### Upcoming Talks

#### Algebra Kolloquium

**Title:**Monoidal Shannon Extensions

**Speaker:**Dr. Lorenzo GUERRIERI (Ohio State Univ., USA)

**Date:**Freitag, 14. 12. 2018, 11:00 c.t.

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

Abstract:

Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d \geq 2$. A \it local monoidal transform \rm of $R$ is a ring of the form $$R_1= R \left[ \frac{\mathfrak{p}}{x} \right]_{\mathfrak{m}_1}$$ where $ \mathfrak{p} $ is a prime ideal generated by regular parameters, $x \in \mathfrak{p}$ is a regular parameter and $ \mathfrak{m}_1 $ is a maximal ideal of $ R[\frac{\mathfrak{m}}{x}] $ lying over $ \mathfrak{m}. $ If $\mathfrak{p}= \mathfrak{m}$ the ring $R_1$ is called a \it local quadratic transform\rm.

Recently, several authors studied the rings of the form $ S= \cup_{n \geq 0}^{\infty} R_n $ obtained as infinite directed union of iterated local quadratic transforms of $R$, and call them \it quadratic Shannon extension\rm.

A directed union of local monoidal transforms of a regular local ring is said \it monoidal Shannon extension\rm.

Here we study features of monoidal Shannon extensions and more in general of directed unions of Noetherian UFDs.

(L. Guerrieri, Directed unions of local monoidal transforms and GCD domains (2018) arXiv:1808.07735 )

#### Algebra Kolloquium

**Title:**Weighted Leavitt path algebras and the normal structure of classical-like groups

**Speaker:**Dr. Raimund PREUSSER (Univ. of Brazilia, Brazil)

**Date:**Freitag, 14. 12. 2018, 16:00 c.t.

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

In the first part of the talk, I will speak about the normal structure of classical and classical-like groups. The description of the normal subgroups for various classes of concrete groups, and especially for classical groups over rings, has been one of the central themes of group theory in the last two centuries, right after Galois introduced the notion of normal subgroups. In the second part of the talk, I will speak about weighted Leavitt path algebras. Weighted Leavitt path algebras are algebras associated to weighted graphs. They generalise in a natural way the usual Leavitt path algebras and also Leavitt's algebras of module type $(n,k)$ where $n,k>0$.

#### Algebra Kolloquium

**Title:**Factorization of matrices over integral domains into products of elementary and idempotent matrices

**Speaker:**Dr.Laura Cossu (Univ. Padova)

**Date:**10. 12. 2018, 9:00 c.t.

**Room:**Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II

**Abstract:**

\noindent

It is well known that Gauss Elimination produces a factorization into elementary matrices of any invertible matrix over a field. A classical problem, studied since the middle of the 1960's, is to characterize integral domains different from fields that satisfy the same property. As a partial answer, in 1993, Ruitenburg proved that in the class of B\'ezout domains, any invertible matrix can be written as a product of elementary matrices if and only if any singular matrix can be written as a product of idempotents.

\noindent

In this talk, after giving an overview of the classical results on these factorization properties, we will present some recent developments on the topic. In particular, we will consider products of elementary and idempotent matrices over special classes of non-Euclidean PID's and over integral domains that are not B\'ezout.

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\begin{itemize}{\footnotesize

\item[\textnormal{[1]}] L.~Cossu, P.~Zanardo, U.~Zannier,

Products of elementary matrices and non-Euclidean principal ideal domains}, J. Algebra 501: 182â€“205, 2018.

\item[\textnormal{[2]}] L.~Cossu, P.~Zanardo,

Factorizations into idempotent factors of matrices over Pr\"ufer domains}, accepted for publication on Communications in Algebra, 2018.

}\end{itemize}

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\noindent{\footnotesize

\textsc{Department of Mathematics ``Tullio Levi Civita'', University of Padova}

{Via Trieste, 63}

{35121, Padova}

{Italy}

}

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\noindent{\footnotesize{E-mail address}:

lcossu@math.unipd.it}

}