dg.differential geometry – Topological implications of curvature singularities

In popular articles on astronomy/physics, singularities are typically described as “holes or rips in the fabric of space”.
Now algebraic topology has a lot of methods for detecting “holes of various dimensions” in a manifold. However, singularities in general relativity and differential geometry are typically described in terms of geodesic incompleteness which $it intuitively $ suggests “holes or rips in the manifold”.

Question: Is this intuitive idea “mathematically detectable” – as in, does the existence/formation of a curvature singularity on a manifold alter its topology somehow ? Or is it purely a “problem with the metric structure” which can’t be detected topologically ?

dnd 5e – What mechanical implications exist when allowing duplicate stronghold types in a castle?

Matt Colville’s supplement Strongholds & Followers details, among other things, rules for building optional strongholds and castles, with castles being complexes of multiple strongholds.

The strongholds are divided into four broad types: keeps, towers, temples, and establishments. A stronghold (of any variety) can be controlled by only one PC at a time, and can only offer its most direct benefits to that character. On page 11 of Strongholds & Followers the rules state that castles cannot contain more than one of each stronghold type.

A castle cannot contain multiple of the same type of
stronghold. If you build a castle with two towers, for
instance, only one of them grants the benefits of spell
research. Also, a stronghold’s benefit only applies
to one character at a time. (Strongholds & Followers, page 11)

I am running a game using these rules in which two PCs (a Wizard and a Sorcerer) will almost certainly want to build a tower each. Even if they don’t, the party has no use for a temple (or variant on a temple, like a Druid’s Grove). With four players that means that unless there is at least one duplicate stronghold type, at least one player will be left out. This is easy enough to just overrule (I plan on allowing two towers), but I’m curious about why the restriction on duplicates exists. I have not found anything in the book itself.

Is there a mechanical implication to duplicating stronghold types within a single castle?

The implications of complex languages becoming obsolete

Seems like everywhere you look in the last few years, the complex languages seem to occupy the top few positions of the programming languages that are becoming less used. In particular, Haskell seems to pop up on a few of those lists, which is a shame because as a programming language it has been both an enigma and fascination with the purists and those who have been around the industry for a while.

Is there really a trend towards less ‘complex’ programming languages because of the need for people to be able to get up to speed quickly and also start delivering and outputting code faster? Is there any indication that the disuse of languages like Haskell and even the object-based (not object oriented) languages are being replaced by the popular and newer programming languages because of their flexibility (e.g. React.js) and suitability for web applications (which appears to take up a significant portion of code being produced) or platform dependency (e.g. Swift for Objective-C).

networking – Security implications of directly connecting a Windows PC to ISP via Network Adapter with Ethernet cable bypassing the Router

When diagnosing Internet connection issues (slow speed for example), an ISP technician may ask a user to connect their ISP-provided Ethernet cable directly to a device (typically a Windows PC) to run speed tests in the browser or pings, etc. (to rule out the possibility of the Router being the culprit).

What are the likely (realistic) as well as theoretical security implications in as far as getting access to the device / retrieving information from it (accessing files, etc.) under the following assumptions:

  • This is done for a short period of time ~ 30 minutes
  • The new network is identified as Public (in Windows UI)
  • Remote assistance is enabled
  • Windows built-in firewall is OFF, but third-party application-level firewall is enabled (restricts Internet access to apps).

And does this compromise saved passwords of network-mapped drives and locations (which are normally only accessible within LAN via the Router)?

docker – Security implications of granting non-root access to privileged ports (

Lots of solutions to this problem e.g. here and here but in order to decide which is best I’d need to know more about the security implications of each solution (or at least in general).

My context: I’m looking into running a rootless Docker/Podman Nginx container (on an Ubuntu Server 20.04 LTS host). Podman gives the following solution with this error message Error: rootlessport cannot expose privileged port 80, you can add 'net.ipv4.ip_unprivileged_port_start=80' to /etc/sysctl.conf (currently 1024) but reading around it doesn’t seem to me like a great solution because it’s giving access to all users.

set theory – What are the known implications of “There exists a Berkeley cardinal”?

Inspired by this question: What are the known implications of “There exists a Reinhardt cardinal” in the theory “ZF + j”?


$delta$ is Berkeley iff for every $alphaltdelta$ and transitive set $M$ such that $deltain M$, there is some $j: Mprec M$ such that $alphalttext{crit}jltdelta$.

$delta$ is club Berkeley iff for every club $Csubseteqdelta$ and transitive set $M$ such that $deltain M$, there is some $j: Mprec M$ such that $text{crit}jin C$.

$delta$ is limit club Berkeley if $delta$ is club Berkeley and a limit of club Berkeley.

Question 1: What is the progress on anaylizing Berkeley cardinals?

Have any of the major open problems since the introduction of Berkeley cardinals been solved? Are club Berkeley cardinals larger than Berkeley cardinals? What is the effect of increasing levels of Choice on Berkeley cardinals?

Berkeley cardinals are infamously large large cardinals. The original paper Large Cardinals Beyond Choice where they were introduced, left open 7 problems, 4 of which were related to Berkeley cardinals. Have any of these been solved? Have any new techniques or results created hope that they might be solved in the foreseeable future?

Question 2: Can we go bigger?

Are there any large cardinals that imply the consistency of (Limit club) Berkeley cardinals? If they exist, are there any interesting results that these larger cardinals imply.

This question is simple. How big can we go? For this particular question, I am willing to accept more “off-the-cuff” answers. Large cardinals developed in private communications, or even developed specifically to answer this question. All the criteria are is that they have to be stronger than club Berkeley cardinals, they can’t be trivial extensions, they can’t be obvious, and they have to either have clear reasoning behind their development, or at least you need to be able to do something interesting with them.

Question 3: Can we rescue Berkeley cardinals?

Are there any variants of Berkeley cardinals that can survive even in the context of Woodin’s Weak $HOD$ conjecture?

Consider the nightmare scenario: Over the next decade or two, more and more results related to Reinhardt and Berkeley cardinals are proved, the open problems are resolved, and an interesting structural theory is developed. Than the weak $HOD$ conjecture is proved and that entire hierarchy is wiped out.

It is really this nightmare scenario that is the motivation for my question. Is there any weakening of Berkeley cardinals that would allow us to save any structure theory developed, in much the same way as set theory was rescued after Russel’s paradox? The obvious answer might be the $HOD$-variants, but I believe even these can be wiped out. Assuming that is not the case, the question is do these really have all the properties that Berkeley cardinals have:

Are $HOD$-Reinhardt cardinals stronger than $ZFC+I0$?

Are $HOD$-Super Reinhardt cardinals stronger than $HOD$-Reinhardt cardinals?

Are $HOD$-Berkeley cardinals stronger than $HOD$-Reinhardt cardinals?

Better yet, perhaps some of the open problems that cannot be solved with classical Berkeley cardinals, can be resolved with these new ones!

web – Security implications of not verifying domain ownership on embeddable website tools like disqus

Context: Disqus and similar tools allow content creators to have an embedded “chat box” so that site visitors can comment on your content.

I noticed recently that when I sign up I can claim ownership of any domain, even if I don’t actually own it.

I found that rather concerning, but personally don’t know what the risks of this could be.

Does anyone know of any known issues that Disqus et al have had as a consequence of this? Or issues that could occur in the future? I assume they don’t do domain validatiion because it creates friction in the sign up process.

301 redirect – SEO/link-juice implications of ‘www.’ links versus links without ‘www.’

Yes it is true and no it hasn’t changed. You should set a redirect from one to the other, or use the canonical tag to reference the canonical URL. (on this page they use an example of a URL with and without www).

As you mention Google is pretty smart and is usually quite good at figuring this out, but I often still see Google indexing both versions of URLs (with and without www prefix) when they are clearly the same page.

So you shouldn’t leave it to chance and you should always make it clear which are your canonical URLs.

spells – Ephemeral Tread’s balance implications

The effects wear off when you return from the Plane

This isn’t spelled out as neatly as I’d like, but seems to be deduced from the fact that “impossible actions” are part of the Plane of Dream’s Highly Morphic trait. A plane’s Morphic trait is defined as

This trait measures how easily the basic nature of a plane can be changed. Some planes are responsive to sentient thought, while some respond to physical or magical efforts. Others can only be manipulated by extremely powerful creatures.

You can gain the benefits of a wish spell, but it will not continue to affect you in the Material Plane. The effect never did affect you at all, you just believed it did to the point that you made it true in the Dream Plane.

Astral Projection

Astral Projection may or may not work how you’re describing. Arguably, it also requires you to be on the Material Plane when cast, up to GM ruling.


Furthermore, assuming that is written off as poor writing and allowed, you are limited by what the GM allows in the Plane of Dreams and the Charisma DC they set for accomplishing it. Magic items DC is given, but Artifacts are not ‘normal’ magic items and should be adjudicated differently. Major Artifact creation may be out of line, but Minor Artifact creation may only be a reasonable DC 25-35 Charisma check. As a GM, I would probably bump the DC by a flat 10 to make an Artifact for your use in the Plane.

Finally, as above, they could easily say that the Dream-Plane-Matter you created with your impossible action is not duplicated by the spell, as it is only looking for things you physically possess (while the impossible item™ is not made of Material matter but Dream matter).

What are the privacy implications of the “Your Phone” app on Windows?

I’m trying to decide whether or not to use the Your Phone app on Windows, and not finding much information on it.

It requires a Microsoft account, but I don’t really see why that should be necessary for a local connection between my phone and my computer. It makes me wonder about the privacy implications.

This page claims the following, but I find it unsatisfactory:

Your Phone relies on local connections through Wi-Fi (the iPhone also needs Bluetooth), but the system never takes your data and stores it on Microsoft’s servers. When disconnected there is no maintained data, but rather a local cache on the user’s PC for some items.

In particular, “relying” on local connections and not “storing” it on Microsoft’s servers is not the same thing as “never sending data over the internet” to begin with. What I want to know is if they transmit any of my phone (or PC’s…) data over the internet in the first place, or whether all data stays local to my network. If they send some of my data over the internet but promise not to look at it, that is not the same thing to me as not sending it in the first place.

So: does the app send any kind of data from my phone over the internet (except my login credentials, obviously)?

Is there a way to ensure it keeps all my data within the LAN (whether it’s app data, SMS, call, whatever)?