I tried to update the core packages using the suggested commands in Bioconductor

> BiocManager::install()

But I get the following error

The downloaded source packages are in
    ‘/tmp/Rtmp6l1eeP/downloaded_packages’
installation path not writeable, unable to update packages: abind, acepack,
  askpass, assertthat, backports, base64enc, BH, bitops, broom, caTools,
  cellranger, cli, clipr, colorspace, crayon, crosstalk, curl, data.table,
  DBI, estimability, fansi, forcats, foreach,.......... rpart, spatial, survival

I tried the way suggested by https://cran.r-project.org/web/packages/BiocManager/vignettes/BiocManager.html#managing-multiple-versions using the method

> BiocManager::valid()

but the answer was the same

I suppose that the problem is: installation path not writeable,

Yes, the path where the files abind, ... are not writeable by you. Probably these packages were installed by some other account, e.g., the administrator, and can't be updated by you. Mostly I would suggest ignoring these messages. Alternatively, investigate ?.libPaths and arrange that only directories you have control of are listed there. You might need to re-install a number of packages, so if you have a deadline tomorrow...