usa Shutdown?Thank you..your interview is over.

What you are interested in is called multi-objective optimization, in your case, of path in a graph.The thing is, when edges' weights are vectors instead of scalars, there might be more than one optimal path. For example, let one path's weight be [math]\left(5,7\right)[/math] and another's weight be [math]\left(3,8\right)[/math]. You can't tell which one is "more optimal" until you throw in some additional metric.So, this is where new terms come in.Let's call path's weight vector's components criteria, and say that path [math]\gamma_1[/math] is better then path [math]\gamma_2[/math] by some criterion if [math]\gamma_1[/math]'s component (corresponding to that criterion) is less than [math]\gamma_2[/math]'s corresponding component. (This is some informal terminology, I guess, but it makes things easier to explain).A path [math]\gamma^0[/math] is called Slater optimal if there's no path which is better than [math]\gamma^0[/math] by all criteria simultaneously. A path [math]\gamma^*[/math] is called Pareto optimal if every path that is better than [math]\gamma^*[/math] by one criterion is worse by another. If a path is Pareto optimal, it is Slater optimal too.On the next picture weights of Pareto optimal paths are displayed as green dots while weights of Slater optimal, but not Pareto optimal paths are displayed as yellow dots. Green dots form Pareto set, green and yellow dots together form Slater set.The algorithms solving the problem of finding an optimal path in a graph with vector weights can be divided into two groups: those finding all optimal decisions, and those finding some of optimal decisions.What other answers suggest is called convolution method (at least in Russia, where I study). Using it, you can find some of Slater optimal solutions. Basically, you select a function which transforms vector weights into scalar weights. Then, you can use Dijkstra's algorithm with those scalar weights, as usual. Depending on function's parameters, you get various solutions. There are some nuances, though.First of all, there's a rule of thumb that the convolution method you select has to correspond with the way you calculate the criteria of a path knowing the weights of its edges. Let me explain.For example, if the meaning of a criterion is total distance traveled, than it is an additional criterion and you sum weights of edges (same for total time spent on the road). If all criteria are additional, the corresponding method is called linear convolution: scalar weight is calculated as linear combination of vector weight's components. Formally speaking, if [math]W(\gamma) \in [/math][math]\mathbb{R}^n[/math] is vector weight, then scalar weight is calculated like this:[math]Q(\gamma) = [/math][math]\sum\limits_{i=1}^n \lambda_i W_i(\gamma),~ \lambda_i \ge[/math][math] 0,~[/math][math]\sum\limits_{i=1}^n \lambda_i= 1[/math].However, what if you want to chose the least dangerous path, and edges' weights represent how dangerous that part of a path is? Then instead of summing weights, you need to select the biggest one to represent the whole path. This is called maximum-type criterion. If all criteria are maximum-type, the corresponding convolution looks like this:[math]Q(\gamma) = \max\limits_{i=1,...,n} \lambda_i W_i(\gamma),~[/math][math]\lambda_i \ge[/math][math] 0,~[/math][math]\sum\limits_{i=1}^n \lambda_i= 1[/math].If you use a non-corresponding convolution method, Bellman's principal of optimality, which is used by Dijkstra's algorithm, won't be fulfilled, and the results might make absolutely no sense.Another issue is that, depending on the problem, there might be optimal (even Pareto optimal) solutions which you won't be able to find no matter how you chose parameters [math]\lambda_i[/math].I'm now going to talk about a generalization of Dijkstra's algorithm which finds all Pareto optimal paths. I guess it's not useful for programming since it must have ridiculous complexity (unfortunately, the source doesn't mention it, so I can't tell exactly how complex it is), so if you aren't into theoretical stuff, you might want to stop reading here (in that case, thank you for your attention).UPD: In comments, Timothy Johnson states that finding all Pareto optimal paths in an NP-complete problem.This generalization was proposed in 1996 by Stanislav Gorodetsky, PhD in Physics and Mathematics, who have worked as an associate professor at the State University of Nizhny Novgorod (Russia) since 1993. I'm not going to give a formal description (it's hard to understand in one try), but instead I'm going to tell what to one needs to change in classic Dijkstra's algorithm in order to generalize it:Instead of one scalar mark, every vertex now has multiple vector marks, each of which is a weight of Pareto optimal path to that vertex. Some part of vertex's marks can be final while the others are not.In order to be able to "remember" optimal paths, for each mark (but the initial one) one needs to store the edge leading to its vertex and the mark of a previous vertex used to calculate it (sounds a bit complicated, but wait for an example, it will make things clearer).Marks become final if they are in Pareto subset of set formed from all non-final marks and all marks of a final vertex. All "newly final" marks are used as starting point for the next iteration.Okay, let's get to an example, shall we? This is our graph (the leftest vertex is a starting point, the rightest is an ending point):Final marks will have grey background, "newly final" marks will have pink background. The color of the mark is the same as the color of the edge leading to it, and the index is its "parent" mark.On zero iteration, we make the initial [math]\left( 0, 0 \right)[/math] mark and flag it final. Here's what happens on first iteration:Pareto subset [math]\mathscr{P} = \{ \left(1,2\right),~\left(5,1\right) \}[/math], so those marks become final. Second iteration:Notice how mark [math]\left(4,6\right)[/math] is forgotten because it's worse then [math]\left(3,5\right)[/math] by both criteria. Pareto subset [math]\mathscr{P} = \{ \left(3,5\right),~\left(4,4\right),~\left(7,3\right) \}[/math]. Third iteration:Mark [math]\left(7,3\right)[/math] doesn't form a new one, since it'll be worse by all criteria than existing ones. Pareto subset [math]\mathscr{P} = \{ \left(3,6\right)\}[/math]. Fourth iteration:Mark [math]\left(3,6\right)[/math] doesn't form a new one, either. After this iteration, all marks are final. So we have 3 Pareto optimal solutions:If you have questions, please ask them in comments and I'll try my best to give you an answer. Thanks for reading!