Local Hadwiger's Conjecture

Lecture Note on Local Langlands Conjecture

Unexpected Local Extrema for the Sendov Conjecture

Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P ) to be the maximum of the distances from each of the roots of a polynomial P to that root’s nearest critical point. In this notation, Sendov’s conjecture asserts that d(P ) ≤ 1 for every P ∈ S(n). Define P ∈ S(n) to be locally extremal if d(P ) ≥ d(Q) for all nearby Q ∈ S(n), and note that identi...

Local Finiteness, Distinguishing Numbers, and Tucker's Conjecture

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a nonlocally finite graph for w...

A Local Version of Szpiro's Conjecture

Szpiro’s Conjecture asserts the existence of an absolute constant K > 6 such that if E is an elliptic curve over Q, the minimal discriminant ∆(E) of E is bounded above in modulus by the K-th power of the conductor N(E) of E. An immediate consequence of this is the existence of an absolute upper bound upon min {vp(∆(E)) : p | ∆(E)}. In this paper, we will prove this local version of Szpiro’s Con...

The local Gan-Gross-Prasad conjecture

in terms of the Langlands parametrization of Irr(G). More precisely, the Gan-Gross-Prasad triples come from certain pairs W ⊂ V of quadratic spaces or hermitian spaces (over a fixed quadratic extension E/F ). We will denote uniformly by h the underlying quadratic or hermitian form on these spaces. To get a GGP triple, you need the following two conditions to be satisfied : – W⊥ (the orthogonal ...