相关论文: A Monstrous Proposal

A conjecture of Boone and Higman from the 1970's asserts that a finitely generated group $G$ has solvable word problem if and only if $G$ can be embedded into a finitely presented simple group. We comment on the history of this conjecture…

We compute the number of orbit types for simply connected simple algebraic groups over algebraically closed fields as well as for compact simply connected simple Lie groups. We also compute the number of orbit types for the adjoint action…

The purpose of this short note is to present a simplified proof of Serre's modularity conjecture using the strong modularity lifting results currently available. This second version includes extra details on definitions and proofs than the…

New cases of the multiplicity conjecture are considered.

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be…

We prove that every non-abelian finite simple group is generated by an involution and an element of prime order.

Monstrous Moonshine was extended in two complementary directions during the 1980s and 1990s, giving rise to Norton's Generalized Moonshine conjecture and Ryba's Modular Moonshine conjecture. Both conjectures have been unconditionally…

We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.

In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie…

We discuss some categorical aspects of the objects that appear in the construction of the Monster and other sporadic simple groups. We define the basic representation of the categorical torus $\mathcal T$ classified by an even symmetric…

An introduction to Joyal's theory of combinatorial species is given and through it an alternative view of Rota's twelvefold way emerges.

The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ is a torsion-free group then the only units of the group ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of group…

We discuss some new results concerning Gap Conjecture on group growth and present a reduction of it (and its *-version) to several special classes of groups. Namely we show that its validity for the classes of simple groups and residually…

In this note, we provide evidence for new (super) moonshines relating the Monster and the Baby monster to some weakly holomorphic weight 1/2 modular forms defined by Zagier in his work on traces of singular moduli. They are similar in…

We present a general conjecture on the divisibility of a certain expression in terms of Kostka numbers and their close variants. This conjecture is closely related to a variant of the period-index problem of noncommutative algebra, with…

We describe and prove uniqueness of a natural homomorphism between some groups associated to finite sets.

Some simple nonlinear recursions which can be completely managed are identified and the behaviour of all their solutions is ascertained.

The study of the additive volume of sets can be reduced to the case of one-dimensional sets. The exact values of the volume of extremal sets are given as a conjecture.

We conjecture that if $G$ is a simple compact Lie group with trivial center, then every $d$-variable non-constant word map with coefficients in $G$ defines a non-constant function on $G^d$. We prove the conjecture for $A_r$, $B_r$, $E_6$,…

We settle an old conjecture of Karrass and Solitar by proving that a finitely generated subgroup of a non-trivial free product $G = A\ast B$ has finite index if and only if it intersects non-trivially each non-trivial normal subgroup of…