Passing arguments from Panel to View without url or using contextual filter from panel to field of taxomony term within view
The easier (semi-automatic) way I can think of
is to


Override the path /taxonomy/term/%taxonomy_term
(enable the *term_view* page provided in system) in Page manager.
Build a page with panels for this path.
Now your content panes, when used in this page, can get their arguments
with the 'From panel argument' option.


You can always have SEO-friendly URLs like
mysite/education/fellowships/surgical or mysite/education/fellowships/transmed
with path (in /admin/config/search/path/add).

The other way is to use the 'Input on pane config' option
for your content panes.
You will have to set the arguments for these panes manually
in their settings dialogue.
In your case, the argument should probably be the (numerical) term ID.
This way your content panes don't need to be tied
to a specific path pattern, and you can have
some logical combinations in your contextual filters,
such as 1+3+5 (term 1 OR 3 OR 5).
This can get quite laborious, though.