1. Graph Theory Open Problems Six problems suitable for undergraduate research projects. http://dimacs.rutgers.edu/~hochberg/undopen/graphtheory/graphtheory.html

Graph Theory Open Problems Index of Problems Unit Distance Graphs-chromatic number Unit Distance Graphs-girth Barnette's Conjecture Crossing Number of K(7,7) ... Square of an Oriented Graph Unit Distance Graphs-chromatic number RESEARCHER: Robert Hochberg OFFICE: CoRE 414 Email: hochberg@dimacs.rutgers.edu DESCRIPTION: How many colors are needed so that if each point in the plane is assigned one of the colors, no two points which are exactly distance 1 apart will be assigned the same color? This problem has been open since 1956. It is known that the answer is either 4, 5, 6 or 7-this is not too hard to show. You should try it now in order to get a flavor for what this problem is really asking. This number is also called ``the chromatic number of the plane.'' A graph which can be embedded in the plane so that vertices correspond to points in the plane and edges correspond to unit-length line segments is called a ``unit-distance graph.'' The question above is equivalent to asking what the chromatic number of unit-distance graphs can be. Here are some warm-up questions, whose answers are known: What complete bipartite graphs are unit-distance graphs? What's the smallest 4-chromatic unit-distance graph? Show that the Petersen graph is a unit-distance graph.

2. OpenProblems Compiled by Jorge Urrutia, University of Ottawa.Category Science Math Geometry open problems......open problems on Discrete and Computational Geometry. Introduction This web pagecontains a list of open problems in Discrete and Computational Geometry. http://www.csi.uottawa.ca/~jorge/openprob/

Open Problems on Discrete and Computational Geometry. Introduction: This web page contains a list of open problems in Discrete and Computational Geometry . Contributions to the list are invited. To contribute problems, submit them to me by e-mail, in html format. For each problem you pose, you may include one or two figures, in gif or jpg format. Make sure they are not too big, as this slows down their downloading time considerably . If any problem posed here is solved, I would appreciate it if you send me an e-mail to jorge@csi.uottawa.ca . In each problem you pose, include, to the best of your knowledge, who posed the problem first, and relevant references. Try to be short, concise and to the point. This will make your problems more attractive, and may increase the chances someone will read and try to solve them. If you detect inaccuracies regarding references, etc. in the problems posed here, please let me know so that I can correct them. At least until the end of this year, the format of this page will be evolving, until a satisfactory final layout is reached. Sorry for the inconveniences this may create. Jorge Urrutia , November, 1996.

3. Some Open Problems Provides problems that describe bounded degree triangulation and combinatorics. Includes tiling puzzles. open problems. Antipodes. Jim Propp asks whether the two farthest apart points, as measured by surface distance, on a http://www.vuse.vanderbilt.edu/~spin/open.html

Send comments or new problems to include to spin@vuse.vanderbilt.edu

4. Open Problems List These are open problems that I've encountered in the course of my research. Not surprisingly, almost all the problems are geometric in nature. http://www.math.sunysb.edu/dynamics/open.html

Open Problems in Dynamical Systems Open problems in holomorphic dynamics Two lists of problems are currently available on-line: Conformal Dynamics Problem List Ed. by B. Bielefeld, IMS at Stony Brook Preprint 1990/1 and more recent Problems in Holomorphic Dynamics Ed. by B. Bielefeld and M. Lyubich, IMS at Stony Brook Preprint 92/7 In addition some open questions may be found in these surveys Open problems in dynamics in several complex variables is a part of Several complex variables problems list available from Several complex variables preprint server in Indiana. Please send your problems to Gregery Buzzard at gbuzzard@iu-math.math.indiana.edu Problems in Low-Dimensional Topology a 380 pages PostScript file edited by Rob Kirby contain some problems on Topological Dynamics. Topology of hyperbolic attractors. A problem submitted by Christian Bonatti. A collection maintained by Sergiy Kolyada and Michal Misiurewicz Is entropy effectively computable? A problem by John Milnor We are soliciting open problems in various areas of Dynamical Systems for posting on this page. You can post a problem by filling out this form or by sending an e-mail to webmaster@math.sunysb.edu

5. Bill Martin -- Open Problems Some open problems in Algebraic Combinatorics Last modified May 4, 1999 Below, I list some of my favourite unsolved problems. But first, a few warnings. Many of these problems have been posed by other people. http://www.uwinnipeg.ca/~martin/RESEARCH/open.html

Some Open Problems in Algebraic Combinatorics Last modified: May 4, 1999 Below, I list some of my favourite unsolved problems. But first, a few warnings. Many of these problems have been posed by other people. I will try to give proper attributions, but I am likely to miss someone's name eventually. Many of the problems I know of were posed by Chris Godsil . Secondly, the problems are all confined to areas in which I work. That is, the list is rather narrow in scope and may not seem thematic. (folklore) For a set C of q -ary n -tuples, let t denote the strength of C as an orthogonal array and let s denote the degree of C as a code. Prove that there exists a constant k such that t <= s+k . Note that no examples are known having t>s+4 (Delsarte, 1973) Do there exist non-trivial perfect codes in the Johnson graphs J(v,k) This question appears in Delsarte's thesis. He ``almost'' conjectures that the answer is NO. The strongest result to date on this is Roos's bound (see Brouwer, et al.). A recent paper by Tuvi Etzion also has nice results (most notably, that the answer is NO if v-2k is prime). I proved that the derived design of a perfect code is always a completely regular code. This gives more leverage to a number-theoretic attack. For example, there are no perfect 2-codes with v

6. Open Problems Collected by Jeff Erickson. Mainly in geometry.Category Science Math Geometry open problems......open problems. These are open problems that I've encountered in the course ofmy research. open problems Jeff Erickson (jeffe@cs.uiuc.edu) 09 Apr 2001 http://compgeom.cs.uiuc.edu/~jeffe/open/

Open Problems These are open problems that I've encountered in the course of my research . Not surprisingly, almost all the problems are geometric in nature. A name in brackets is the first person to describe the problem to me; this may not be original source of the problem. If there's no name, either I thought of the problem myself (although I was certainly not the first to do so), or I just forgot who told me. Problems in bold are described in more detail than the others, and are probably easier to understand without a lot of background knowledge. If you have any ideas about how to solve these problems, or if you have any interesting open problems you'd like me to add, please let me know . I'd love to hear them! Existence Problems: Does Object X exist? Unfolding convex polytopes Joe O'Rourke and Komei Fukuda Combinatorial Problems : How complex is Object X? Degenerate facets of polytopes Faces of intricate polytopes - Updated! Point-hyperplane incidences Union of intersecting unit balls - Solved! Line traversals of Delaunay triangulations Nina Amenta - Solved!

7. Open Problems For Undergraduates A collection of open problems in Discrete Mathematics which are currently being researched by members of the DIMACS community. http://dimacs.rutgers.edu/~hochberg/undopen/

Open Problems for Undergraduates Open Problems by Area Graph Theory Combinatorial Geometry Geometry/Number theory Venn Diagrams Inequalities Polyominos This is a collection of open problems in Discrete Mathematics which are currently being researched by members of the DIMACS community. These problems are easily stated, require little mathematical background, and may readily be understood and worked on by anyone who is eager to think about interesting and unsolved mathematical problems. Some of these problems are quite hard and have been open for a long time. Others are newer. For further information on a particular problem, you may write to the associated researcher. Although these problems are intended for undergraduates, it is expected that high school students, teachers, graduate students and professional mathematicians will be drawn to this collection. This is not discouraged. Each of these problems is associated with some member of DIMACS. If you have any questions, comments, insights or solutions, please send email to the researcher who is listed with the problem. These pages are maintained by Robert Hochberg Last modified Feb. 5, 1997.

8. The Geometry Junkyard: Open Problems Compiled by David Eppstein of the University of California at Irvine.Category Science Math Geometry open problems......The Geometry Junkyard. open problems. Geombinatorics Making Math Fun Again. A journalof open problems of combinatorial and discrete geometry and related areas. http://www.ics.uci.edu/~eppstein/junkyard/open.html

Open Problems Antipodes . Jim Propp asks whether the two farthest apart points, as measured by surface distance, on a symmetric convex body must be opposite each other on the body. Apparently this is open even for rectangular boxes. Bounded degree triangulation . Pankaj Agarwal and Sandeep Sen ask for triangulations of convex polytopes in which the vertex or edge degree is bounded by a constant or polylog. Centers of maximum matchings . Andy Fingerhut asks, given a maximum (not minimum) matching of six points in the Euclidean plane, whether there is a center point close to all matched edges (within distance a constant times the length of the edge). If so, it could be extended to more points via Helly's theorem. Apparently this is related to communication network design. I include a response I sent with a proof (of a constant worse than the one he wanted, but generalizing as well to bipartite matching). The chromatic number of the plane . Gordon Royle and Ilan Vardi summarize what's known about the famous open problem of how many colors are needed to color the plane so that no two points at a unit distance apart get the same color. See also another article from Dave Rusin's known math pages.

9. Past Open Problems Columns - Douglas B. West By Douglas B. West, from the SIAM Activity Group Newsletter in Discrete Mathematics.Category Science Math Combinatorics Graph Theory open problems......Past open problems Columns Douglas B. West. From the readers. Pleasesend me the open problems you would like to see solved! Email http://www.math.uiuc.edu/~west/pcol/pcolink.html

Past Open Problems Columns - Douglas B. West From the SIAM Activity Group Newsletter in Discrete Mathematics These columns are the pre-publication input format sent to the Newsletter editor. In making this archive available more broadly, I am hoping also for input from readers. Please send me the open problems you would like to see solved! Email contributions to west@math.uiuc.edu DBW home page Eventually, we hope to establish a more flexible archive of open problems, searchable by keywords in various fields, with direct links from the problem pages to updates about full or partial solutions. Webmaster volunteers to help establish the searchable archive system are eagerly solicited! Open Problems #25 postscript , Winter-Spring 1998. Open Problems #24 postscript , all seasons, 1997. Open Problems #23 postscript , Summer-Fall 1996. Open Problems #22 postscript , Spring 1996. Open Problems #21 postscript , Winter 1996. Open Problems #20 postscript , Fall 1995. Open Problems #19 postscript , Summer 1995. Open Problems #18 postscript , Spring 1995.

10. Open Problems In Mathematical Physics open problems in Mathematical Physics Quick Access Soft phases in 2D O(N) models Quantum Hall Conductance Exponents dimensions LRO for Quantum Heisengerg Ferr. Extended states Impossibility theorems Fermi gas Entropy production Separatrix http://www.math.princeton.edu/~aizenman/OpenProblems.iamp

Open Problems in Mathematical Physics Quick Access: Soft phases in 2D O(N) models Quantum Hall Conductance dimensions ... IAMP This page leads to a collection of significant open problems gathered from colleagues during the academic year 1998/99. They are offered in the belief that good challenges stimulate our work, tempered by the dictum that preformulated questions should not discourage one from seeking new perspectives. All are invited to send pertinent comments, references to solutions, and contributions for this page to M. Aizenman (Editor): aizenman@princeton.edu List by Contributors By order of submission General Framework Quantum Field Theory Statistical Physics Quantum Many Body Systems Geometry and Physics Schroedinger Operators Disordered Systems Non-equilibrium Relativity and Gravitation Dynamical Systems Fluid Dynamics (Layout webmaster: aizenman@fas.harvard.edu

11. OpenInTopology Compiled by the Algebra and Topology group, Faculty of Mechanics and Mathematics, Lviv, Ukraine. http://www.franko.lviv.ua/faculty/mechmat/Departments/AlgTop/Seminars/OpenInTopo

Back to Homepage OPEN Problems in Topology (The reader needs to be familiar with TeX) Infinite-Dimensional Topology Infinite-Dimensional Manifolds and related Topics Category Topology Hyperspaces, Superextensions etc. Topological Algebra Topological Groups, Semigroups, Rings etc. Back to Homepage of Algebra and Topology

12. Kézdy -- Some Open Problems Modulo n, Cyclic Neofields, and Tree Embeddings. These problems arise from some of my work with Hunter Partitioning Permutations. The problems in this section arise from some of my http://www.louisville.edu/~aekezd01/open/open.html

Sums Modulo n, Cyclic Neofields, and Tree Embeddings These problems arise from some of my work with Hunter Snevily (University of Idaho at Moscow, ID). Z n is alternating if f(i,j) = - f(j,i) (mod n), for all i,j. Permutations are viewed as sequences, so the permutation in S n is viewed as the sequence (n). For i,j, define the distance in from i to j, denoted d(i,j), as the quantity (j) - (i). Clearly d(i,j) = -d(j,i) (i.e. d is an alternating function). Conjecture A: f: [k] x [k] Z n , there exists a permutation in S k , such that d(i,j) f(i,j) (mod n), for all distinct i,j in [k] We have proven Conjecture A when n is prime. For a = (a ,a ,...,a k ) in Z n k , let (n, a ) denote the number of permutations in S k such that (n, a a in Z n k n ``, by H. Snevily, Amer. Math. Monthly, No. 6, June-July (1999), 584-585). Conjecture B : N(n,k) is monotone in n and k. Specifically, N(n,k) and N(n,k) Conjecture C : For n sufficiently large with respect to k, N(n,2k) = (k!) and N(n,2k+1) = (k+1)(k!) Note that, if true, Conjecture C would be sharp because a =(0,...0,n-1...n-1) achieves the bound (where the number of 0's is floor(k/2) and the number of n-1's is ceiling(k/2)).

13. Open Problems In Discrete Mathematics From the SIAM DM activity group newsletter and other sources. http://www.siam.org/siags/siagdm/siagdmopen.htm

search: Open Problems Open Problems Columns from the SIAM DM activity group newletter, by Doug West Graph Coloring Problems the Archive from the Jensen/Toft book. Unsolved problems from Bondy and Murty text with comments from Steven Locke Unsolved Mathematics Problems list of pages on the web by Steve Finch of Mathsoft Questions/Comments about our Web pages? Use our suggestion box or send e-mail to the Online Services Manager. About SIAM Membership Journals SIAM News ... Laura B. Helfrich , Online Services Manager Updated: LBH Links Member Directory Books and Journals Conferences Open Problems ... Miscellaneous Links

14. Links To Open Problems In Mathematics, Physics And Financial Econometrics Long standing open problems and prizes P versus NP The Hodge Conjecture The PoincaréConjecture The Riemann Hypothesis YangMills Existence and Mass Gap Navier http://www.geocities.com/ednitou/

OPEN QUESTIONS August 19th, 2002 MATHEMATICS Long standing open problems and prizes P versus NP The Hodge Conjecture The Poincaré Conjecture The Riemann Hypothesis Yang-Mills Existence and Mass Gap Navier-Stokes Existence and Smoothness The Birch and Swinnerton-Dyer Conjecture Mathematical challenges of the 21st century including moduli spaces and borderland physics Langlands Program Difficult to understand areas: Long to learn: quantum groups motivic cohomology , local and micro local analysis of large finite groups Exotic areas: infinite Banach spaces , large and inaccessible cardinals Unsolved problems: Goldbach conjecture Normality of pi digits in an integer base Polynomial-time algorithm determining if a number is prime Unsolved problems and difficult to understand areas ... Fields Medal and Rolf Nevanlinna Prize PHYSICS Physics Today (NRC) Survey of quantum gravity Required mathematics Required physics ... Important unsolved problems in physics Quantum gravity Explaining high-Tc superconductors Complete theory of the nucleus Realizing the potential of fusion energy Climate prediction Turbulence Glass physics Solar magnetic field Complexity, catastrophe and physics

15. Open Problems In Linear Analysis And Probability Problems taken from workshop lectures given at Texas A M University. http://www.math.tamu.edu/research/workshops/linanalysis/problems.html

Open Problems in Linear Analysis and Probability The problems here were either submitted specifically for the purpose of inclusion in this page, or were taken from talks given during the Workshop in Linear Analysis and Probability. Click here to view the postscript version of the file. (Submitted by G. Pisier) Let $1 (Submitted by G. Pisier) Describe the Schur multipliers which are bounded on $S_p$ for $0 < p If you have any open problems you would like to publicize, please contact cherylr@math.tamu.edu and I will add them to the list.

16. Open Problems In Group Theory Part of the Magnus project. Contains over 150 problems in group theory, both well known and relatively new. http://zebra.sci.ccny.cuny.edu/web/nygtc/problems/

Open Problems in combinatorial and geometric group theory This page has been accessed times since 10/16/97. We have collected here over 150 open problems in combinatorial group theory, and we invite the mathematical community to submit more problems as well as comments, suggestions, and/or criticism. Please send us e-mail at sphynx@rio.sci.ccny.cuny.edu This collection of problems has been selected by G.Baumslag, A.Miasnikov and V.Shpilrain with the help of several members and friends of the New York Group Theory Cooperative. In particular, we are grateful to G.Bergman G.Conner W.Dicks R.Gilman ... I.Kapovich , V. Remeslennikov, V.Roman'kov E.Ventura and D.Wise for useful comments and discussions. Our policy Hall of Fame We have arranged the problems under the following headings: Outstanding Problems Free groups One-relator groups Finitely presented groups ... Algorithmic problems Periodic groups (under construction) Groujps of matrices Hyperbolic and automatic groups Nilpotent groups Metabelian groups ... Group actions

17. Open Problems Articles in PostScript format. http://www.cecm.sfu.ca/personal/pborwein/CA_MOSAIC/PROBLEMS/A_PROBLEMS.html

Problems Each entry includes a postscript version of the paper and a discussion section. The problems are available online only in postscript form. If you have solutions, references or additional comments please send them to pborwein@cecm.sfu.ca. Online Problems P95-1: Some old and new problems in approximation theory P95-2: Two problems on interpolation P96-1: A convergence problem for rational interpolants P96-4: Bivariate segment approximation and free knot splines ... P97-2: Conjectures around the Baker-Gammel-Wills conjecture

18. Some Open Problems open problems and conjectures concerning the determination of properties of families of graphs.Category Science Math Combinatorics Graph Theory open problems......open problems and conjectures concerning the determination of propertiesof families of graphs. Some open problems and Conjectures. http://www.eecs.umich.edu/~qstout/constantques.html

Some Open Problems and Conjectures These problems and conjectures concern the determination of properties of families of graphs. For example, one property of a graph is its domination number. For a graph G , a set S of vertices is a dominating set if every vertex of G is in S or adjacent to a member of S . The domination number of G is the minimum size of a dominating set of G . Determining the domination number of a graph is an NP-complete problem, but can often be done for many graphs encountered in practice. One topic of some interest has been to determine the dominating numbers of grid graphs (meshes), which are just graphs of the form P(n) x P(m) , where P(n) is the path of n vertices. Marilynn Livingston and I showed that for any graph G , the domination number of the family G x P(n) has a closed formula (as a function of n ), which can be found computationally. This appears in M.L. Livingston and Q.F. Stout, ``Constant time computation of minimum dominating sets'', Congresses Numerantium (1994), pp. 116-128. Abstract Paper.ps

19. Open Problems On Model Categories Problems on model categories listed by Mark Hovey at Wesleyan University. http://claude.math.wesleyan.edu/~mhovey/problems/model.html

Model categories This is part of an algebraic topology problem list , maintained by Mark Hovey I am not sure working on model categories is a safe thing to do. It is too abstract for many people, including many of the people who will be deciding whether to hire or promote you. So maybe you should save these until you have tenure. A scheme is a generalization of a ring, in the same way that a manfold is a generalization of R^n. So maybe there is some kind of model structure on sheaves over a manifold? Presumably this is where de Rham cohomology comes from, but I don't know. It doesn't seem like homotopy theory has made much of a dent in analysis, but I think this is partly due to our lack of trying. Floer homology, quantum cohomologydo these things come from model structures? Every stable homotopy category I know of comes from a model category. Well, that used to be true, but it is no longer. Given a flat Hopf algebroid, Strickland and I have constructed a stable homotopy category of comodules over it. This clearly ought to be the homotopy category of a model structure on the category of chain complexes of comodules, but we have been unable to build such a model structure. My work with Strickland is still in progress, so you will have to contact me for details. Given a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the category of monoids in C is again a model category (with underlying fibrations and weak equivalences). On the other hand, the category of commutative monoids seems to be much more subtle. It is well-known that the category of commutative differential graded algebras over Z can not be a model category with uinderlying fibrations and weak equivalences (= homology isos). On the other hand, the solution to this is also pretty well-knownyou are supposed to be using E-infinity DGAs, not commutative ones. Find a generalization of this statement. Here is how I think this should go, broken down into steps. The first step: find a model structure on the category of operads on a given model category. (Has this already been done? Charles Rezk is the person I would ask). We probably have to assume the model category is cofibrantly generated.

20. Open Problems In Algebraic Topology Problems in algebraic topology, listed by Mark Hovey, mathematician at Wesleyan University. http://claude.math.wesleyan.edu/~mhovey/problems/

Mark Hovey's Algebraic Topology Problem List This list of problems is designed as a resource for algebraic topologists. The problems are not guaranteed to be good in any wayI just sat down and wrote them all in a couple of days. Some of them are no doubt out of reach, and some are probably even worseuninteresting. I ask that anybody who gets anywhere on any of these problems, has some new problems to add, or has corrections to any of them, please keep me informed (mhovey@wesleyan.edu). If I mention a name in a problem, it might be good to consult that person before working too hard on the problem. However, even if the problems we work on are internal to algebraic topology, we must strive to express ourselves better. If we expect our papers to be accepted in mathematical journals with a wide audience, such as the Annals, JAMS, or the Inventiones, then we must make sure our introductions are readable by generic good mathematicians. I always think of the French, myselfI want Serre to be able to understand what my paper is about. Another idea is to think of your advisor's advisor, who was probably trained 40 or 50 years ago. Make sure your advisor's advisor can understand your introduction. Another point of view comes from Mike Hopkins, who told me that we must tell a story in the introduction. Don't jump right into the middle of it with "Let E be an E-infinity ring spectrum". That does not help our field. Here are the problems: Major problems in the field Morava K- and E-theory Elliptic cohomology Applications ... Miscellaneous Unstable modules and algebras over the Steenrod algebra (coming soon)

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