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can't wait for vacation to resume the classic activity of "walk in some direction (not actually necessarily in one direction, can make any turns) aimlessly for hours until i am too tired, then rest and try to find my way back with no usage of phone maps"

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My faith is based on only one axiom...... that every category is a locally small category in some Grothendieck universe, and therefore the Yoneda Lemma always applies......

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Your personal rant is much appreciated. Thank you for the words in the end, they truly are comforting.

The PhD application process is so dreadful. And you are telling me I might have to do this again in 3 years for postdoc and again and again and again...

Part of me wants to quit this shit show. I love the idea of doing a phd but the application process makes me want to puke. What do you mean that I will have to vouch for myself, tell great things about me when I don't even believe it.

Moreover, getting a rejection hurts. Even when I know that there could be a lot of reasons for it, I can't help taking it personally.

Even when I try to distract myself by doing some math, I feel guilty that I am not doing enough for my applications. Maybe I should search more for open positions. Maybe I should change my cover letter completely. Maybe I should start cold emailing people.

Ugh, it's so frustrating.

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The PhD application process is so dreadful. And you are telling me I might have to do this again in 3 years for postdoc and again and again and again...

Part of me wants to quit this shit show. I love the idea of doing a phd but the application process makes me want to puke. What do you mean that I will have to vouch for myself, tell great things about me when I don't even believe it.

Moreover, getting a rejection hurts. Even when I know that there could be a lot of reasons for it, I can't help taking it personally.

Even when I try to distract myself by doing some math, I feel guilty that I am not doing enough for my applications. Maybe I should search more for open positions. Maybe I should change my cover letter completely. Maybe I should start cold emailing people.

Ugh, it's so frustrating.

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everytime i open ncatlab its a whole new experience

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Failing my set theory seminar because I only studied videos from a beetlegirl on youtube

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two words that really shouldn't be the same word:

inductive in "inductive reasoning" (philosophy term)

inductive in "inductive proof" (math term)

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What is the proper pace for mathematical research, and how does one know if you are pushing too much versus relaxing too much? I feel that I have this narrative that things take time in my head, but I'm not sure how much that is becoming a self-fulfilling prophecy versus simple realism.

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I'm interested in forming a sort of...math & physics reading group network. well, with some very important modifications to the concept of "reading group".

for example, right now, I'd like to learn algebraic geometry, qft, and/or refresh myself on representation theory with someone—maybe just one or two people—meaning that traditionally, we'd pick a text for one of these topics, discuss the material (asynchronously or synchronously?), exchange exercises, etc.

but currently (being Between Institutions), my best bet is posting on tumblr. and that's a pretty good bet, tbh! there are a lot of us here!

though, wouldn't it be great if there were a way to coordinate groups like this across institutions? you make a post proposing a group, specifying your goals and constraints...

even that would be a boon. but I think the concept of a "reading group" itself could be changed in interesting ways. this is what I'm really interested in.

there are variations among reading groups themselves already. sometimes you have directed reading groups, where someone already knows the material and "leads" it; some people are looking for more or less people involved; and there are probably things to explore for making sure that reading groups stick through it instead of falling apart when some motivation flags. default meeting times help with this, for example.

there are many experiments to be done! I think lessons for group-making can be taken from a maybe-surprising source: theater. there are a lot of things that make groups which put on shows more robust and rewarding than reading groups. a sense of building to something; many factors that create informal group cohesion (e.g. such a structure should make sure it creates more-informal "cafe" time in addition to more-formal "practice" time, just as rehearsal in physical spaces facilitates that casual sort of interaction on its periphery); ways to get into the right headspace during discussions (just as warm-ups do in theater; the engagement with this material is an event); clear goals (e.g. "understand ___"); successive shared accomplishments...to that end I wonder if it makes sense to form math troupes, which do successive reading groups together, drawn from its members.

it might be useful to envision some sort of public-facing artifact created as the culmination of this learning, whether a presentation, or an article, or some novel application or research...the crucial question is: how do we choose a goal that we find meaning in?

one idea, for example, is to have a collection parallel reading groups learning different things, and end by presenting to each other! that way we know what we learn will be meaningful to others, too, from the beginning. in general, I think it's important to feel that our own development of insight and understanding can be meaningful to others and to the group. it's nice to participate; it's nice to be able to offer something that is valued. what form can this take? how can you set up the interactions such that everyone has a part to play, and so this meaningfulness is tangled up in participation in the group?

I've also got a couple of ideas for "activities" that let us engage, re-engage, and play with the concepts we're learning with each other, beyond the text itself. how can we give ourselves the opportunity to toss around the concepts we're learning? I believe that the fun ultimately comes from the understanding itself, and therefore that any group exercise which lets us effectively play with the ideas will be fun.

it's a lot to ask people to come up with structure like that themselves, but using a pre-existing structure is not so difficult! sort of like how it's hard to make a TTRPG itself, so simply saying "go off and roleplay" isn't that helpful, but it's easy to use the structure of an existing one to run a game.

you might say, well, the existing form of a reading group is fine. okay! existing reading group structures can be low-stakes, relaxed, and accessible...but they can also fall apart easily (especially when not tied to an institution, in my experience), and you have to get lucky to find a truly rewarding one. I find reimagining our mechanisms of learning pretty exciting, and I think the space of ways to learn math with each other is underexplored at this level (emphasis on the with each other). there's a lot of potential!

anyway! reply or tag with "!" if this is something you could maybe be interested if done well? or if you're at least curious! I'm just taking a temperature. :)

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You’ll ask an algebraic geometer what a topological space is and they’ll go on a tangent about how really a topological space is just something you can put a sheaf on and that a sheaf is really just something that gives you a good cohomology so really what you should be asking is what is cohomology and then not answer the question.

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You’ll ask an algebraic geometer what a topological space is and they’ll go on a tangent about how really a topological space is just something you can put a sheaf on and that a sheaf is really just something that gives you a good cohomology so really what you should be asking is what is cohomology and then not answer the question.

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choosing a PhD advisor is not unlike marriage reality shows. making a multi year commitment on the basis of only a handful of awkward, perfunctory interactions. very small pool of options. something kinda wrong with all of them

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I do feel like the hardest part of category theory so far is keeping track of what exactly all of that notation is actually saying

like, if you make an itemized list of all of these letters, what objects they are referring to and what the types of those objects are, then this diagram (and by extension the proof of the yoneda lemma) actually does become totally sensible and not actually that hard to understand but like. that's a huge hurdle to clear at the start!

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I think math teacher don't focus enough on history. Like, yeah, Abel results on the convergence of series are great and all, but did you know Cauchy hated Abel and is indirectly responsible for his death ?

Galois was a math wizard and died in a duel. But did you know he also was kind of a leftist, got thrown out of École Normale Supérieure because of that, joked about killing Louis Philippe and probably died because of that ?

Gauss is cool, but did you know he was saved by Sophie Germain, discovered at the same time Sophie Germain was a woman, and wrote one of the coolest letter blaming "our customs and prejudices" for not allowing women in math ?

Urysohn's lemma is cool, but did you did you know he tragically died while swimming with Alexandrov ? Oh, also they were both gay and in love. And later Alexandrov had a relation with Kolmogorov

There's tons of other stories, even minor ones (like how Italian mathematicians argued against some notations in vector calculus because they were used by Germans). Focusing only on how "abstract" and "pure" math is, is pedantic. Mathematicians, on the other side, are way funnier and not just lost in their abstract and complicated world

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now if you thought "group" is an unhelpful and nondescriptive name for a mathematical concept. just remember that the broader category of mathematical concepts that groups belong to is called, a "category". so it could always be worse

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The Topology Game

Here's an updated list of topological properties that I use to play @topoillogical's Topology Game. The rules are as follows. You roll a random integer between -N and N inclusive, where N is the number of properties below (currently N = 114). If the number is negative, you get the negation of the property whose number is the magnitude of the number. You come up with a topological space satisfying this property. This was round 1.

For all future rounds, you start by rolling another property. If the new property follows from the previous properties, prove this. If the new property contradicts the previous properties, prove this. If the new property is independent of all the previous, come up with two spaces satisfying all the previously rolled properties, where one space has the new property and one space has its negation.

You 'win' the game if you manage to solve round 10. You can never truly lose as you can always come back to a given game.

List of 114 properties:

A note on terminology: The space a property refers to is always denoted by X. A neighbourhood of a subset A of X is a subset N of X such that A is contained in the interior of N. Any subset of X is implicitly equipped with the subspace topology.

0. Add your favourite property here to get a slightly higher chance of rolling it.

Property modifiers: These are listed separately, but some specific incarnations (like 'locally connected') are also in the rest of the list because they are more common. If you roll a property modifier, roll again to get the property that is modified. If you roll another modifier, your modifiers stack, so you can get something like 'locally retracts onto a space that contains a compact space'. 1. Locally P: Every point has a neighbourhood basis of P subsets. 2. Somewhere locally P: Some point has a neighbourhood basis of P subsets. 3. Semilocally P: Every point has a P neighbourhood. 4. Somewhere semilocally P: Contains some P subset with non-empty interior. 5. Has a basis of P sets. 6. Hereditarily P: Every subset of X is P. 7. Weakly hereditarily P: Every closed subset of X is P. 8. Contains a P space: X contains some P subset. 9. Retracts onto a P space: X contains a P subset A as a retract, i.e. the inclusion A ↪ X has a continuous left inverse. 10. Is the image of a P space: There is a P space Y and a continuous surjection Y -> X. 11. Homotopy-equivalent to a P space. 12. Weakly homotopy-equivalent to a P space: There is a P space Y and a continuous map X -> Y that induces an isomorphism on all homotopy groups. 13. Is a projective limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical limit. 14. Is an inductive limit of P spaces: There is a categorical diagram of P spaces of which X is the categorical colimit. 15. Is homeomorphic to a space Y: Here Y can be any topological space, for which you will need to find some other way to generate it. Reroll this property if you did not in this round first roll a modifier before.

Classes of spaces 16. Euclidean space. Included for completion, but probably not very interesting to roll. 17. Locally Euclidean. 18. Locally Euclidean with boundary. 19. Homeomorphic to a CW-complex. 20. Homotopy-equivalent to a CW-complex. 21. Homeomorphic to a linearly ordered set with the order topology. 22. Homeomorphic to a topological group. 23. Homeomorphic to a space of continuous maps with the compact-open topology. 24. Spectral: Is homeomorphic to the prime spectrum of a commutative ring with its Zariski topology. 25. Homeomorphic to a vector bundle of positive rank. 26. Homeomorphic to a non-trivial principal bundle.

Separation properties Two subsets A, B are said to be 'separated by P sets' if they have disjoint P neigbourhoods. A 'Urysohn function' for A and B is a function f from X to the real numbers such that A is contained in the preimage of 0 and B is contained in the preimage of 1. A 'perfect Urysohn function' for A and B is a Urysohn function such that A and B equal the preimages of 0 and 1 respectively. If you want you can replace some of the following properties with a weaker version that only holds between topologically distinguishable points, so that your property plus Kolmogorov becomes the property as written. 27. Indiscrete. Included for completion, but probably not very interesting to roll. 28. Point-distinguishing (Kolmogorov): No two points are contained in precisely the same open sets, i.e. any two points are topologically distinguishable. 29. Accessible (Fréchet): Any two points have open neighbourhoods not containing the other point. Equivalently, all singletons are closed. 30. Sober: Any irreducible closed set is the closure of a unique point. 31. Weakly Hausdorff: Any image of a compact Hausdorff space is closed. 32. All compact subsets are closed. 33. Hausdorff: Any two points are separated by open neighbourhoods. 34. Closed Hausdorff: Any two points are separated by closed neighbourhoods. 35. Urysohn Hausdorff: Any two points have a Urysohn function. 36. Perfectly Hausdorff: Any two points have a perfect Urysohn function. 37. Semiregular: The regular open sets, i.e. open sets equal to the interior of their closure, form a base of the topology. 38. Quasiregular: Any non-empty open set contains a non-empty regular open set. 39. Regular: Any point and any closed set not containing it are separated by open neighbourhoods. Not to be confused with being a regular open or closed set. 40. Urysohn regular: Any point and any closed set not containing it have a Urysohn function. 41. Perfectly regular: Any point and any closed set not containing it have a perfect Urysohn function. 42. Normal: Any two disjoint closed sets are separated by open neighbourhoods. Urysohn's lemma states exactly that this is equivalent to Urysohn normal. 43. Hereditarily normal. 44. Fully normal: Any open cover has an open star refinement. 45. Perfectly normal: Any two disjoint closed sets have a perfect Urysohn function. 46. Discrete. Included for completion, but probably not very interesting to roll.

Metrizability properties 47. Metrizable: Is homeomorphic to a metric space. 48. Completely metrizable: Is homeomorphic to a complete metric space. 49. Completely uniformizable: Is homeomorphic to a complete uniform space. Note that 'uniformizable' is equivalent to 'Urysohn regular'. 50. Developable: Admits a countable collection of open covers such that for any closed subset F and point x not in F, one of these covers satisfies that no set in the cover that contains x also intersects F.

Connectedness properties 51. Connected: Has exactly one connected component. 52. Locally connected. 53. Semilocally connected. 54. Path-connected: Any two points are connected by a path. 55. Simple path-connected: Any two distinct points are connected by a simple path, i.e. one that is injective. 56. Locally path-connected. 57. Contractible: Is homotopy-equivalent to a point. 58. Hyperconnected: No two non-empty open subsets are disjoint. 59. Ultraconnected: No two non-empty closed subsets are disjoint. 60. Irreducible: Not the union of two proper closed sets. 61. Dense-in-itself: There are no isolated points. 62. Scattered: Every non-empty subspace has at least one isolated point. 63. Zero-dimensional: The topology has a basis of clopen sets. 64. Totally disconnected: The only connected components are points.

Countability properties 65. Finite: Contains finitely many points. 66. Countable: Contains countably many points. 67. Continuum-sized: Admits a bijection with the real numbers. 68. Supercontinuum-sized: Larger than continuum-sized. 69. First countable: Every point has a countable neighbourhood basis. 70. Second countable: The topology has a countable basis. 71. Separable: Contains a countable dense subset. 72. Sequential: Any set closed under limits of sequences is closed. 73. Baire: If a countable union of closed sets has non-empty interior, then one of the closed sets had non-empty interior. 74. Meagre: Is a countable union of nowhere-dense subsets, i.e. subsets whose closures have empty interior. 75. Gδ space: Every closed set is a countable intersection of open sets. 76. Cosmic: Is the continuous image of a separable metric space.

Compactness properties 77. Compact: Any open cover has a finite subcover. 78. Semilocally compact. Note that this is usually called 'locally compact', but this clashes with my terminology. 79. Lindelöf: Any open cover has a countable subcover. 80. Sequentially compact: Any sequence has a convergent subsequence. 81. Limit point compact: Every infinite subset has a limit point. 82. Pseudocompact: Every real-valued continuous function on X is bounded. 83. σ-compact: Is a countable union of compact subsets. 84. Rim-compact: The topology has a basis of open sets with compact boundaries. 85. Core-compact: For every point x and every neighbourhood U of x there exists a smaller open neighbourhood whose closure in U is compact. 86. Paracompact: Any open cover has a locally finite refinement. 87. Metacompact: Any open cover has a refinement such that every point is contained in only finitely many sets of the refinement. 88. Orthocompact: Any open cover has a refinement such that for every point, the intersection of all open sets in the refinement containing that point is open. 89. Compactly generated: A subset U is open if and only if its intersection with any compact subset K is open in K. 90. Noetherian: Any downward chain of closed sets stabilizes. 91. Admits a partition of unity.

Algebraic-topological properties Several properties to do with homotopy groups require selecting a base point. You can interpret each of these properties as being required to hold at every base point, at some base point, or just requiring that the space is path connected. 92. Simply connected: The fundamental group is trivial. 93. Weakly contractible: All homotopy groups are trivial. 94. Locally contractible. 95. Semilocally semisimply connected: Any point has a neighbourhood N such that the inclusion N ↪ X induces a trivial homomorphism on the fundamental group based at that point. This is usually called 'semilocally simply connected', but this clashes with my terminology. 96. Has an abelian fundamental group. 97. Has a finitely generated fundamental group. 98. Has an uncountable fundamental group.

Miscellaneous properties 99. Alexandrov-discrete: Any intersection of open subsets is open. 100. Door space: Any subset is open or closed or both. 101. Submaximal: Every subset is locally closed, i.e. the intersection of an open set and a closed set. 102. Contains a simple curve: Admits an injective continuous map from the closed interval. 103. Has a cut-point: Contains a non-isolated point x such that if you remove x from its connected component, it is no longer connected. 104. Has a generic point: Contains a dense singleton subset. 105. Homogeneous: For any two points x, y, there is a homeomorphism X -> X carrying x to y. 106. Weakly homogeneous: For any two points x, y and any neighbourhoods U, V of x, y respectively, there exist smaller neighbourhoods U' and V' that are pointedly homeomorphic w.r.t. the base points x, y. 107. Totally heterogeneous: X admits only one homeomorphism X -> X. 108. Uniformly based: Has a basis of homeomorphic subsets. 109. Self-based: Has a basis of subsets homeomorphic to itself. 110. Resolvable: Is the union of disjoint dense subsets. 111. Strongly discrete: Every non-isolated point is an accumulation point of a subset whose points can be simultaneously separated by open neighbourhoods. 112. L-space: Hereditarily Lindelöf and not hereditarily separable. 113. S-space: Hereditarily separable and not hereditarily Lindelöf. 114. Polyadic: Is the continuous image of a power of a one-point compactification of a discrete space.