1. 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! 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?

2. 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.

3. The Geometry Junkyard: 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.

4. Links To Open Problems In Mathematics, Physics And Financial Econometrics OPEN QUESTIONS. April 13th, 2004. MATHEMATICS. Fields Medal and Rolf Nevanlinna Prize. PHYSICS. Important unsolved problems in physics. Quantum gravity. Explaining highTc superconductors. Complete http://www.geocities.com/ednitou

OPEN QUESTIONS April 13th, 2004 MATHEMATICS Lists of unsolved problems 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 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 Related links Sir Michael Atiyah's Fields Lecture (.ps) 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 Polynomial-time algorithm determining if a number is prime 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 Unified theory: Langlands Program , Theory of "motives"

5. Some Open Problems Please mail me with any corrections, or any other comments. The list is very rough at the moment. dimensional partial orders open problems include several variants of scheduling problems (e http://www.vuse.vanderbilt.edu/~spin/open.html

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

6. Open Problems List open problems in Dynamical Systems. open problems in holomorphic dynamics. Two lists of problems are currently available online We are soliciting open problems in various areas of Dynamical Systems 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. Open Problems Other sites with this page. open problems. in Dynamical Systems Ergodic Theory. Welcome! This collection of open problems in Dynamical Systems Ergodic Theory originated from the Katsiveli 2000 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?

8. 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.

9. Open Problems Some questions compiled by Jon Perry. http://www.users.globalnet.co.uk/~perry/maths/openproblems/openproblems.htm

Open Problems [Home] [Other Maths] Open Problems This is a collection of my own open problems. These already have their own page: Triangular Transforms - see Triangular Transforms Lucas-Carmichaels - see Lucas-Carmichaels Prime Puzzles conjecture 33 These have no write-up as yet: Consider an AP, with gcd(a,b)=1, and define an extra parameter, p, p prime, and gcd(a,p)=gcd(b,p)=1. So the AP is represented (a,b,p). Does (a,b,p) have an infinite number of primes for k!=0modp? Do we need all the integers to form all primes of the form 1mod4 as the sum of 2 squares? This can be stated as; for every x does there exists a y such that x +y is prime? Let A(n) be the set of integers less than n such that n+a_i is prime. Does A(n)=A(m) imply n=m? Goldbach Slim : Can every odd integer be represented as T(n)+p, with T(0)=0 allowed, and p optional? Playstation Numbers : Can every positive integer can be represented as some combination of 1, T(n), x and y (representing the circle, triangle, square and X)? Perfect Partition Problem : The conjecture states that the proper divisors of the perfect number 2^(p-1).(2^p-1)=T(2^p-1) always have a sumwise representation via the integers 1..T(2^p-1). e.g. the proper divisors of 28 are 1,2,4,7,14, and this can be written as 1,2,4,7,3+5+6.

10. Kézdy -- Some Open Problems 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). 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)).

11. 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

12. 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.

13. 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 daly@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

14. 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

15. My Favorite Open Problems open problems in Discrete Math. Prizes. More online collections of open problemsin discrete math Dan Archdeacon s problems in topological graph theory; http://www.math.princeton.edu/~matdevos/open/open.html

Open Problems in Discrete Math Prizes Flows on Graphs Tutte's 3-flow, 4-flow, and 5-flow conjectures The 2+epsilon flow conjecture Jaeger's modular orientation conjecture Bouchet's 6-flow conjecture ... Unit vector flows Cycle Covers The cycle double cover conjecture The circular embedding conjecture Conjectures on (m,n)-cycle-covers Faithful circuit covers ... Circuit decompositions of Eulerian graphs Choosability for Ax=y A nowhere-zero point in a linear mapping The additive basis conjecture The permanent conjecture The Alon-Tarsi basis conjecture Edge Coloring The Berge-Fulkerson conjecture The Petersen coloring conjecture A generalization of Vizing's Theorem Packing T-joins ... Conjectures on edge-connectivity Vertex Coloring Homomorphisms of planar graphs Homomorphisms to the circuit of length 5 Directed Graphs Seymour's second neighborhood conjecture Reversing edges in digraphs Packing dijoins Topological Graph Theory Grunbaum's conjecture Embedding digraphs Conway's thrackle conjecture Matroid Theory Bases of many weights Stable bases Group Theory / Additive Number Theory Freiman's theorem for arbitrary groups Sets with distinct subset sums Miscellaneous Problems The lonely runner conjecture Graham's conjecture on tree reconstruction please feel free to email me if you know of unmentioned progress (possibly a solution) to any of these problems.

16. Some 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

17. 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.

18. 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)

19. Mesh Generation: Open Problems ICS Theory Group ICS 280G, Spring 1997 Mesh Generation for Graphicsand Scientific Computation. open problems. We proved (4/3/97) the http://www.ics.uci.edu/~eppstein/280g/open.html

ICS 280G, Spring 1997: Mesh Generation for Graphics and Scientific Computation Open Problems We proved ( ) the existence of triangulations of any polygon or straight line graph, and of convex quadrilateralizations of any orthogonal polygon. What about curved objects? Do spline-polygons have spline-triangulations? An example formed by connecting four quarter-ellipses shows Steiner points may be needed, even for quadratic splines, but maybe they only need to be added in the interior of the splinegon. On we went over dynamic programming techniques for optimal triangulation (e.g. minimum total edge length) of simple polygons, in O(n ) time or O(E ) if the visibility graph has E edges. So the slowest case is seemingly the most simple, when the polygon is convex. Can we find the minimum length triangulation of convex polygons in o(n ) time? Steve S. suggested Frances Yao's generalization of Knuth's speedup to optimal binary search tree construction (which has the same general dynamic programming form) but it doesn't seem to work. The same dynamic programming methods also work for optimal quadrilateralization, in time O(n

20. OpenProblems open problems on Discrete and Computational Geometry. Introduction This web pagecontains a list of open problems in Discrete and Computational Geometry. http://www.site.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.

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