Domestic shorthair, neutered male, 9 years · 9 kg → 7 kg · Status: stable, at home

What this document is

Cake had megacolon that kept coming back. The standard fix is surgery, and by the time it was needed he was too unwell — and too heavy — to come through it safely. Enemas, a full manual clear-out, laxatives and a wet-food diet all failed to hold. At that point the realistic option given to me was putting him to sleep.

We found another option – my favorite vet suggested we try neostigmine, a drug that makes a weak colon push harder. He is still here – eating, grooming, using his litter tray, and happier than he was before this nightmarish adventure began. He just needs an injection every few weeks, and probably always will.

Neostigmine is not an established treatment for megacolon in cats. There is no trial behind it, no licensed indication, nothing in the standard textbooks. It is a folklore treatment — known to some vets, used quietly, never formally written up. That is exactly why it may not have been offered to you, and it is the first thing section 01 explains.

This is not a claim that it works for every cat, and it is not a protocol to follow at home. It is a written account of one cat’s treatment, detailed enough that you can take it to your own vet and ask whether it is worth considering.

01 · A folklore treatment, not an established one

When Cake’s vet suggested the neostigmine she used to treat baby horses, she was clear about what she was proposing: this is not a scientifically established treatment for megacolon in cats. It is not licensed for it. There is no clinical trial. It does not appear in the textbook chapters or the treatment guidelines on feline constipation.¹ A vet who says they have never heard of using it this way is not being careless — there is very little for them to have heard.

What does exist is elsewhere. Neostigmine is well established in horses for large colon impaction, where it is the most commonly chosen drug of its kind.²³ It is used in human medicine for a condition in which the colon dilates and stops moving.⁴ The principle — restore the push to a bowel that has stopped pushing — is old and well understood. It is the specific application to cats that nobody has written down.

Mathematicians have a name for a result that specialists know, pass around and rely on, but which nobody ever formally published: a folklore theorem. It is true, it is used, and you will not find it by looking it up. Neostigmine for feline megacolon sits in roughly that position. Some vets know of it. Some have used it and seen it work. But because it was never written up, knowledge of it spreads by word of mouth between practitioners rather than through the literature — which means whether it gets offered to your cat depends largely on who you happen to be standing in front of.

That is the gap this document is trying to close, and it cuts both ways. A folklore treatment is not a secret good one that the profession has ignored. It is a treatment nobody has tested properly, which is a real reason for caution and a legitimate reason for a vet to decline. Your vet may have grounds you are not aware of, specific to your cat. The point is only that the conversation should be possible.

⚠ An honest caveat

Sometimes euthanasia genuinely is the right decision. A cat in unrelieved pain, with a colon that has stopped responding, or with other serious illness alongside, may be at the end of what medicine can reasonably offer. This document is not an argument that nobody should ever make that choice.

It is an argument that you should get to make the choice knowing what the alternatives were.

02 · What megacolon actually is

The colon is the last stretch of bowel. It draws water out of waste and pushes what is left toward the exit in waves of muscular contraction.

In megacolon the colon stretches wide and loses much of its ability to push.¹ Waste sits there. The longer it sits, the more water is drawn out of it, and the harder it becomes. Harder waste is harder to move, so it sits longer still.

Almost everything that follows in this document is an attempt to break that loop at one point or another. It is worth holding onto, because it explains why single treatments keep failing: an approach that softens the waste but does nothing for the pushing, or improves the pushing while the waste is already concrete, addresses half a problem.

In most cats no cause is ever identified. This is called idiopathic megacolon.¹ When a cat can no longer empty the bowel at all, the word used is obstipation.

⚠ The single most important number in this document

In a 2022 study, cats whose signs had lasted less than six months responded to medical management in 66.7% of cases. Cats whose signs had lasted six months or more responded in 5.6% — the rest needed surgery.⁸

If your cat is early in this, that is the window. It is the strongest argument in the literature for pushing hard on diagnosis and treatment now rather than managing conservatively and seeing how it goes.

03 · The part that was missed: pain

Cake’s X-rays showed spondylosis — small bony spurs growing along the vertebrae of his lower spine. It is common in older cats and often causes no trouble at all, which is exactly why it is easy to dismiss.¹¹

But consider what a cat has to do to pass a stool: crouch, curl the lower back, hold that position, and strain. If that hurts, the cat puts it off. And putting it off is precisely what dries the waste out and hardens it. Pain and constipation feed each other.

This is not merely a plausible story. In a radiographic study of 1,365 cats, those with abnormal lumbosacral vertebrae were significantly more likely to have large bowel problems (p=0.0057, OR 1.73), and for acquired abnormalities specifically — the category spondylosis falls into — the odds ratio was 4.11 (p<0.0001).¹²

How we knew it was real

In my entire acquaintance with Cake, he had never washed the back half of his body. His front half was immaculate. This was put down to his size — he was 11 kg when I adopted him, and was around 9 kg by the time the symptoms of megacolon began. Even a 9 kg cat may simply not be able to bend that far.

He started grooming his back end again shortly after beginning pain relief every 6 weeks, before he had lost any meaningful weight. That timing is the whole argument. If bulk had been the obstacle, the behavior would have returned gradually as the weight came off. It did not. It returned when the pain was treated.

Worth checking on your own cat

A scruffy or matted coat over the lower back, dandruff you only see behind the ribs, or a dirty bottom on a cat who used to be fastidious. Grooming is one of four validated behavioral domains used to assess musculoskeletal pain in cats — alongside mobility, activity and temperament — and all four improve measurably after analgesia.¹³ It is easy for everyone in the room to attribute to age or weight. It is worth saying out loud at the appointment.

04 · What was tried, in order

Stage 1 — failed · A superficial enema

It cleared only the waste nearest the exit. Some stool came out, which looked like progress. The bulk of the blockage sat further forward, in the widened part of the colon, and fluid introduced from the rectum never reached it.

This is a common and instructive failure. If your cat has an enema and still seems wrong afterwards, or the belly still feels full, say so — partial output is easily mistaken for resolution.

Stage 2 — worked, did not hold · Full manual deobstipation under anesthetic

The vet emptied the entire colon by hand under general anesthetic. At the same time she confirmed that nothing was physically blocking the way — no narrowed pelvis, no growth, nothing swallowed.

That check turns out to be the single most important event in this whole account, for reasons that become clear in section 05.

Stage 3 — failed · Daily lactulose and exclusively wet food

Lactulose is a sweet liquid laxative that draws water into the bowel and keeps stool soft. Cake had 5 ml daily in his food, and ate wet food only. Dietary modification and hyperosmotic laxatives are the standard first-line components of multimodal management.¹

The blockage rebuilt anyway. This is the pivotal observation: his colon had been completely emptied, his stool was softened, his water intake was good, and waste still accumulated. The failure was therefore not one of hydration and not a one-off mechanical event. His colon was not pushing.

Stage 4 — unavailable · Subtotal colectomy

Where megacolon keeps recurring, the definitive treatment is an operation removing most of the colon. Cats generally do well afterwards — 72.2% favorable outcomes in the 2022 cohort,⁸ and a high rate of owner satisfaction across 18 hospitals in a 2021 series¹⁵ — and if it is genuinely available to your cat it is usually the better long-term answer.¹¹⁶

It was not available to Cake. By the time the question was live he was both markedly obese at 9 kg and unwell from the obstipation itself, and the combination made a long abdominal surgery an unacceptable risk. This is worth being precise about, because “not a surgical candidate” covers two very different situations: a cat who is too well to need it yet, and a cat who has become too unwell to survive it. Cake was the second.

There is a bitter logic in that which is worth naming. The disease that makes surgery necessary is the same disease that makes surgery dangerous, and the longer it runs the more true that gets.⁸ Waiting is not neutral.

On not having the surgery

For most of this document the non-surgical route reads as a consolation prize, and I want to correct that, because it isn’t only one.

Subtotal colectomy is major abdominal surgery, and the recovery is not nothing. Soft stools and sometimes frank diarrhea are expected for the first few months afterwards, stools become firmer with time but rarely return fully to normal, and some cats have occasional soft-stool leakage.¹⁷ Outcomes are good and most cats end up with an excellent quality of life¹⁵¹⁷ — I am not arguing against the operation. But a cat who never has it also never has the anesthetic, never has the laparotomy, never has the months of loose stool, and never carries the small risk of the anastomosis failing.

Cake has none of that. He has a dose every few weeks, a spoonful of syrup in his dinner, and an otherwise ordinary life. Measured against a cat who sailed through colectomy, that is second best. Measured against what was actually on the table for him — which was repeated anesthetics or euthanasia — it is a good outcome by any standard, and it is allowed to be one.

05 · Neostigmine, and why the timing is everything

Nerves in the bowel wall release a chemical messenger that tells the muscle to contract. Neostigmine slows the breakdown of that messenger, so the signal persists and the contraction is stronger.¹⁴ In plain terms: it helps a weak colon push. It is given by injection, because it is not absorbed if swallowed.¹⁸

⚠ The critical point

A drug that makes the bowel contract harder is dangerous if the bowel is genuinely blocked. Mechanical obstruction of the gastrointestinal tract is a listed contraindication to neostigmine.¹⁴ Contracting against a hard mass that will not move does not expel it. It raises pressure inside a colon that is already stretched and thin-walled. That is painful, and at worst the bowel can be damaged or perforated. The same objection is made explicitly in the equine literature: a prokinetic given in the presence of a physical obstruction pushes against an impassable point and increases pain.¹⁹

This is why the order of Cake’s treatment matters more than any individual element of it. The same drug is either useful or harmful depending entirely on the state of the colon when it is given.

Neostigmine was used only:

• after the colon had been completely emptied by hand;
• after mechanical obstruction had been excluded;
• and thereafter early, while the contents were still soft enough to move.

Why the human comparison only partly holds

Neostigmine is used in people for acute colonic pseudo-obstruction, which is defined as marked colonic dilatation in the absence of mechanical obstruction.⁴ But in that condition the distended colon is full of gas, which compresses and vents the moment motility returns. A megacolonic cat’s colon contains a solid mass whose deformability decreases by the day. Same drug, same organ, different mechanics — and the difference is the entire reason timing governs the outcome.

06 · Cake’s regime

Neostigmine, intermittently

• We started with a small test dose, under observation, to check he did not react badly to the drug itself.
• Then, proceeded to an induction course: twice daily for three to four days, started after the manual clear-out.
• Ongoing Maintenance: I feel his abdoment around twice a week to check if waste has built up. If it has built up, he gets a dose. If not, no dose. His current dosing rate is one dose every few weeks (recorded how much and when administered), two at the absolute maximum.
• As his weight fell from 9 kg to 7 kg, the dose was reduced — proportionally more than the weight loss alone would require, and it still works.

⚠ The stop rule

Now that he is in ongoing treatment: If two doses do not produce a bowel movement, he does not get a third. He goes in for imaging and manual clearance instead. So far this has not occured. However, failure to respond to dosing is the signal that the contents have already hardened past the point where more contraction helps — and at that point further dosing is pushing against something that will not move. This rule is not a formality. It is what keeps the treatment on the safe side of the contraindication.¹⁴¹⁹

No doses are given in this document

Deliberately. Neostigmine is supplied at several different concentrations, and a volume that is correct for one cat and one product can be a significant overdose with another. Dosing is calculated per kilogram by a vet who knows which vial is in their hand. This document exists to start a conversation, not to replace one.

Lactulose, daily

5 ml once daily, mixed into wet food. This is not an optional extra alongside the neostigmine — the two are co-essential, and this is the mechanistic heart of why the combination works where either alone did not:

• Lactulose keeps the stool soft enough to be moved.
• Neostigmine supplies the force to move it.

Soft stool with no propulsion simply sits there — that is what the months on lactulose alone demonstrated. Strong propulsion against a hardened mass only builds pressure — that is the danger in section 05. Neither element is sufficient. The aim is soft but formed stool; watery diarrhea means the dose needs revisiting with the vet rather than adjusting yourself.

Solensia, monthly

An injection for pain every 6 weeks. It is a felinized monoclonal antibody that binds nerve growth factor, a substance that sensitizes nerves to pain.²² It is licensed for osteoarthritis pain in cats;²³ using it for spinal pain is a slightly different application.

Its role here is not only comfort. If defecation stops hurting, the cat stops deferring it — which breaks the pain–avoidance loop from section 03 at the other end from the laxative and the prokinetic.

07 · Feeling the abdomen

Cake’s re-dosing is triggered by palpation — feeling the abdomen for waste building up, and treating before a hard mass has formed rather than after. That is what keeps the drug on the right side of the line described in section 05.

⚠ Read this before you try it

Ask your vet to show you on your own cat first, with their hands over yours. A description on a page is not a substitute for being taught, and the two structures you are distinguishing between feel more alike than this illustration suggests.

Never press hard, and never squeeze. A distended bladder can be ruptured by firm pressure. If your cat tenses, cries, or resists, stop.

The colon sits high and toward the cat’s left, running backwards through the rear half of the abdomen. The bladder sits low, central and further back. Both can feel firm. Telling them apart is the reason to be taught this in person rather than from a diagram.

How it is done

• Cat standing, or relaxed on your lap. Never when agitated.
• Fingers flat, using the pads rather than the tips. One hand each side, or one hand spanning the belly.
• Start behind the last rib, in the rear half of the abdomen, and work gently backwards until you are directly under the spine.
• Press slowly and lightly — enough to feel through the body wall, no more. The pressure is closer to resting your hand than to gripping.

What you are feeling for

• Normal: soft and yielding. Nothing firm that holds its shape.
• Filling up: a firm, sausage-shaped tube high on the cat’s left, running front to back. It may feel lumpy, like a row of hard beads through a sleeve.
• Not what you want: a hard, round, distinct ball sitting low and well back. That is more likely the bladder. Leave it alone.

⚠ A cat straining may not be constipated

A cat straining in the litter tray may be unable to urinate rather than defecate. From the outside the two look almost identical.

In a male cat, a urinary blockage is life-threatening within hours, not days. If you feel a tense, firm, round structure low and far back in a cat who is straining, do not press on it — phone the vet immediately and say you cannot tell which it is.

08 · Weight

Cake went from 9 kg to 7 kg. For an overweight cat this helps in several directions at once: less load on a painful spine, an easier crouch to pass stool, and better odds if surgery is ever reconsidered.

⚠ Do not do this quickly

Overweight cats must lose weight slowly and under veterinary supervision. Obesity and a period of reduced food intake are the classic setup for hepatic lipidosis, a serious and potentially fatal liver condition in cats.²⁵ No crash diets. If your cat stops eating, phone your vet. Weight loss that happens without you trying is also a reason to be seen, not reassured.

09 · Day to day, and what to watch

• Lactulose in wet food, daily. Aiming for soft but formed.
• Wet food only. Water always available.
• A written note of when he goes, what it looks like, and whether he strained. This is the single most useful thing to bring to an appointment.
• Regular abdominal palpation, to catch accumulation early rather than late.

The number that matters most

How often the injection is needed. If the gaps between doses stay roughly constant, the regime is holding. If they start shortening, the colon is likely deteriorating — and that is the moment for a fresh conversation about what comes next, including whether surgery has become possible again at a lower weight.

Writing that down honestly is uncomfortable, and it is the reason this document is a record rather than a recommendation. A treatment that is working is not the same as a cat that is cured. The duration data in section 02 make clear that this disease has a direction of travel.⁸

⚠ Phone the vet now if

• Straining in the tray with nothing coming out
• Nothing passed for more than a couple of days
• Vomiting, especially with a tight or swollen belly
• Not eating, or gone quiet and withdrawn
• Crying or yowling in the tray
• After a neostigmine dose: weakness, collapse, or difficulty breathing

10 · Taking this to your own vet

If your cat is where Cake was, these are the questions that opened up the useful parts of the conversation:

1. Has anything physical been ruled out — a narrowed pelvis, a stricture, a growth?
2. Could pain be part of why my cat avoids the tray? Has the spine, hips and pelvis been assessed, and would a trial of pain relief be reasonable?
3. If the colon has been fully emptied and it still rebuilds on laxatives, does that point to a propulsion problem rather than a hydration one?
4. Is my cat a surgical candidate — and if not, what specifically rules it out, and could that change?
5. If surgery is off the table, are there prokinetic options worth trying, including ones that are not standard?

References

My first play, Impaction, was performed in LA as part of Nothing to See Here, directed by my amazing friend Jacques Manjarrez. The last night was recorded and can be seen here.

Impaction

by Rin Ray

Cast

• Main Character (MC)
• Sick Creature — a human wrapped in blankets to look vaguely unidentifiable, but certainly alive
• Doctor — wearing a rhinoceros mask and a pink lab coat
• Creatures — a crowd in costumes

Props

• Lots of yellow and pink ribbon
• Two chairs
• Two phones
• A banana

Scene One

MC and Sick Creature on stage. A phone near the back of the stage.

MC: (Standing, staring out into space at the mostly lifeless body.)

Sick Creature: (Lying on their side, crying slightly, breathing labored.)

MC: (Backs up slowly and sinks down next to the phone without looking at it, still staring at Sick Creature. Tension slowly builds. Through shaky breaths, holding back tears, dials.) “Hi, does your clinic offer the procedure?”

Voice on the phone: (Disappointed) “Ah, I see. Okay, thank you.”

(MC crosses a name off a list. Approaches the creature and strokes its hair while calling once more. Increasingly agitated, MC throws the phone down.)

Sick Creature: (Flinches and curls in on itself, breathing faster and shallower, building toward panic.)

(MC picks the phone back up and calls the Doctor.)

MC: “Doctor, I can’t reach anyone who does the procedure. Do you know someone?”

(MC gets up to go outside. A flash of different styles and genres of music begins to play.)

(A parade emerges from behind the audience and weaves through the seats: minions, clowns, demons, nuns, a flamingo — all shitfaced drunk. Each holds a pink or yellow ribbon trailing behind them. The music keeps switching rapidly as they float by. Strings and ribbons flow through the space.)

(The characters pull to the side and play rock paper scissors in an uncanny, chittering clown language. The next batch comes out with ribbons in the same way, in front of MC, freestyling mischievously in their mayhem.)

(Suddenly MC is caught in a web of ribbons, dodging side to side, running their hands through them. They “trip” and grab one. The being holding that ribbon — the flamingo — stops, walks up to MC, and hugs them. MC holds on tight, then looks at the ribbon in their hand, then back at the flamingo. MC lets go of the ribbon, and the flamingo immediately turns around and rejoins the parade.)

Scene Two

(The parade mostly clears the stage as MC exits. A small parade keeps passing parallel to the stage, ribbons leading offstage. Upbeat, quickly shifting music continues.)

(Two chairs. The Doctor, in a pink coat, sits on one next to a phone. Hidden behind them: another phone and a banana.)

(MC opens a door, arriving.)

MC: “Help me.”

Doctor: “What? Okay.”

(MC and Doctor both start calling frantically while the parade continues, alternating the same line.)

MC & Doctor: “Hi, do you offer the procedure?”

(More characters join the parade in the background. MC and Doctor keep calling, now in different languages, building to a frantic pace.)

Doctor: “Hi, do you offer the procedure?”

MC: “Hallo, bieten Sie die Operation an?”

(More characters enter.)

Doctor: “Bonjour, proposez-vous l’OPÉRATION ?”

MC: “Hola, ¿ofrecen la OPERACIÓN?”

(The Doctor raises the banana to their other ear, as if on two phones at once.)

Doctor: “Salve, operationem offersne?”

Doctor: “They have always abandoned us. We must do it ourselves.”

(The parade grows louder and closer until it is walking in front of and around them, club music pounding. MC and Doctor are obscured.)

Scene Three

(While covered, the Doctor exits with their chair and phone. Sick Creature enters and lies on the floor, two ribbons already tied around its throat, each held by a costumed person.)

Sound: (The upbeat music continues, but slowed down, as if playing in another room.)

(The costumed people in front of MC part down the middle and move left and right behind them, revealing Sick Creature on the floor. MC sits blindfolded on a chair. Behind MC, the costumed people split the ribbons into two paths — yellow and pink — like a fork in the road.)

Doctor: (Walks out to stand behind MC.) “Here are the options.”

Doctor: “You can end its suffering — kill it.” (Points to the pink ribbon.) “Or you can try the procedure yourself, but act as its tormentor as you try.” (Points to the yellow ribbon.)

MC: “What will they lead to?”

Doctor: “You cannot know the future.”

(The two ribbons around Sick Creature’s neck feed into the two paths behind MC, so that MC sits inside a triangle of ribbon. The people holding the neck ribbons begin to pull.)

Sick Creature: (Fights the ribbons at its neck, clawing wildly and whimpering, in clear agony, growing more intense by the second.)

MC: (Breathing heavily, still blindfolded, gets up and crawls on hands and knees to stroke the creature’s face as it whimpers. The ribbon-holders move closer; the triangle shrinks and forces MC back into the chair.)

Sick Creature: (Screaming and seizing in pain, still clawing at the ribbons.)

(MC turns to the audience.)

MC: “Fuck! What do I do? Fuck!”

Offstage Narrator: “Those who want yellow, raise your hand. Those who want pink, raise your hand.”

(A TV in the background shows yellow on one side, pink on the other. The audience votes.)

In this post, I conjecture that the Frobenius eigenvalues of Artin-Schreier-Witt Curves (\( \mathbb{Z}/p^k \)-covers) are Gauss sums, generalizing an old and classic theorem of Davenport and Hasse (fuck nazis though). I explain and expand (i.e., make the paper type check and fill in details) a gorgeous reproof of this theorem using local systems by Robert Coleman.

This playful note is toward exploring this force, and outlining a conjectural approach for further exploiting it. Enjoy and click. Read about my Gauss Sums :P

Code newton (Newton polygons of Artin–Schreier–Witt curves)

• the euler characteristic \( \chi(SL_2(\mathbb{Z})) = \zeta(-1), \) and
• the tamagawa measure of \( \mu(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}) = \zeta(2)\).

I repeat, these apriori unrelated invariants of our beloved example case \( SL_2(\mathbb{Z}), \) are dual zeta values! Is it a general pattern? Let’s define and explain these results to spiritually prepare ourselves for the general picture!

Euler Characteristics of Arithmetic Groups – star of the show: \( \chi(SL_2(\mathbb{Z})) = \zeta(-1) \)

An arithmetic group is the integer points of an algebraic group (i.e. a group object in abelian varieties). For example, \( SL_2(\mathbb{Z}) \) is the arithmetic group associated to the algebraic group \( \mathbb{SL}_2 \). In general, \( \chi(G) \) for arithmetic groups is related to L-functions and zeta functions.

The euler characteristic is defined for a torsion free group \( G’ \) as follows: \(\chi(G') = \sum_i (-1)^i H^i(G', \mathbb{Z}).\) This is supposed to generalize the notion of euler characteristic for surfaces of polyhedra where \( \chi(P) = V - F + E \), the analogy is to consider \( \dim H^0(P, \mathbb{Z}) = V \), \( \dim H^1(P, \mathbb{Z}) = E \), and \( \dim H^2(P, \mathbb{Z}) = F\).

For example:

• if we consider \( F_n \) the free group of rank \( n \geq 0 \) then \( BF_n \) has one vertex and n \( 1 \)-cells, therefore, \( \chi(F_n) = 1-n \)
• free abelian group of rank \( n \geq 1 \) then \(n\)-torus \( (S^1)^n \) has euler characteristic \( \chi(S^1), …, \chi(S^1)=0.\)

The euler characteristic is defined for any group \( G \), in terms of \( G’ \) a torsion free group, \(\chi(G) = [G: G']^{-1}\chi(G')\)

Note: This doesn’t depend on choice of \( G \), consider a two different torsion free subgroups \( \Gamma’ \) and \( \Gamma’’\), and define \( \Gamma_0 := \Gamma’ \cap \Gamma’’ \) , then

It’s time to stare at the Gauss Bonnet Thm: Let \( Y \) be closed Riemannian manfiold, then there’s a canonical measure \( \mu \) in \( Y \) such that \(\chi(Y) = \mu(Y).\)

There’s a version for orbifolds, \( Y = BG\) and let \( X \) be its universal cover, there’s a canonical measure such that

Further, \( \chi(BG) = \chi(G). \)

Let’s look at \( SL_2(\mathbb{Z}) \). The free group \( F_2 \) is a torsion free group living inside of \( SL_2(\mathbb{Z}) \) via the ping pong lemma. It has index 12, not sure why tbh.

Thus, \(\chi(SL_2(\mathbb{Z})) = [F_2: SL_2(\mathbb{Z})]^{-1} \chi(F_2) = \frac{1}{12} (1-2) = -\frac{1}{12} = \zeta(-1).\)

The Tamagawa Number of Arithmetic Groups – star of the show: \( \mu_\infty(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{Q})) = \zeta(2) \)

Given an arithmetic group \( G \), it comes equipped with a canonical measure on its adelic points \( G(\mathbb{A}) \). All other measures are a constant times this measure. It is defined by considering the measure associated to a differential form.

Given an algebraic group \( G \), pick any \( \omega \) which is a top degree invariant differential form of \( G \) defined over \( \mathbb{Q}. \) Then, define the measure \( \mu \) place-wise (a la the adelic perspective of Tate) for each \( G (K_v) \) by \(\mu_v := |\omega|_v \) on \((\mathbb{Q}, +)\). We then renormalize such that:

• for finite places \(v \), \( \mu_v(\mathbb{Z}_v)=1 \),
• and at infinite places \(\mu_\infty([0,1]) = 1.\)

You might be like – we chose \( \omega \) there, seems noncanonical to me. Well it is noncanonical, if we consider it individually over each place, but all together, no matter which choice of \( \omega \) we take, we will get the same answer because for any number \( f \), the product of its valuation over global fields will always be one: \( \prod |f|_v = 1. \)

Non-archimedian case (family friendly) \( \prod_p \mu_p(SL_2(\mathbb{Z}_p)) = \zeta(2)^{-1} \)

In this section we wish to show that \(\prod_p \mu_p(SL_2(\mathbb{Z}\_p)) = \prod_{p} (1-p^{-2})^{-1} = \zeta(2)^{-1}.\)

We may compute the Tamagawa measure for \( SL_2(\mathbb{R}) \) using Iwasawa decomposition. We get that the Haar measure on \( SL_2(\mathbb{Q}_p) \) is \( |x^{-1}|_pdxdydz, \) where \( |x^{-1}|_p = p^{-v_p(x)} \). A nice reference on this is the notes of Paul Garrett.

Now consider the reduction map:

\(\mu\_p(SL_2(\mathbb{Z}\_p))= \# SL_2(\mathbb{F}\_p)\mu\_p(\text{ker } f)\) and \(\mu_p(\text{ker } f) = \int_{x,y,z} \|x^{-1}\|_p dxdydz = p^{-3}.\)

Lemma: \( \# GL_2(\mathbb{F}_p) = (p^2-1)(p^2-p), \),

Proof: This because when forming a matrix in \( GL_2 \), the first column \( r_1 \) of the matrix can be anything but the 0 vector (giving us \( p^2 - 1\) choices), and second column of the matrix can be anything other than a multiple of the first vector (it can’t be \( r_2 = cr_1 \) ).

Lemma: \( \# SL_2(\mathbb{F}_p) = 1-p^2, \)

Thus, we have proven: \( \mu(SL_2(\mathbb{Z}_p)) = \frac{1}{1-p^{-2}}, \) and we may conclude:

Let’s get Archimedian \( \mu_\infty(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{Q})) = \zeta(2) \) (to be continued…)

The Bernoulli numbers were defined by Faulhaber in terms of the following generating series.

But why? Where did this come from? Well, the mathematicians of the time were contemplating the following sorts of patterns:

The general formula requires \( B_k \) :)

To give you an idea of how hard this is, I don’t even know of a proof of the \( s = 2 \) case that that doesn’t involve guessing the formula then showing inductively (other than the Bernoulli version I’m about to show).

Today I’m giving a modern explanation of Faulhaber’s trick. This is based on notes of John Baez.

Consider \( \mathcal{E} \) to be the space of entire functions on \( \mathbb{C} \). We can consider \( \mathcal{E} \subset \mathbb{C}[[z]] \)

Next, consider the finite difference operator

Discrete Fundamental Theorem of Calculus: If \( \Delta F = f \), then \(\sum_{i = 0}^{n-1} f(i) = F(n) - F(0)\)

For example, here’s a big clue: If \( f(z) = z^s \), then if \( \exists F \) such that \( \Delta F = f \), then \(\sum^{n-1}_{i=0} i^s = F(n) - F(0).\)

Let’s set some groundwork before proceeding, we consider an operator (called the ‘‘annihilation operator” in conformal field theory):

Let us further define our friend the exponential

Lemma: \(e^{ta}f(z) = f(z+t)\)

Proof: I can’t help myself, I just love power series. We prove this by expanding both sides and observing that they are the same. Take \( f(z) := \sum_i c_iz^n) \), then

\(\begin{aligned} e^{ta}f(z) &= f(z) \\& + (t\frac{d}{dz})f(z) \\ & + (t\frac{d}{dz})^nf(z) \\ & + \cdots \end{aligned}\) i.e., \(\begin{aligned} e^{ta}f(z) &= c_0 + c_1z + c_2z^2 + \cdots + c_nz^n + \cdots \\& + c_1t + 2c_2tz + \cdots + tnc_nz^{n-1} + \cdots \\ & + n!\frac{c_nt^n}{n!} + (n+1)!\frac{c_{n+1}t^nz}{n!} + \cdots \\ & + \cdots \end{aligned}\)

Let’s expand the other guy, \(\begin{aligned} f(z+t) &= c_0 + c_1(z+t) + c_2(z+t)^2 + \cdots + c_n(z+t)^n + \cdots, \\ &= (c_0 + c_1t+ c_nt^n + \cdots) + (c_1t + 2c_2t + \cdots + nc_nt^{n-1} + \cdots)z + \cdots \end{aligned}.\)

It’s the same yay. That means we can write \(\Delta = e^{a} - 1.\)

So, given our big clue above, if we can find \( \Delta^{-1} \), we can solve our addition problem.

Let’s play…GUESS THAT INVERSE!!

If we consider \( e^x - 1 \) as a function, we want to find another function \( f(x)g(x) = 1 \), then:

GUESS #1: \( \frac{1}{e^x-1} \) beeping noise nope this is not entire, has a pole when \( x= 0 \).

GUESS #2: \( \frac{x}{e^x-1}) \) we can plug that pole, and get an entire function!! However, we have this extra \( x \) factor floating around (since we want to compose to 1, i.e., the identity operator).

If we were working non-commutatively, then, we’d want to look at either

NON-COMMUTATIVE GUESS #3 : \( x^{-1} \frac{x}{e^x - 1} \) or \( \frac{x}{e^x - 1} x^{-1} \)

Alright, we have some candidates. Let’s get back to operators to make this into real math. We define

Note, \( a a^{-1} f = f \), but the other order is not always true \( a^{-1} a f \neq f \) (because of the icky \( + c \) ).

That means our noncommutative guess \( \frac{x}{e^x - 1} x^{-1} \) is a good bet! In fact, we define the following operator:

and finally, we claim \(\Delta^{-1} = \frac{a}{e^a-1}a^{-1}.\)

Tying it all together to finish the job

We take: \( f(z) = z^s \), we need to calculate \( \Delta^{-1}f := \frac{a}{e^a-1}a^{-1} \).

Note in particular that \( \Delta^{-1}f(0) = 0 \). Using our discrete fundamental theorem of calculus, we see the following:

Since \( \Delta(\Delta^{-1}f) = f\), let \( f = z^s \), then

Tada! We did it!

There is also the following fabulous quick derivation trick of the Bernoulli numbers, which we discussed several years ago in Umbral Calculus Derivation of the Bernoulli Numbers.

A quick way to remember the sum formula is to set \( B^i \) equal to the Bernoulli number \( B_i \). This can be earnestly done if one uses umbral methods.

Graph Laplacians in More Generality

The definition of the graph Laplacian for a graph \( G \) is the following. Let \( v \in V \) be the vertices in the graph, and let \( N(v) \) be the set of vertices neighboring the vertex \( v \).

The directed graph Laplacian rather looks at \( N^+(v) \), the set of vertices the vertex \( v \) maps to with edge length 1:

We can define the graph Laplacian to act on entire functions on \( \mathbb{C} \):

Notice that if we consider the lattice of \( \mathbb{Z} \) in \( \mathbb{R} \) to be directed (going toward positive numbers), we must make a slight type change.

This is quite interesting because the left inverse of a graph Laplacian is the Green’s function associated to the uniform random walk on the graph where you hop to each of your neighbors of distance one away with probability one. The Bernoulli numbers are derived from the right inverse of \( \Delta = -\nabla_{\mathbb{Z}, 0} \). That’s a big difference in description of the inverses on both sides!

Let’s use this language to boogie. Take the lattice \( \Lambda := \mathbb{Z} + \tau\mathbb{Z} \subset \mathbb{C} \) , with the directions pointing toward the positive imaginary and positive real directions. The neighbors of the vertex

If we allow ourselves full license to fuck around, let’s set \( \tau = i \) and play GUESS THAT INVERSE. We consider the function \(1-\frac{1}{2}(e^x + e^{\tau x})\) And look at it’s inverse: \(\begin{aligned} \frac{1}{1-\frac{1}{2}(e^x + e^{\tau x})} &= \frac{2}{e^x + e^{\tau x}-2} \\ &= \frac{2}{\sum_n \frac{(1+\tau^n)x^n}{n!}-2}\end{aligned}\)

This function has a pole of order 3 at 0, and for the simplest case \( \tau = i \), the remaining poles are of order one at roots of \( i \). We can plug those poles, no problem:

I’m not sure where to go from here, I’d love for this to in some way compare to the Kronecker-Eisenstein series (which tantalizingly involves the character \( \phi(z) = e^{\frac{z-\overline{z}}{A}} \)). Here, \( A = \frac{\text{im } \tau}{\pi} \) is the area of the fundamental domain of the lattice \( \Lambda \) divided by \( \pi \). The Kronecker-Eisenstein series is defined as follows, (typically the role of the letter \( b \) is played by \( a \), but we are already using \( a \) to stand for annihilation operator).

where the \( * \) means that the summation excludes \( \lambda = -z \) if \( z \in \Lambda \).

A good reference on these is Elliptic Functions according to Eisenstein and Kronecker : An Update. See in particular the elliptic analogs of Bernoulli numbers discussed on page 8.

This is a nostalgic topic for me which feels like home to step back to. I previously contemplated the analogs of such shifts in 2015 while I was still a robotics inventor sitting in the back of lectures at Berkeley On Detiling Polynomials: A Generalization of the Euler MacLaurin Formula.

Let \(X\) be a curve over a complete local ring, and let us decompose it into two affine pieces \(X = A \cup B.\)

Given a sheaf \(\mathcal{F}\) over \(X\), we have the following Mayer-Vietoris sequence.

In particular, if take the decomposition of \(X\) into the point at infinity and its complement \(X = U \cup {\infty}\), we get the following sequence, let \(T\) be a divisor at \(\infty\).

A global section \(H^0(X, \Omega^1_X)\) can be specified by either a basis in terms of \(\mathcal{F}(U)\), in terms of \(\mathcal{F}^\wedge_\infty.\) This is because we are considering \(H^0(X, \mathcal{F})\) as a subset of \(\mathcal{F}(U) \times \mathcal{F}^\wedge_\infty\).

Lyrics:

I’ll cut you into manageable pieces
I hope you’re not too hard to glue back together
There’s so many ways to form an affine cover
Yet I keep slicing open your stomach
And I pull out your heart
I don’t know why
that’s where I like to start

It’s an easy thing to fall in love with a hand
When its lifeless and the fingers are missing
I can trace the truth
As it sits motionless
beneath my glass, slowly twitching

And when I stitch it back together
It jumps and motions toward the sky
There’s a hanging sequence of all of your pieces
And your other hand is waving hi

I’ll cut you into manageable pieces
I hope you’re not too hard to glue back together
There’s so many ways to form an affine cover
Yet I keep slicing open your stomach
And I pull out your heart
I don’t know why
that’s where I like to start

Those intersections between your skin
Reveal raw flesh that I’m kissing
There’s so many ways to stitch you back
I GIVE YOU MORE EYES MORE EYES MORE EYES MORE EYES
YOUR DICK IS MISSING, i needed it to make more eyes

I love to cut you apart to find the ways you combine
How many times can I do this before you fall apart forever?

If I was braver then, I would have added this drawing to the paper which I wrote at an Arizona Winter School. I still think that paper is good resource for learning quickly about the Fargue Fontaine curve. A Global Crystalline Period Map

All of this is on my mind because I have had the great privledge and joy of returning to this topic after years away.

While we are sharing old things, in 2018 or 2019, Artem, Kolya and I made an outline of an approach to rational Chromatic Vanishing \(H^*(J_h, W(k)) \times \mathbb{Q} \simeq H^*(J_h, W(k)[[u_1, ..., u_{h-1}]][u^{\pm}]) \otimes \mathbb{Q}\) using the two tower isomorphism which reduces the problem to directly calculating the \( GL_h(\mathbb{Q}_p) \) action on constant functions on the Drinfeld projective space Chromatic Vanishing Approach.

At the time, the tools to compute the p-adic cohomology of the Drinfeld projective plane were not available to us so we ulitmately were scooped. I was initially sad about this because I felt then I could not prioritize thinking about these awesome and gorgeous concepts. However, recently I have begun with many others a spin-off project, which I am delighted and healed by. :)

From 5 to 9 August 2024, the Young Topologists Meeting came to the University of Münster. YTM is a conference run by and for early-career topologists: PhD students and postdocs give most of the talks, and two lecture series anchor the week. This year they were Ishan Levy (Copenhagen) on Telescopic computations in algebraic K-theory and Arunima Ray (MPI Bonn) on From knots and links to 4-manifolds.

I was on the organizing committee, together with Konrad Bals, Etienne Batelier, Edith Hübner, Marin Janssen, Alexander Kutzim, Lukas Stöveken, Maximilian Tönies and Thomas Tony. I also drew the logo, and Natalia Pacheco-Tallaj did the graphic design around it.

The logo

The letters Y, T and M are drawn as flames, hot pink with cyan shadows, and the T in the middle is split by a lightning bolt. They sit inside a yellow seal with a scalloped edge, somewhere between a sun and a sticker. Everything has a heavy black outline, so the logo reads both large on the banner and small in the corner of each day of the schedule.

The schedule

Natalia and I made the schedule together. Each day is its own lime-green card with the logo in the corner, speakers in cyan, and titles in black; the coffee breaks, lunches and the poster session (“with finger food!”) get their own bands. Yellow smiley stickers and pink and cyan bursts are scattered across the page, and a last panel says where things are: talks in M1 and M2, coffee breaks and the poster session in the Cluster Lounge, and a silent room in the SRZ. Here is the full schedule, page by page:

The palette is the same everywhere: hot pink, cyan, lemon yellow and lime on teal and lilac.

YTM 2024 was supported by the Cluster of Excellence Mathematics Münster, the K-theory Foundation and the Foundation Compositio Mathematica. Banner: © Natalia Pacheco-Tallaj (graphic design) & Rin Ray (logo). Schedule: © Natalia Pacheco-Tallaj & Rin Ray. Photos of the week, by Arye Deutsch, are on the conference page.

We propose there is collection of disorders characterized by sensory processing issues. This collection of symptoms affects many and is rarely viewed holistically. We propose to view them holistically and approach their treatment as such – keeping in mind that autoimmune disorders usually are best detected genetically, and medication-wise sensory processing is best treated using tricylic antidepressants, GABA analogues, and glasses prescribed by a behavioral optometrist.

Sensory processing disorders and connectivity

People who react strongly to perfumes and strong smells (chemical intolerance) are such that they perceive a smell at a constant level of intensity, the signal is processed at the same intensity over time. Normal people perceive the signal intensity to decrease over time, they “get used to it”. Signal intensity is monitored via brain blood flow (measured using EEG and fMRI). The same blood flow pattern as those with chemical intolerance is found in those with chronic pain. [(2012)Summary of Linuss Andersson (Sick of Smells: Empirical Findings and a Theoretical Framework for Chemical Intolerance)] [(2018) Multiple Chemical Sensitivity Review of the State of the Art in Epidemiology, Diagnosis, and Future Perspectives].

Sensory processing disorders and glutamate

If you want to imagine what life is like as a highly sensitive person recall what it is like to have a hangover — when you have a hangover you have an excess of free glutamate which causes overexcitability of sensory processing.

Children with sensory processing disorders have disconnected white matter (measured using DTI) [(2016)Brain’s Wiring Connected to Sensory Processing Disorder].

It is worth nothing that Gabapentin (600mg) seems quite effective in cases of patients who tend toward alcohol to calm overexcitability and anxiety. Gabapentin is structurally similar to the neurotransmitter glutamate and competitively inhibits branched-chain amino acid aminotransferase (BCAT), slowing down the synthesis of glutamate.

Excitable Mental Illness and the role of Glutamate

A sensory processing disorder makes excitable mental illnesses more likely. Mental illness wise, adhd, aspergers, bipolar, anxiety and autism have “ring of fire” blood flow patterns in the brain. Too much glutamate causes overexcitability of the brain (measured using SPECT). [Amen Clinics Articles] Their clinic uses SPECT imaging as a diagnostic tool for psychiatric treatment [Amen Clinics: Multi-site six month outcome study of complex psychiatric patients evaluated with addition of brain SPECT imaging]. Some examples (all photos are from the Amen Clinic):

SPD causing Visual Tunneling and Autoimmune conditions affecting Eyesight

Visual tunneling is a particularly pervasive issue that affects all senses, it results from an overload of visual sensation in the brain. Tunneling is a subconscious spatial adaptation to reduce the information the individual handles. The individual is either overwhelmed with informatio or has trouble sorting out the significant area for attention.[(2014)Visual Tunneling: A Pervasive Vision disorder]

“Cognitively speaking, for these individuals to succeed, they subconsciously fragment space or tunnel, i.e. reduce the amount of information processed in order not to be overwhelmed in dealing with abstract or conceptual activities. In the physiological aspect, the complications of this adaptation for the hyperopic patient include moving the pelvis backward and the forehead forward, and the eyes feel like they are turned out.” This visual tunnelling affects their ability to control their attention, as well as motor and postural changes. For example, one might walk with smaller steps, round their shoulders, lock their knees when standing, and develop upper back, shoulder, or neck tension.

Nutrient deficiences such as low vitamin D as well as being sensorily overwhelmed cause eyesight issues (autoimmune diseases of the eye and visual tunneling respectively) [source?].

Sensory processing disorders and Autoimmune Disease

People who are highly sensitive (a genetic trait) are more likely to have autoimmune diseases such as diabetes, and I would postulate, celiac. [(2018)Sensory Processing Sensitivity and Type 1 Diabetes] [Celiac and Diabetes are often comorbid]

This study does not show causalty, it shows that 50% of children with autism also suffer gastrointestinal symptoms, such as celiac disease or constipation, comorbidly. Children with ASD have altered gut microbial composition, with decreased gut bacterial variety and underestimation of potentially helpful bacteria such as Bifidobacterium, Prevotella, and Desulfovibrio, as well as a decrease in short-chain fatty acids. Screening of gastrointestinal symptoms and celiac disease in children with autism spectrum disorder

Enteric Nervous System and Excitable Mental Illness

Our guts produce 90% of serotonin, if you are low on folic acid or any other basic vitamins/nutrients you literally cannot create serotonin, dopamine, and other neurotransmitters. [add] Would a severe serotonin deficiency show up in a SPECT scan?

There is ongoing work to directly detect and monitor this for diagnostic and treatment purposes. [UMD: building an ingestible capsule that can monitor and model gut microbiome serotonin activity]

Altered composition and function of intestinal microbiota in autism spectrum disorders: a systematic review. The overall changing of gut bacterial community in terms of β-diversity was consistently observed in ASD patients compared with HCs. Furthermore, Bifidobacterium, Blautia, Dialister, Prevotella, Veillonella, and Turicibacter were consistently decreased, while Lactobacillus, Bacteroides, Desulfovibrio, and Clostridium were increased in patients with ASD relative to HCs in certain studies.

A cross-sectional study of gastrointestinal symptoms, depressive symptoms and trait anxiety in young adults

There has been some recent research on fecal transplants curing some forms of mental illness. The inverse is also true “with the transmission of depressive and anxiety-like symptoms and behaviors resulting from the transplantation of microbiota from psychiatrically ill donors to healthy recipients”. We collect some examples here:

Effect of fecal microbiota transplant on symptoms of psychiatric disorders: a systematic review.

Fecal Microbiota Transplantation in Depression

Posttraumatic stress disorder is associated with altered gut microbiota that modulates cognitive performance in veterans with cirrhosis.

Examining Changes in Microbiota Over the Course of PTSD Treatment.

Open-label placebo vs double-blind placebo for irritable bowel syndrome: a randomized clinical trial

Enteric Nervous system/SPD causing Lowered Immune System causing Autoimmune Disease

If you have a lower immune system (low igA or igG or igM) you are more likely to have an autoimmune disease.

Remark: As a note, you are also more likely than average to have an autoimmune disease with an abnormally strong immune system. This is not the case we are discussing here.

The enteric nervous system generates igA and igG in our body. There is a link between igA and levels with a person’s perception of stress, implying the nervous system has the ability to control igA levels. EFFECTS OF PSYCHOLOGICAL STRESS ON SERUM IMMUNOGLOBULIN, COMPLEMENT AND ACUTE PHASE PROTEIN CONCENTRATIONS IN NORMAL VOLUNTEERS.

People with IgA deficiency should be tested for celiac disease because they are 10 to 20 times likely to develop an autoimmune response to gluten than the general population. ]

Using tests of raised igA and igG to detect for celiac disease is harmful and ineffective, as having naturally lowered igA and igG is usually comorbid with celiac disease. Thus, the rasied igA and igG register as normal levels on tests, giving many patients false negatives. The most effective test is a DNA test for the celiac markers. [(2017)Lack of Utility of Anti-tTG IgG to Diagnose Celiac Disease When Anti-tTG IgA Is Negative].

Immunosenescence or the aging of the immune system is linked to the increase of autoantibody frequency while the antibodies decline [(2004) The Neuroendocrine Immune Network in Aging ].  One way to frame the connection between SPD and being immunocompromised is that higher sensitivity leads to greater stress / perceived stress, which leads to a decrease in immunity which then naturally causes an increase in autoantibodies that would otherwise have occured during someones natural immunosenescence. [(2004) Inflamm-aging: autoimmunity, and the immune-risk phenotype ]

Mechanisms of Functional Dyspepsia and Treatment with Tricyclic Antidepressent

A proposed mechanism is that there is original inflammation, and then continued faulty pain signaling when input is gone. That is, a central sensitization model. [(2002) Functional dyspepsia–a psychosomatic disease ][Functional Dyspepsia: A Review of the Symptoms, Evaluation, and Treatment Options ] [(2005-2015) Antidepressent Therapy for Function Dyspepsia ] [(2018) Antidepressants improve functional dyspepsia symptoms, but how remains unclear ]

Autoimmune Disease is Under Direct Enteric Neural Control

The following are quotes from this paper: Fast acting nervous regulation of immunoglobulin A secretion from isolated perfused porcine ileum

“As in the salivary gland, sIgA is rapidly secreted in mucosal explants from porcine distal colon in response to both cholinergic and adrenergic receptor agonists, but remains unaltered by acute exposure to an enteroadherent microbial pathogen. The enteric neurotransmitters acetylcholine and NE appear to act on muscarinic cholinergic and alpha-adrenergic receptors expressed by crypt epithelial cells to promote transcytosis of the polymeric Ig receptor and enhance vectorial secretion of both sIgA and free SC toward the mucosal surface. In its early stages, this action appears to be temporally coincident with, but mechanistically independent of active ion transport. The precise receptor types and associated signal transduction mechanisms mediating sIgA secretion by these and possibly other enteric neurotransmitters at this mucosal immune effector site remain to be characterized.”

“In a more complex, vascularly-perfused porcine ileum preparation however, electrical stimulation of extrinsic nerves innervating the intestine increases sIgA secretion only after atropine and phentolamine pretreatment; this effect is inhibited by hexamethonium, a blocker of ganglionic neurotransmission. This result suggests that there may exist important inhibitory reflex circuits containing muscarinic cholinergic and alpha-adrenergic receptors which modulate the enteric neural control of sIgA output. Additional investigations will be necessary to establish how basal sIgA secretion onto the mucosal surface in vivo is modulated by the release of acetylcholine from enteric nerves.”

Virome Mediates Stress and Memory Responses

This will be a topic of a future post, it is critical to note that in fact the virome controls the gut biome entirely. Bacteriophages maintain biodiversity of a healthy gut.

The vital viruses that shape your microbiome and your health

Keyword summary:

• visual tunneling
• chemical intolerance
• chronic pain
• excess free glutamate
• high sensitivity
• autism/ocd/add/ptsd/bipolar
• functional dyspepsia
• sensory processing disorder.
• low autoimmune system
• likelihood to have autoimmune disease
• enteric nervous system
• lowered ability to create serotonin
• joint hyperflexibility