1. 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.

2. Open Problems Collected by Jeff Erickson. Mainly in geometry. 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! 30 Jul 2003: Complete or partial solutions for several of these problems have been discovered in the two years since I last updated this site. Over the next few weeks, I'm planning to add pointers to these new results, as well as descriptions of several new open problems. (Search for "soon" on this page.) Stay tuned! Existence Problems: Does Object X exist?

3. 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.

4. Links To Open Problems In Mathematics, Physics And Financial Econometrics Lists of unsolved problems Long standing open problems and prizes P versus NP List of open questions including condensed matter problems http://www.geocities.com/ednitou/

RESEARCH OPEN QUESTIONS September 16th, 2005 GENERAL Lists of unsolved problems Science magazine 125 big questions MATHEMATICS (PHYSICIST'S PERSPECTIVE) Sir Michael Atiyah's Fields Lecture (.ps) 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 Some recent links between mathematics and physics Number theory and physics Conjectured links between the Riemann zeta function and chaotic quantum-mechanical systems Deep and relatively recent ideas in mathematics and physics Standard model and mathematics: Gauge field or connection Dirac operators or fundamental classes in K-theory ( Atiyah-Singer index theorem String theory and mathematics: Mirror symmetry Conformal field theory Mathematics behind supersymmetry Mathematics of M-Theory Chern-Simons theory Unified theory: Langlands Program Theory of "motives" Lists of unsolved problems Long standing open problems PRICE 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 Mathworld list Mathematical challenges of the 21st century including moduli spaces and borderland physics Goldbach conjecture Normality of pi digits in an integer base Unsolved problems and difficult to understand areas PRICES Fields Medal and Rolf Nevanlinna Prize Abel Prize PHYSICS Important unsolved problems in physics Quantum gravity Explaining high-Tc superconductors

5. Some Open Problems 9) Problems on 2 dimensional partial orders open problems include several This is a famous open problem, but even the following is unknown and must be http://www.vuse.vanderbilt.edu/~spin/open.html

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

6. Open Problems List A collection of papers outlining unsolved problems maintained at Stony Brook. 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

7. The Geometry Junkyard: Open Problems Compiled by David Eppstein of the University of California at Irvine. 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.

8. Open Problems Originally from the Katsiveli 2000 open problems Session, now maintained by Sergiy Kolyada. PDF/PS. http://www.math.iupui.edu/~mmisiure/open/

Other sites with this page O pen P roblems in D ynamical S ystems E rgodic T heory Welcome! Katsiveli - 2000 Open Problems Session. New problems are being added to it. If you would like to submit some open problems to this page, please send them to Sergiy Kolyada If you have any remarks about this page, please write to Sergiy Kolyada or Michal Misiurewicz Geometric models of Pisot substitutions and non-commutative arithmetic Submitted by Pierre Arnoux (corrections - November 29, 2001) Ergodic Ramsey Theory - an update Submitted by Vitaly Bergelson (see also here Dense periodic points in cellular automata Submitted by Francois Blanchard Non-discrete locally compact second countable groups Submitted by Sergey Gefter Martingale convergence and ergodic theorems Submitted by Alexander Kachurovskii The problem is closed (October 21, 2002) Entropy, periodic points and transitivity of maps Submitted by Sergiy Kolyada and Lubomir Snoha (corrections - October 26, 2002) Natural spectral isomorphisms Submitted by Jan Kwiatkowski Density of periodic orbit measures for piecewise monotonic interval maps Submitted by Peter Raith Polygonal billiards: some open problems Submitted by Pascal Hubert and Serge Troubetzkoy Is any kind of mixing possible in "ToP" N-actions?

9. 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 PDF version of the file. Problem #1 (Submitted by Leonid Kovalev) Problem #2 (Submitted by Gilles Pisier) The answer to this question is known to be "YES" in the cases: Problem #3 (Submitted by Gilles Pisier) Problem #4 (Submitted by Roger Smith) Problem #5 (Submitted by Deguang Han) Problem #6 (Submitted by Michal Johanis) The answer to this question is known to be yes if $B$ is separable space.

10. Past Open Problems From the SIAM Activity Group Newsletter in Discrete Mathematics. In PostScript.Compiled by Douglas B. West. 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.

11. 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.

12. Open Problems In Combinatorics Features links to newsletters, workshops, and papers. http://www.combinatorics.net/problems/

13. OpenProblems This web page contains a list of open problems in Discrete and ComputationalGeometry. Contributions to the list are invited. To contribute problems, submit 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.

14. Open Problems In Algebraic Topology Problems in Algebraic Topology, compiled by mathematician Mark Hovey of 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)

15. Some Open Problems open problems and conjectures concerning the determination of properties offamilies of graphs. 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

16. Kézdy -- Some Open Problems Kézdy s open problems. 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)).

17. The Geometry Junkyard Open Problems open problems Antipodes. Jim Propp asks whether the two farthest apart points, as measured by surface distance, on a symmetric convex body must http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

18. The Open Problems Project This is the beginning of a project1 to record open problems of interest to To begin navigating through the open problems, you may select from a category http://cs.smith.edu/~orourke/TOPP/

Next: Numerical List of All The Open Problems Project edited by Erik D. Demaine Joseph S. B. Mitchell Joseph O'Rourke Introduction This is the beginning of a project to record open problems of interest to researchers in computational geometry and related fields. It commenced with the publication of thirty problems in Computational Geometry Column 42 [ ] (see Problems 1-30 ), but has grown much beyond that. We encourage correspondence to improve the entries; please send email to TOPP@cs.smith.edu . If you would like to submit a new problem, please fill out this template Each problem is assigned a unique number for citation purposes. Problem numbers also indicate the order in which the problems were entered. Each problem is classified as belonging to one or more categories. The problems are also available as a single Postscript or PDF file. To begin navigating through the open problems, you may select from a category of interest below, or view a list of all problems sorted numerically Categorized List of All Problems Below, each category lists the problems that are classified under that category. Note that each problem may be classified under several categories.

19. Unsolved Problems A good paper on open and solved cageproblems is P.K. Wong, Cages-A Survey , JGT, Vol.6 (1982) 1-22 Newer results can be found on Gordon http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

20. Open Problems On Discrete And Computational Geometry open problems on Discrete and Computational Geometry. Introduction http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

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