Below are some questions that members of the mathematical community at IISER find interesting and/or challenging.

- 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
A
_{n}. - 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 π
_{n}(S^{m}) 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 determinant 1?
- If G is a non-Abelian group and g is any element, then an inner automorphism
ι
_{G,g}of G is defined by hghg^{-1}. Clearly, if G′ is a group that contains G, then ι_{G′,g}restricts to ι_{G,g}on G′. In other words, the automorphism i_{G,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 = ι

_{G,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 × ℝ
^{n}(as a vector bundle over M). Given a manifold, one can define a sequence of manifoldsIs it true that one of these is parallelisable? Iterated tangent bundle is parallelisable.