Notes to the Feit-Thompson conjecture

A proof of the Thompson moonshine conjecture

Notes on the 2n-conjecture∗

A sign pattern is an n × n matrix, A, with entries in {+,−, 0}. If A is a real n× n matrix for which each entry has the same sign as its corresponding entry in A, then A is a realization of A, and we write A ∈ A. The 2n-conjecture is related to the study of the spectral properties among the matrices in A. The n × n sign pattern A is a spectrally arbitrary pattern (or a SAP, for short), provided...

Notes on Dickson’s Conjecture

In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson’s conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson’s conjecture to the higher order integral polynomial case. However, they did not generalize Dickson’s conjecture to the multivariable case. In 2006, Green and Tao [13] considered Dickson’s conjecture in the multivariable case and gave direct...

Notes on Chvátal's conjecture

Using Kleitman’s lemma and results of Sch2 onheim and Mikl os it is shown that if w(D)= |D|=2, then every maximum-sized intersecting family in D contains all base elements of D. Then, the converse of this statement is conjectured and shown that this is equivalent to that of Chv atal. c © 2002 Elsevier Science B.V. All rights reserved.

Doran–Harder–Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves

We prove the Doran–Harder–Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi– Yau manifold X degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi–Yau manifold of X can be constructed by gluing the two mirror Landau– Ginzburg models of the quasi-Fano manifolds. The two crucial i...