Related papers: The A-theoretic Farrell-Jones Conjecture for virtu…

Conjugacy separability of any group of the class of one-relator groups given by the presentation $<a, b; [a^m,b^n]=1>$ ($m,n>1$) is proven.

We study the link between stably finiteness and stably projectionless-ness for $C^*$-algebras of solvable Lie groups. We show that these two properties are equivalent if the dimension of the group is not divisible by $4$; otherwise, they…

We give a survey of recent developments in the investigation of the various local-global conjectures for representations of finite groups.

We develop a version of controlled algebra for simplicial rings. This generalizes the methods which lead to successful proofs of the algebraic K- theory isomorphism conjecture (Farrell-Jones Conjecture) for a large class of groups. This is…

In this note, we initiate the study of $\mathcal{F}$-weights for an $\ell$-local compact group $\mathcal{F}$ over a discrete $\ell$-toral group $S$ with discrete torus $T$. Motivated by Alperin's Weight Conjecture for simple groups of…

In this paper, the definition of Hom-Lie groups is given and one conntected component of Lie group $GL(V)$, which is not a subgroup of $GL(V)$, is a Hom-Lie group. More, we proved that there is a one-to-one relationship between Hom-Lie…

We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity $m$, and use it to prove Wilf's conjecture holds whenever $m \le 18$. Our algorithm utilizes techniques from polyhedral…

We show that strong approximate lattices in higher-rank semi-simple algebraic groups are arithmetic.

We prove a generalization of the Fulton-Hansen connectedness theorem, where ${\mathbb P}^n$ is replaced by a normal variety on which an algebraic group acts with a dense orbit.

We show an invariance result for the L2-torsion of groups under uniform measure equivalence provided a measure-theoretic version of the determinant conjecture holds. The measure-theoretic determinant conjecture is discussed and, for…

This thesis aims to serve as an introduction to the theory of quasitilings for amenable groups. In order to showcase the power of this theory, we focus on the study of the Sofic L\"uck Approximation Conjecture, which can be proven for…

We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove the main Uniqueness Theorem, analogous to the Bender method in…

We reprove and generalize the result that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a…

In this paper, we prove Sp\"ath's Character Triple Conjecture for $p$-solvable groups. This is a conjecture proposed by Sp\"ath during the reduction process of Dade's Projective Conjecture to quasisimple groups. In addition, as suggested by…

We establish a categorical version of Vogan duality for quasi-split real groups. This proves a conjecture of Soergel in the quasi-split case.

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite…

We review the Kohno-Drinfeld theorem as well as a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection D on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes…

In this paper we prove the theorem on freedom for relatively free groups with a single relation (analogous with the well-known result of Magnus) and the theorem on freedom for relatively free Lie algebras with a single relation (analogous…

We prove Schlichting's theorem for approximate subgroups: if $\mathcal{X}$ is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with…

We prove that if $G$ is a finite simple group of Lie type and $S$ a subset of $G$ of size at least two then $G$ is a product of at most $c\log|G|/\log|S|$ conjugates of $S$, where $c$ depends only on the Lie rank of $G$. This confirms a…