appendix_ws2324.tex

\chapter{Lectures in winter semester 2023/2024}

\begin{enumerate}
\item
2023-10-10: The Kaplansky conjectures and their implications, residually finite groups are directly finitely.

\item
2023-10-12: Unique product property, left orderability, statement of Burns--Hale, \cref{corollary:LI_kaplansky}.

\item
2023-10-17: Proof of Burns--Hale theorem, dynamical point of view, diffuse groups.

\item
2023-10-19: Hyperbolic groups, Delzant's theorem.

\item
2023-10-24: Primality of group rings (Connell's theorem) up to the proof of \cref{lemma:fc_is_locally_finite_by_free_abelian}.

\item
2023-10-26: Rest of proof of Connell's theorem.

\item
2023-10-31: Exercises (on torsion, FC-groups, $\F_2[\Z/3]$), trace map, traces of idempotents and nilpotents for finite $G$.

\item
2023-11-02: Inner product on $\C[G]$, alternative proof for trace of complex idempotents for finite $G$, start of proof for arbitrary $G$ (Kaplansky's theorem) with generalized Cauchy--Schwarz.

\item
2023-11-07: Rest of approximation-based proof of Kaplansky's theorem on complex idempotents, generalization to arbitrary characteristic zero fields, direct finiteness of $K[G]$ when $\operatorname{char}(K) = 0$, places and valuation rings.

\item
2023-11-09: Extension theorem for places, valuations, zero divisors over $\C$ give zero divisors over $\F_{p^n}$ for some $n$, lemma on test elements for places with finitely many elements guaranteed to be in valuation ring.

\item
2023-11-14: The zero divisor conjecture is equivalent to the group of normalized units always being torsion-free, the power map, trace-like functions, traces of nilpotents, ``normal closure traces'' of nilpotents.

\item
2023-11-16: Zalesskii's theorem that the trace of an idempotent is in the prime subfield (using number theoretic black box to get required places to deduce characteristic $0$ case from characteristic $p$ case), Formanek's theorem on idempotents, corollaries of Formanek's theorem.

\item
2023-11-21: Big picture overview of results in the course, the Hantzsche--Wendt group $P$ as an abstract finitely presented group and as a subgroup of $D_\infty \times D_\infty \times D_\infty$.

\item
2023-11-23: Two proofs that $P$ is torsion-free, $P$ is none of the following: bi-orderable, locally indicable, left-orderable, diffuse.

\item
2023-11-28: Ravels and corresponding universal groups (with and without free factor of $\Z$), Bowditch's ravel defines $P$, symmetries of the ravel give group automorphisms.

\item
2023-11-30: Duplexes, the symmetry of Promislow's duplex and its utility in verifying the failure of the unique product property.

\item
2023-12-05: Automorphism group of the direct product of a centreless and an abelian group, trivial-unit-preserving group ring automorphisms, twisted-unitary and symmetric units.

\item
2023-12-07: The ``2 out of 4 cosets'' lemma for units of $K[P]$, reformulation in Boolean satisfiability, start of Mirowicz's computation of $(\F_2[D_\infty])^\times$ (embedding in matrix ring and determinant condition)

\item
2023-12-12: Ping pong lemma, subgroup of units of the form $\ast_\Z \oplus_\N \Z / 2$.

\item
2023-12-14: End of computation of $(\F_2[D_\infty])^\times$ (semi-direct product structure and proof that we found all units), the subgroup of trivial units is self-normalizing under mild assumptions.

\item
2023-12-19: Bi-orderability, torsion-free nilpotent groups are bi-orderable, bi-orderable groups are locally indicable.

\item
2023-12-21: Malcev--Neumann theorem embedding $K[G]$ in a skew field for bi-orderable $G$.

\item
2024-01-09: Stable finiteness of $K[G]$ from direct finiteness of $K[G \times H]$ with $H$ finite, amenability and soficity, cellular automata.

\item
2024-01-11: Surjunctive groups are directly finite, amenability is detected by the Ore condition (Bartholdi--Kielak).

\item
2024-01-16: Semisimplicity.

\item
2024-01-18: Property (T) (cancelled due to snow).

\item
2024-01-23: Revision and Q\&A.

\item
2024-01-25: Revision and Q\&A.

\item
2024-01-30: Live coding demo (using SAT for left-orderability and the unique product property).

\item
2024-02-01: Guest lecture by Grigori Avramidi on conjectures around 2-complexes.
\end{enumerate}