On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface Π21 equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any n, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface Πn1, does not imply the existence of such a pair with the associated relation in Σn1 or in a lower class.

An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited

A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and [Formula: see text]o-large generic extensions of the constructible universe L. More specifically, if [Formula: see text] is either Sacks generic or [Formula: see text]o generic real over L, then it is true in L[Formula: see text] that there is a lightface [Formula: see text] equivalence relation Q on the [Formula: see text] set [Formula: see text] with exactly two equivalence classes, and both those classes are non-OD sets.

Extended Scalar Sectors

Extended scalar sectors appear in various extensions of the Standard Model of particle physics, such as supersymmetric models. They are also generic extensions of the Standard Model and can address a number of its shortcomings. Direct searches for additional Higgs bosons and measurements of the 125-GeV Higgs boson, both of which provide insights into the different possible sectors, are carried out at the LHC. This review gives an overview of searches for the additional Higgs bosons and their implications for different models. The discussed analyses comprise searches for neutral and charged Higgs bosons that decay in various final states. In addition, the review summarizes the constraints from precision measurements, including in particular the observed couplings of the 125-GeV Higgs boson. While several models naturally incorporate a Higgs boson with couplings that are similar to the ones in the Standard Model, the measurements of the 125-GeV Higgs boson provide constraints on all considered extensions.

Remarks on PBW bases of Ringel–Hall algebras of cyclic quivers

In this paper, we give a recursive formula for the interesting PBW basis [Formula: see text] of composition subalgebras [Formula: see text] of Ringel–Hall algebras [Formula: see text] of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of [Formula: see text] from the monomial bases [Formula: see text] following [Multiplication formulas and canonical basis for quantum affine, [Formula: see text], Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of [Formula: see text] associated with modules of Loewy length [Formula: see text]. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras.

COHERENT SYSTEMS OF FINITE SUPPORT ITERATIONS

We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń’s diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a ${\rm{\Delta }}_3^1$ well-order of the reals.