Below are some questions that members of the
mathematical community at IISER find interesting and/or
- The “Hawaiian Earring”
is the union of circles that pass through the origin and have
centres on a ray at distances of 1∕n from the origin as n ranges over the natural numbers.
Is there Torsion in the fundamental group
of the Hawaiian earring?
- Galois Theory associates
a finite group to every polynomial equation. Noether’s inverse
Galois problem asks whether, given a finite group G, one can find a polynomial whose Galois
group is G. It is already
interesting to find such a polynomial for the alternating group
- The homotopy groups of
spheres are finitely generated abelian groups. In fact, they
are often finite. Given natural numbers n and m with
n > m > 1, one would like to
compute properties of the group πn(Sm)
like its size and the fundamental divisors.
- Abel showed that a
general equation of degree 5 over rationals cannot be solved
using radicals (n-th roots). Can
“good” approximate solutions be found using radicals? Here
“good”-ness is measured not only in terms of usual distance but
also in number-theoretic terms by p-adic norms.
- What can we say about
the finite subgroups of the group of square matrices of size
n with integer coefficients and
- If G is a non-Abelian group and g is any element, then an inner automorphism
ιG,g of G is
defined by hghg-1. Clearly,
if G′ is a group that contains
G, then ιG′,g restricts to ιG,g
on G′. In other words, the automorphism
iG,g extends to every larger group G′.
Is the converse true? Given that an
automorphism τ of G extends to every group G′ that
contains G, is it true that
τ = ιG,g
for some g in G.
- The only group without
automorphisms is ℤ∕2ℤ. What
kinds of groups have larger automorphism groups than themselves
(“Abundant Groups”) and smaller automorphism groups than
themselves (“Deficient Groups”).
- A manifold
M is said to be parallelisable
if its tangent bundle T(M) is
isomorphic to M × ℝn (as a vector bundle over M). Given a manifold, one can define a
sequence of manifolds
Is it true that one of these is
parallelisable? Iterated tangent bundle is