The History of Combinatorial Group Theory: A Case Study in the History of Ideas


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During the early part of the last century, F. Frobenius raised, in his lectures, the following problem called the Diophantine Frobenius Problem FP : given relatively prime positive integers a1,. It turned out that the knowledge of g a1,. A number of methods, from several areas of mathematics, have been used in the hope of finding a formula giving the Frobenius number and algorithms to calculate it. This book aims to provide a comprehensive exposition of what is known today on FP.

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P.

Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.

Graph theory is now an established discipline but the study of graph homomorphisms has only recently begun to gain wide acceptance and interest. This text is devoted entirely to the subject, bringing This text is devoted entirely to the subject, bringing together the highlights of the theory and its many applications. It looks at areas such as graph reconstruction, products, fractional and circular colourings, and constraint satisfaction problems, and has applications in complexity theory, artificial intelligence, telecommunications, and statistical physics.

It has a wide focus on algebraic, combinatorial, and algorithmic aspects of graph homomorphisms. A reference list and historical summaries extend the material explicitly discussed. The book contains exercises of varying difficulty. Hints or references are provided for the more difficult exercises.

This book provides an introduction to the concept of fixed-parameter tractability. The corresponding design and analysis of efficient fixed-parameter algorithms for optimally solving combinatorially The corresponding design and analysis of efficient fixed-parameter algorithms for optimally solving combinatorially explosive NP-hard discrete problems is a vividly developing field, with a growing list of applications in various contexts such as network analysis or bioinformatics. The book emphasizes algorithmic techniques over computational complexity theory.

It is divided into three parts: a broad introduction that provides the general philosophy and motivation; followed by coverage of algorithmic methods developed over the years in fixed-parameter algorithmics forming the core of the book; and a discussion of the essentials of parameterized hardness theory with a focus on W[1]-hardness which parallels NP-hardness, then stating some relations to polynomial-time approximation algorithms, and finishing up with a list of selected case studies to show the wide range of applicability of the presented methodology.

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory.

By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics.

These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups.

For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang—Baxter equation.

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OSO version 0. University Press Scholarship Online. Sign in. Not registered? Sign up. Publications Pages Publications Pages. Advanced Search. Search my Subject Specializations: Select Browse Print Email Share. Community Ecology Gary G. Essentials of Landscape Ecology Kimberly A. Search within results Search within results. Well, the original intuition behind the NZ-conjecture was clearly wrong. Many nice examples is not a good enough evidence. But the conjecture was so plausible! Where did the intuition fail? Consider this example.

All small examples confirm this.


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Planar cubic graphs do have very special structure. The fact that Tutte found a counterexample is no longer surprising. Confused enough? Back to the patterns. Same story here. When you look at many small cases, everything is P-recursive or yet to be determined. This suggests that small examples are not representative of complexity of the problem. Time to think about disproving ALL conjectures based on that evidence.

The more you know, the better you get. Pay attention to small crumbs of evidence. And think negative!

Visual Group Theory, Lecture 2.1: Cyclic and abelian groups

However, being so positive has a drawback — sometimes you get things badly wrong. In fact, even polynomial Diophantine equations can be as complicated as one wishes. I understand why they do this, just disagree with the reasoning. If anything, we should encourage thinking in the direction where there is not enough research, not in the direction where people are already super motivated to resolve the problem. In general, it is always a good idea to keep an open mind. If you think a conjecture might be false, ignore everybody and just go for disproof.

There are some journals who do that, but not that many. Hopefully, this will change soon…. When we were working on our paper, I wrote to Doron Zeilberger if he ever offered a reward for the NZ-conjecture, and for the disproof or proof only? He replied with an unusual award , for the proof and disproof in equal measure. When we finished the paper I emailed to Doron. And he paid. He did surprisingly well, but I am a tough grader and possibly biased about some of the predictions. Judge for yourself…. It is a truth universally acknowledged that humans are very interested in predicting the future.

People tend to forget wrong predictions unless they are outrageously wrong. Sometimes these predictions are rather interesting see here and there , but even the best ones are more often wrong than right…. Although rarely done, analyzing past predictions is a useful exercise, for example as a clue to the truthiness of the modern day oracles. Of course, one can argue that many of the political or technology predictions linked above are either random or self-serving, and thus not worth careful investigation.

And the fact that they were made so long ago makes them uniquely attractive, practically begging to be studied. A quick review of the last few quotes on this list I assembled shows how much things have changed. Basically, the area moved from an ad hoc collection of problems to a degree panorama of rapidly growing subareas, each of which with its own deep theoretical results, classical benchmarks, advanced tools and exciting open problems. Note the aerodynamic shape of the engine. Both the silent cinema and phonograph were invented by ; the sound came to movie theaters only in So the invention here is of a home theater for movies with sound.

Some points are lost due to the lack of widespread popularity in This is an electric appliance for floor cleaning. Well, they do exist, sort of, obviously based on different principles. Roller skates have been invented in 18th century and by became popular. So no credit for the design, but extra credit for believing in the future of the mean of transportation now dominated by rollerblades.

Grade: B-. There are however electric shavers and hair cutters which are designed very differently. The author assumes that personal air travel will become commonplace, and at low speeds and heights. This is almost completely false. The author imagines that the public spectacle of horse racing will move underwater and become some kind of fish racing. Here are my biased and possibly uninformed grades for each problem he mentions.

I think it is fair to say that since combinatorics made no contribution in this direction. While physicists and probabilists continue studying this problem, there is no exact solution in dimension 3 and higher. Grade: F.


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  7. The study of percolation completely exploded since and is now a subject of numerous articles in both probability, statistical physics and combinatorics, as well as several research monographs. Grade: A. As far as I can tell, there are various natural and interesting generalizations of necklaces, but none surprising. Simply put, the g.

    Grade: D. Despite strong interest, until recently there were very few results in the two-dimensional case some remarkable results were obtained in higher dimensions. Combinatorialists did of course contribute to the study of somewhat related questions on enumeration of various classes of polyomino which can be viewed as self-avoiding cycles in the grid, see e. Grade: C. This is a fundamental problem in optimization theory, connected to the study of Hamiltonian cycles in Graph Theory and numerous other areas.

    It is also one of the earliest NP-hard problems still playing a benchmark role in Theoretical Computer Science. No quick of summary of the progress in the past 45 years would give it justice. In this light, his emphasis on algorithmic complexity and allusions to Computability Theory e. So are his briefly mentioned connections to topology which are currently a popular topic. Well done! Grade: A-. What a missed opportunity! But this was an easy prediction to make given the ongoing effort by Carlitz, Polya, Riordan, Rota himself and many other peope. The subject was already popular and it did continue to develop but perhaps at a different pace and directions than Rota anticipated Hadamard matrices , tools from Number Theory.

    Rota gives a one-sentence desctiption:. There are various important connections between Logic and Combinatorics, for example in Descriptive Set Theory see e. Either way, Rota was too vague in this case to be given much credit. One can argue this all to be too vague or misdirected, but the area does indeed explode in part in the directions of problems Rota mentions earlier.

    So I am inclined to give him benefit of the doubt on this one. In total, Rota clearly got more things right than wrong. He displayed an occasional clairvoyance, had some very clever insights into the future, but also a few flops.

    Hans Heinrich Wilhelm Magnus

    Note also the near complete lack of self-serving predictions, such as the Umbral Calculus that Rota was very fond of. Since predictions are hard, successes have a great weight than failures. Overall, good job, Gian-Carlo! Eventually Rota wrote me a letter of recommendation for a postdoc position. Recently, there has been plenty of discussions on math journals, their prices, behavior, technology and future.

    Should all existing editorial boards revolt and all journals be electronic? Be patient — a long explanation is coming below. I would like to argue that the debate over the second question is the general misunderstanding of the first question in the title. In fact, I am pretty sure most mathematicians are quite a bit confused on this, for a good reason. If you think this is easy, quick, answer the following three questions:. Is this a fault of author, referee s , handling editor, managing editor s , a publisher, or all of the above? Same question.

    Would the answer change if the author lists the paper in the references? Sections are disorganized and the introduction is misleading. Now that you gave your answers, ask a colleague. In theory, a lot. In practice, that depends. However, as any editor can tell you, you never know what exactly did the referee do. Some reply within 5 min, some after 2 years. The anonymity is so relaxing, doing a poor job is just too tempting.

    The whole system hinges on the shame, sense of responsibility, and personal relationship with the editor. They help the authors. They do their best, kind of what ideal advisors do for their graduate students, who just wrote an early draft of their first ever math paper.

    Combinatorics | mathematics | abymedoxuhav.tk

    They do what they can and as much work as they want. To make a lame comparison, the referees are like wind and the editors are a bit like sailors. While the wind is free, it often changes direction, sometimes completely disappears, and in general quite unreliable. But sometimes it can really take you very far. Of course, crowd sourcing refereeing is like democracy in the army — bad even in theory, and never tried in practice. The editor was very sympathetic if a bit condescending, asking me not to lose hope, work on my papers harder and submit them again.

    So I tried submitting to a competing but equal in statue journal, this time under my own name. The paper was accepted in a matter of weeks. You can judge for yourself the moral of this story. A combinatorialist I know who shall remain anonymous had the following story with Duke J. A year and a half after submission, the paper was rejected with three! The authors were dismayed and consulted a CS colleague.

    That colleague noticed that the three reports were in. Turns out, if the cropping is made straightforwardly, the cropped portions are still hidden in the files. Using some hacking software the top portions of the reports were uncovered. The authors discovered that they are extremely positive, giving great praise of the paper. Now the authors believe that the editor despised combinatorics or their branch of combinatorics and was fishing for a bad report.

    A year after submission, one of my papers was rejected with the following remarkable referee report which I quote here in full:. Needless to say, the results were new, and the paper was quickly published elsewhere. Three little things, really. They choose referees, read their reports and make the decisions. But they are responsible for everything. And I mean for everything , both 1 , 2 and 3. If the referee wrote a poorly researched report, they should recognize this and ignore it, request another one. They should ensure they have more than one opinion on the paper, all of them highly informed and from good people.

    If it seems the authors are not aware of the literature and referee s are not helping, they should ensure this is fixed.

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    It the paper is not well written, the editors should ask the authors to rewrite it or else. Contacting the author s is also a good idea, but sometimes the anonymity is helpful — the editor can be trusted to bring bad news and if possible, request a correction. Neumann described here how he thinks the journal should operate. I wish his views held widely today. Now, the reason most people are confused as to who is responsible for 1 , 2 and 3 , is the fact that while some journals have serious proactive editors, others do not, or their work is largely invisible.

    In theory, managing editors hire associate editors, provide logistical support, distribute paper load, etc. In practice they also serve as handling editors for a large number of papers. Publishing is a business, after all…. Good mathematicians. Great mathematicians. Mathematicians who write well and see no benefit in their papers being refereed. Mathematicians who have many students and wish the publishing process was speedier and less cumbersome, so their students can get good jobs. In general, these are people who see having to publish as an obstacle, not as a benefit.

    Publishers of course , libraries, university administrators. In general, people who need help with their papers. For some mathematicians, having all journals to be electronic with virtually no cost is an overall benefit. While I imagine that in the future many excellent top level journals will be electronic and free, I also think many mid-level journals in specific areas will be run by non-profit publishers, not free at all, and will employ a number of editorial and technical stuff to help the authors, both of papers they accept and reject. This is a public service we should strive to perform, both for the sake of those math papers, and for development of mathematics in other countries.

    Free journals can do only so much.

    Without high quality editors paid by the publishers, with a large influx of papers from the developing world, there is a chance we might loose the traditional high standards for published second tier papers. My view was gloomy but mildly optimistic. In fact, since that post was written couple more combinatorics papers has been accepted.

    But let me give you a quiz. Here are two comparable highly selective journals — Duke J. In the same period, Duke published 8 combinatorics papers of total. Q: Which of the two Composito or Duke treats combinatorics papers better? The reasoning is simple. Forget the anecdotal evidence in the previous interlude. Here is what Compsito website says with a refreshing honesty:. By tradition, the journal published by the foundation focuses on papers in the main stream of pure mathematics.

    This includes the fields of algebra, number theory, topology, algebraic and analytic geometry and geometric analysis. Papers on other topics are welcome if they are of interest not only to specialists.

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    Translation: combinatorics papers are not welcome as are papers in many other fields. I think this is totally fair. Nothing wrong with that. Clearly, there are journals which publish mostly in combinatorics, and where papers in none of these fields would be welcome. In fact there is a good historical reason for that. Compare this with what Duke says on its website :. Without specializing in a small number of subject areas, it emphasizes the most active and influential areas of current mathematics.

    See the difference? How are the authors supposed to guess which are these? Note also, that things used to be different at Duke. How come? I understand the temptation for each country, or university, or geographical entity to have its own math journal, but nowadays this is counterproductive and a cause for humor. This parochial patriotism is perhaps useful in sports or not , but is nonsense in mathematics. Eventually all these journals will reorganize into a non-profit institutions or foundations.

    This does not mean that the journals will become electronic or free. While some probably will, others will remain expensive, have many paid employees including editors , and will continue to provide services to the authors, all supported by library subscriptions. If you have two similar looking free electronic journals, which do not add anything to the papers other than their. This is not enough. All journals, except for the very top few, will have to start limiting their scope to emphasize the areas of their strength, and be honest and clear in advertising these areas.

    Alternatively, other journals will need to reorganize and split their editorial board into clearly defined fields. While highly efficient, in a long run this strategy is also unsustainable as it leads to general confusion and divergence in the quality of these sub-journals. See e. But at least those of us who have been in the area for a while, have the memory of the fortune of previously submitted papers, whether our own, or our students, or colleagues.

    A circumstantial evidence is better than nothing. For the newcomers or outsiders, such distinctions between journals are a mystery. Neither paper is in Combinatorics, but both are in Discrete Mathematics , when understood broadly. Do you think you know the answer? Do you think others have the same answer? Imagine you could go back in time and ask this question to a number of top combinatorialists of the past 50 years. What would they say? Would you agree with them at all?

    The History of Combinatorial Group Theory: A Case Study in the History of Ideas The History of Combinatorial Group Theory: A Case Study in the History of Ideas
    The History of Combinatorial Group Theory: A Case Study in the History of Ideas The History of Combinatorial Group Theory: A Case Study in the History of Ideas
    The History of Combinatorial Group Theory: A Case Study in the History of Ideas The History of Combinatorial Group Theory: A Case Study in the History of Ideas
    The History of Combinatorial Group Theory: A Case Study in the History of Ideas The History of Combinatorial Group Theory: A Case Study in the History of Ideas
    The History of Combinatorial Group Theory: A Case Study in the History of Ideas The History of Combinatorial Group Theory: A Case Study in the History of Ideas
    The History of Combinatorial Group Theory: A Case Study in the History of Ideas The History of Combinatorial Group Theory: A Case Study in the History of Ideas
    The History of Combinatorial Group Theory: A Case Study in the History of Ideas The History of Combinatorial Group Theory: A Case Study in the History of Ideas
    The History of Combinatorial Group Theory: A Case Study in the History of Ideas The History of Combinatorial Group Theory: A Case Study in the History of Ideas
    The History of Combinatorial Group Theory: A Case Study in the History of Ideas

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