@joshuagrochow@mathstodon.xyz avatar

joshuagrochow

@joshuagrochow@mathstodon.xyz

Assoc. Professor @ CU Boulder Comp. Sci. & Math; Research interests: theoretical computer science, pure mathematics, and complex systems.
Other interests: climate change; accurate (long)COVID info; equity, inclusion, & accessibility; improving academia. #BlackLivesMatter #TransRightsAreHumanRights #LandBack #CovidIsAirborne #CleanTheAir #StillMasking #MaskUp #TCS #TheoryCS #math #ComplexSystems #complexity (both kinds) #academia

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TaliaRinger, to random
@TaliaRinger@mathstodon.xyz avatar

Any mathematicians looking for CS partners for the NSF proposal on AI for Math? We are thinking of doing something related to discovering relations as well as automatically formalizing proofs from both natural language and diagrams drawn by hand. We have a few other ideas too.

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@TaliaRinger sounds like something @vcvpaiva (and Topos Inst?) would be interested in

highergeometer, to random
@highergeometer@mathstodon.xyz avatar

Spotted in the wild:

"We understand that a career change is a big decision. This role may even have you considering a move to Adelaide to embark on your next career chapter?"

What's the question?

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@johncarlosbaez @highergeometer but in this case it really feels like they meant it to be ! and not ?

MartinEscardo, to random
@MartinEscardo@mathstodon.xyz avatar

What is a topological space?

It is a mathematical device to define what a continuous function is, in a general setting.

  1. A topological space is a set X together with a collection of subsets of X, called open, such that finite intersections of open sets are open, and arbitrary unions of open sets are open.

  2. A function of topological spaces is continuous if inverse images of open sets are open.

What is the intuition behind (1) and (2)?

I claim that it is better to ask, instead, how mathematicians came up with (1) and (2).

1/

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@MartinEscardo do you know, was Grothendieck's issue essentially that the Zariski topology just doesn't have very many open sets? Or was there something more to it?

ProfKinyon, to random
@ProfKinyon@mathstodon.xyz avatar

A mathematics paper without open problems is like... uh... a ma(th)stodon post without a punchline.

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@MartinEscardo @ProfKinyon or associated open source software, preferably in your community's language of choice

funcrunch, to Wikipedia
@funcrunch@me.dm avatar

Thanks to noticing a @wikipedia edit by @11011110 on my watchlist, I realized I should see if this year's Fellowships had been awarded yet. They had indeed, a week ago. So I created a list of this year's fellows, as I have for the previous two years:

https://en.wikipedia.org/wiki/List_of_Guggenheim_Fellowships_awarded_in_2024

https://www.gf.org/news/fellows-news/announcing-the-2024-guggenheim-fellows/

If you're looking for editing experience, please have a look and create or improve articles for the talented people on this list!

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@funcrunch @wikipedia @11011110 Thanks for your service! It'd be really great if that could be sorted by field, institution, etc., rather than having a separate page for each year (so at the "top level" you are forced to pick a year, rather than, say, a field, and then see all awards in that field over time). But I understand that'd be a really big table if they were all in one page. Any ideas on how to resolve this?

andrejbauer, to random
@andrejbauer@mathstodon.xyz avatar

What is the correct reply to a mathematician, who has never formalized anything, asking how they would go about formalizing a research paper of theirs?

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@11011110 @MartinEscardo @andrejbauer This is something some of us have been thinking about. We organized a reading group this semester on some potentially relevant background material.

Even for more "discrete" domains like graph algorithms, one would like to prove things about their complexity, and formalizing that complexity ultimately ends up related to some "base" computational model that needs to be polynomially related to TMs.

Formalizing TMs is a huge pain, but I don't yet know how to get around it. It'd be nice to have a "synthetic" theory of computation... I'm currently thinking to revisit some of Gurevich's writings about "what is an algorithm" in thinking about this. Would be very happy to chat about it at some point.

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@johncarlosbaez @11011110 @MartinEscardo @andrejbauer I will look at them, for sure, thanks!

My experience so far has been that while categorical approaches to computation are nice in a similar way to functional programming being nice, they can be difficult to work with when one wants to do more advanced stuff with computational complexity. Maybe I just don't have enough experience working with them though...I guess we'll see.

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@MartinEscardo @11011110 @andrejbauer Ooh, this makes me really glad I said something here (which I was initially wavering on). I was not aware of this work, and it looks very interesting. I will definitely check it out, thanks!

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@andrejbauer @MartinEscardo @11011110 @yforster Yes, that's one line of research we've already been reading. Didn't realize he was on here though, thanks.

joshuagrochow, to Matrix
@joshuagrochow@mathstodon.xyz avatar

No one defines a #matrix as "a thing that transforms like a matrix". Why define tensors that way?

Array=numbers in a (possibly high-dim) grid
Matrix=array representation of a linear map* in a chosen basis
Tensor=array representation of a multilinear map in a chosen basis

(* or linear endomorphism, or bilinear function, but we'll get there.)

Vectors=1-tensors, but not all 1-index arrays are vectors
Matrices=2-tensors, but not all 2-ary arrays are matrices

Similarly, not all k-ary arrays are tensors. Some examples:

Christoffel symbols aren't a tensor because they aren't (multi)linear in all of their arguments.

Most "tensors" in #MachineLearning #AI aren't tensors b/c they aren't multilinear - they are just multi-dim arrays of numbers. To say an array is (or represents) a tensor is to endow it with additional multilinear structure, same as with arrays vs matrices vs linear structure.

(1/4)

#tensors #matrix #algebra

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

Now, how do we get the whole "a tensor is a thing that transforms like a tensor"? Well, let's start with matrices. How a matrix changes under change of basis tells you what kind of multilinear thing the matrix is representing, and the same is true of tensors. Examples:

If a matrix M represents a linear map L:V→W, then when we change basis in V by an invertible matrix A in GL(V), and change basis in W by an invertible B in GL(W), then M changes to B M A^{-1} (where I'm writing my inputs as column vectors on the right).

In contrast, if a matrix M represents a linear endomorphism L:V→V, then when we change basis in V by an invertible matrix A in GL(V), M becomes AMA^{-1}.

If a matrix M represents a bilinear map V⊗V→F (by (x,y)→x^t M y), then under change of basis A^{-1}, M becomes A^t M A.

(2/4)

#tensors #matrix #algebra

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

How the matrix transforms is "equivalent data" to "what kind of multilinear thing the matrix represents."

(3/4)

#tensors #matrix #algebra

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

e.g. if I tell you I have a matrix M and under change of basis it transforms as A^t M A, then I know it's representing a bilinear map of the form V⊗V→F. etc.

Similarly, if I tell you what kind of multilinear "thing" a tensor T is representing, then that tells you how it transforms under change of basis, and vice versa. For 3-tensors, there are several natural possibilities (up to permuting indices):

U⊗V⊗W→F
U⊗U⊗V→F
U⊗U⊗U→F (trilinear map)
U⊗V→W (bilinear map)
U⊗U→V (bilinear map)
U⊗V→U (linear action of V on U)
U⊗U→U (algebra, not nec. associative)
U→V⊗W
U→U⊗V (coaction)
U→U⊗U (coalgebra, not nec. coassociative)
F→U⊗V⊗W
F→U⊗U⊗V
F→U⊗U⊗U

(4/4)

#tensors #matrix #algebra

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@mrdk I don't think so (you can define linear transformation etc. without choosing a basis, but a matrix involves a choice of basis), but I see your point.

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

Oops.

I just found a great chapter in the Handbook of Linear Algebra by Lek-Heng Lim: https://www.stat.uchicago.edu/~lekheng/work/hla.pdf

where he says "tensor" is the generic term for multilinear things, and "hypermatrix" should be what I called "tensor" in this thread. Maybe I agree w/ that!

joshuagrochow, to random
@joshuagrochow@mathstodon.xyz avatar

Best April Fools I've seen so far is the website from the Lab for Atmospheric & Space Physics at CU Boulder: https://lasp.colorado.edu/

It's worth clicking & scrolling, they really went all out!

(h/t Liz Bradley)

#AprilFools #AnimalsInSpace

MartinEscardo, to random
@MartinEscardo@mathstodon.xyz avatar

I don't think it is unreasonable to demand confidence from a proof assistant. But some people seem to be saying it is fine to live without the confidence. I am very puzzled about this.

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@MartinEscardo While I don't share their opinion, I can imagine it. Not everyone wants proof assistants to add confidence, some people just want them as tools that aid in finding proofs. Similar to how many people currently use computer algebra systems (but in a different way/domain).

joshuagrochow, to random
@joshuagrochow@mathstodon.xyz avatar

Excited to have @shriramk in CU Boulder's CS Dept Colloquium (in-person & online) April 23, 2024! Presenting on the Human Factors of Formal Methods.

https://calendar.colorado.edu/event/cs-colloquium-shriram-krishnamurthi-on-the-human-factors-of-formal-methods

#FormalMethods

TheConversationUS, to ArtificialIntelligence
@TheConversationUS@newsie.social avatar

Don’t get fooled by all the hype.

“The field of artificial intelligence has been running through a boom-and-bust cycle since its early days. Now, as the field is in yet another boom, many proponents of the technology seem to have forgotten the failures of the past – and the reasons for them. While optimism drives progress, it’s worth paying attention to the history.”

https://theconversation.com/weve-been-here-before-ai-promised-humanlike-machines-in-1958-222700
#artificialintelligence #AI #science #technology #history

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@TheConversationUS This is a nice article, but unlike previous AI boom-bust cycles, in this case many people are not being fooled by the hype, they are using tools like commercially available generative AI because they find them useful.

And I don't mean business execs being fooled into thinking it'll revolutionize their business, cut costs, etc. I mean individual people using them in their daily lives.

The issue is that many people either don't know about or are ignoring the negative externalities or even sometimes dangers in doing so.

There is an urgent need for education at all levels around the ethical aspects of using (currently commercially available) generative AI.

#ai #ethics

joshuagrochow, to random
@joshuagrochow@mathstodon.xyz avatar

"If all experts communicated clearly about the continued risks of the virus [covid, SARS-CoV-2], Dr. Lara Jirmanus [MD, MPH, Harvard FXB Center for Health & Human Rights] thinks people might be more open to precautions like masking, staying home when sick, and getting vaccinated" -TIME https://time.com/6960789/covid-19-cautious-americans/

With quotes from @luckytran and @lindyg.bsky.social

anton_hilado, to random
@anton_hilado@mathstodon.xyz avatar

Let us now sketch an argument that bun is good

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@anton_hilado If ever there was an argument that was "obvious" or "could be left to the reader"

(More srsly, what paper is that? It looks interesting)

bkeegan, to psychology
@bkeegan@hci.social avatar

An op-ed on moral #psychology and land use policy from yours truly.

“Few cities have the resources, expertise and values of #Boulder to meet the challenges of climate change. A Boulder that remains open to change will manage these disruptions better than a Boulder that puts up walls. We need moral engagement on behalf of the future rather than moral disengagement on behalf of the past.”

https://www.dailycamera.com/2024/03/21/guest-opinion-brian-c-keegan-we-must-end-patterns-of-moral-disengagement-in-our-policy-debates/

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@bkeegan ❤️ "we need moral engagement on behalf of the future rather than moral disengagement on behalf of the past" applies to so many things!

joshuagrochow, to random
@joshuagrochow@mathstodon.xyz avatar

(in most of North America) if you wake up today and your clock hasn't changed, it will say N when it "should" say N+1. In other words, careful not to be late for your Monday work/school/appts.

Also double-check international mtg times since most other DST countries won't switch for a few weeks.

#timechange

mc, to random
@mc@mathstodon.xyz avatar

there must be something really profound hiding in the fact no good definition of 0*oo is possible, as if the structure of the extended reals wanted to be something more

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

@MartinEscardo @mc but there is a topological monoid structure on [0,1].

One natural way to try to give a monoid structure to [0,oo] is to extend ordinary multiplication of reals. The issue then is that both 0 and oo want to act like zeros of the monoid, but a semigroup can have at most one zero element.

joshuagrochow, (edited ) to Pubtips
@joshuagrochow@mathstodon.xyz avatar

Can anyone explain to me how the "Subscribe to Open" (S2O) model (like MSP's, here: https://msp.org/publications/s2o/) is supposed to work and be sustainable?

I understand the mechanics, what I don't understand is what the equilibrium is supposed to be. If a journal regularly goes open because of S2O, won't ppl come to expect that, and then there's a tragedy of the commons issue?

#openaccess #publishing #academia

joshuagrochow,
@joshuagrochow@mathstodon.xyz avatar

I got a answer to this from @tomkalei on bsky!

The key (in this case) is that MSP subscription is super-cheap! A subscription to their whole bundle for a year is comparable in price to a single article publication fee at many Springer or Elsevier journals. So big unis subscribing those journals wouldn't bother unsubscribing even if it were regularly going open because of S2O. Love it!

#openaccess

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