Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.
The famous 0-1 Knapsack problem is: Given and integers, determine whether or not there are values so that The best known worst-case algorithm runs in time times a polynomial in . Is there an algorithm that runs in time ?
Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
Let be a graph. If and are two integers, a -colouring of is a function from to such that for each edge . Given a list assignment of , i.e.~a mapping that assigns to every vertex a set of non-negative integers, an -colouring of is a mapping such that for every . A list assignment is a --list-assignment if and for each vertex . Given such a list assignment , the graph G is --colourable if there exists a --colouring , i.e. is both a -colouring and an -colouring. For any real number , the graph is --choosable if it is --colourable for every --list-assignment . Last, is circularly -choosable if it is --choosable for any , . The circular choosability (or circular list chromatic number or circular choice number) of G is
Problem What is the best upper bound on circular choosability for planar graphs?
Conjecture For every prime , there is a constant (possibly ) so that the union (as multisets) of any bases of the vector space contains an additive basis.
Conjecture If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
Conjecture For every graph without a bridge, there is a flow .
Conjecture There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.
Problem has the homotopy-type of a product space where is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of .
Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.
Conjecture Let be a sequence of points in with the property that for every , the points are distinct, lie on a unique sphere, and further, is the center of this sphere. If this sequence is periodic, must its period be ?
Conjecture Let be a simple graph with vertices and list chromatic number . Suppose that and each vertex of is assigned a list of colors. Then at least vertices of can be colored from these lists.
Problem (2) Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .
Conjecture If is a bridgelesscubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .
Problem Let be an indexed family of filters on sets. Which of the below items are always pairwise equal?
1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .
2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .
Conjecture \item If is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph of a locally finite graph is 4-connected, then is hamiltonian.
Conjecture If is the adjacency matrix of a -regular graph, then there is a symmetric signing of (i.e. replace some entries by ) so that the resulting matrix has all eigenvalues of magnitude at most .
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such that for every graph 
, ![\[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]](/files/tex/989db06683633e86605c26e7d9f0bffc7e46a496.png)
and 
integers, determine whether or not there are 
values 
so that 
The best known worst-case algorithm runs in time 
times a polynomial in 
. Is there an algorithm that runs in time 
?
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $](/files/tex/f5fa90ccf8c1dde9b9bf1f21399b3b5428dab81d.png)
, whenever 
.
, is it still valid?
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
be a graph. If 
and 
are two integers, a 
-colouring of 
from 
to 
such that 
for each edge 
. Given a list assignment 
of 
a set of non-negative integers, an 
such that 
for every 
. A list assignment 
-
and 
for each vertex 
. Given such a list assignment 
, the graph 
(possibly 
) so that the union (as multisets) of any 
contains an additive basis. 
. 
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
.
so that antipodal points of 
receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero. 
has the homotopy-type of a product space 
where 
is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of 
-degenerate graph (a forest) and a 
-degenerate graph is 
-colourable. 
be a set of ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
has an acyclic induced subdigraph of order at least 
. 
be a sequence of points in 
with the property that for every 
, the points 
are distinct, lie on a unique sphere, and further, 
is the center of this sphere. If this sequence is periodic, must its period be 
? 
is a 2-transition system for 
has a compaible decomposition. 
. Suppose that 
and each vertex of 
vertices of 
and 
it holds
if 
if 
; \item 
; \item 
. 
, whether there exists a homomorphism from 
for some constant 
or 
which divides both 
(see 
(see 
of 
be an indexed family of filters on sets. Which of the below items are always pairwise equal?
. 
-graph is an 
for every 
with odd size.
for every 
of a locally finite graph 
is the adjacency matrix of a 
-regular graph, then there is a symmetric signing of 
entries by 
) so that the resulting matrix has all eigenvalues of magnitude at most 
. 
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