- American Dream Serialization (Early Chapters)
- Introduction to Jim Chaffee's Studies in Mathematical Pornography by Maurice Stoker
- Introduction to Jim Chaffee's Studies in Mathematical Pornography by Tom Bradley
- Studies in Mathematical Pornography: American Dream Title Page by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 1 by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 2 by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 3 by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 4 by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 5 by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 6 by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 7 by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 8 by Jim Chaffee
- Studies in Mathematical Pornography: Chapter 9 by Jim Chaffee
- The Apocalypse of St. Cleo, Part V by Thor Garcia
- The Apocalypse of St. Cleo, Part IV by Thor Garcia
- The Satyricon of Petronius Arbiter Volume 2 Translation by W. C. Firebaugh
- The Apocalypse of St. Cleo, Part I by Thor Garcia
- The Apocalypse of St. Cleo, Part II by Thor Garcia
- The Apocalypse of St. Cleo, Part III by Thor Garcia
- The Satyricon of Petronius Arbiter Volume 1 Translation by W. C. Firebaugh
- DADDY KNOWS WORST: Clown Cowers as Father Flounders! by Thor Garcia
- RESURRECTON: Excerpt from Breakfast at Midnight by Louis Armand
- Review of The Volcker Virus (Donald Strauss) by Kane X Faucher: Excerpt from the forthcoming Infinite Grey by Kane X Faucher
- Little Red Light by Suvi Mahonen and Luke Waldrip
- TEXECUTION: Klown Konfab as Killer Kroaked! by Thor Garcia
- Miranda's Poop by Jimmy Grist
- Paul Fabulan by Kane X Faucher: Excerpt from the forthcoming Infinite Grey by Kane X Faucher
- Operation Scumbag by Thor Garcia
- Take-Out Dick by Holly Day
- Patience by Ward Webb
- The Moon Hides Behind a Cloud by Barrie Darke
- The Golden Limo of Slipback City by Ken Valenti
- Chapter from The Infinite Atrocity by Kane X. Faucher
- Support the Troops By Giving Them Posthumous Boners by Tom Bradley
- When Good Pistols Do Bad Things by Kurt Mueller
- Corporate Strategies by Bruce Douglas Reeves
- The Dead Sea by Kim Farleigh
- The Perfect Knot by Ernest Alanki
- Girlish by Bob Bartholomew
- The Little Ganges by Joshua Willey
- The Invisible World: René Magritte by Nick Bertelson
- Honk for Jesus by Mitchell Waldman
- Red's Dead by Eli Richardson
- The Memphis Showdown by Gabriel Ricard
- Someday Man by John Grochalski
- I Was a Teenage Rent-a-Frankenstein by Tom Bradley
- Only Love Can Break Your Heart by Fred Bubbers
- Believe in These Men by Adam Greenfield
- The Magnus Effect by Robert Edward Sullivan
- Performance Piece by Jim Chaffee
- Injustice for All by D. E. Fredd
- The Polysyllogistic Curse by Gary J. Shipley
- How It's Done by Anjoli Roy
- Ghost Dance by Connor Caddigan
- Two in a Van by Pavlo Kravchenko
- Uncreated Creatures by Connor Caddigan
- Invisible by Anjoli Roy
- One of Us by Sonia Ramos Rossi
- Storyteller by Alan McCormick
- Idolatry by Robert Smith
- P H I L E M A T O P H I L I A by Traci Chee
- They Do! by Al Po
- Full TEX Archive
Studies in Mathematical Pornography: American Dream - 5
By Jim Chaffee
Tuesday opened late in the day, noonish or so, my dick and mouth coated with the remains of my guests nether excretions seasoned by leftover onion and mustard. It occurred to me I'd be running low on funds with the fucking credit card to pay on, likely facing short rations, needing to curtail my habit of eating out every night.
I smoked a cigarette and a joint and drank a cup of instant coffee, shit, showered, shaved, and trudged off to the A&P where I gathered coffee and basic cheap foodstuffs to fit my limited galley potential: canned sardines, canned soup, canned tomato sauce and noodles, Kraft macaroni and cheese in a box, carrots and potato chips for when I got munchies, Dixie beer. I piled into a queue at a register behind a spidery chick the color of blackstrap molasses, hair pulled back in braids woven with beads and other colorful tidbits hanging down over a high arch of ass. Spindly legs ending in feet shod in spangled pink flip flops tumbled from a short denim skirt. Pronounced ribs demarcated a stretch of torso barely covered in short orange bandana halter-top tied in back and weighted down with mammoth flesh mounds hanging via suspensions of skin stretched from the sides.
I ached to reach out and grab that orb of a narrow ass. She must have read my mind, turned and sized me up. The round face didn't harmonize with the elongated body and pendulous bosoms sequestered in their tenuous halter; she'd painted it up so the eyes appeared as bone-black almonds, lashes dense and inky, the extended line of narrow lips amplified beyond their natural boundaries with lipstick of fire-engine red. Her forehead domed high, naked with the pulled-back braids; her flattened nose might once have been smashed by a two-by-four.
"Misty likes tall white boys," she said, her voice a honeyed drawl not from New Orleans.
"Really? What's she like about them?"
"White dicks. Misty likes sucking white dicks."
No one turned to listen, as if standing in line in the A&P at the corner of Royal and St. Peter talking about sucking dicks with a half naked black woman was the most natural thing in the world; but then the guy in front of her wore lipstick and hung on to the man beside him as if they were long-lost lovers.
"So who's Misty?"
She pointed to herself.
I'd have been suspicious she was a guy in drag had one of the pendant hangers in the halter not shown the upper arc of a giant nipple. And nary a hint of facial hair.
"Misty likes them tall like you. And that white hair. You are a pretty white boy."
"Thanks. You're not so bad yourself."
"You want to visit? I take a small contribution."
"What kind of contribution?"
"Money, baby. Just a little for you. Almost free."
"Well, I'm a tapped out white boy. No money till payday, and then not much. A poor white boy."
She gave me another once over, noting my clothes, and smiled. "I guess I can see that. Well, you come on by sometime anyway. Maybe we can work out a sliding scale." She handed me a pink card with a phone number and address in the upper reaches of Royal, likely around Barracks, the name Misty smeared across in the same color as her lipstick. She held her tits up with both hands, the giant gap diverging like a fissure, and asked, "You like these, baby?"
"They seem admirable, but I'd need to inspect them to be sure."
"Well, you call me before you come by, so's I'm not pre-occupied."
She turned, made her purchase, and walked straight-backed out of the store, flip-flops flapping the concrete, high ass playing its own rhythm trailing a wake of disturbance behind.
I closed myself up in my apartment and played from my worn out record collection, boxes of LPs kept for me by a high school friend while I survived the Green Machine, spinning them on my only furniture, Mcintosh amp, Thorens turntable, and a pair of BIC speakers. Moving would have been a breeze had it not been for shipping this set. Everything else expendable, consigned to the ash-heap of university scrounges.
I smoked dope and listened to Miles Sketches of Spain, then Monk, Billie Holliday, Dolphy at the Five Spot, Trane, Sonny Rollins, Parker and Gillespie on a Canadian gig taped on a shit tape recorder by Mingus. As it darkened I smoked one joint after the other and cried, falling asleep lying flat on my back in bed until Bill Evans emptied out and thumping against the spindle roused me. I ate a can of sardines, some crackers and carrots, drank a beer, passed into sleep.
Falling backwards into a hole, panicked, sat bolt upright. "Cocksucker," said or thought. I sat for a while, heart racing, cold with matted sweat, swung my legs over the edge of the bed and got up, sun rising over Bourbon Street, thinking I'd review for my exams. I smoked a joint and looked at McKean's Stochastic Integrals, drifted asleep reading his proof of Feller's test for explosions, understated perfection eschewing the obvious.
Then it was late and I had to rush to get to class, barely time for the iced coffee I kept for such emergencies, jolted awake and quick shit shower shave forsaking the lulling streetcar for the Freret Street bus, flanking the math department on maneuver directly to the class late, ambushed by the bell. Five minute rule in effect. My beetle browed lady fan not in attendance. I sang a dry lecture on sampling statistics, their first brush with confusing abstractions. The entire class fled at the bell, not even one question from the baffled cheerleader.
Not in a chatty mood, I checked mail and headed directly for the seminar. The Frenchman greeted me at the door, his hello sliding above the film of accent enough to let the listener know he didn't hail from these parts.
"Your exam tomorrow afternoon," he said. "I wanted to come, but was told only the PDE and topological algebra will be there."
"Interesting. Supposed to be open to the public."
"Yes, but I think it means less than what it says. It's so?"
"I have heard it is a formal affair, this exam. They have decided in your favor already. But don't tell anyone I have said this."
"No problem. I'm not going to believe it anyway. If I fuck up big time, they'll either drop me like a hot potato or make me redo some of it. That's how it works. Lots of times need to repeat some fucked up part of the performance."
"You'll be fine. Don't worry."
The first day he'd handed out a mimeographed set of lecture notes in French from some Séminaire de Probabilités on Géométrie Différentielle Stochastique by a bunch of hotshots I'd heard of, Paul Meyer and Laurent Schwartz among others, including himself. He also gave me a preprint of a some stuff on martingales on analytic complex variétés, but I hadn't taken much of a look at it. I could read mathematics in French, had passed a standardized exam in French for science that wiped out one language requirement, but still found it a pain. I wasn't sure if variété meant algebraic variety or manifold, I assumed the latter, but needed to invest in the French mathematics dictionary I knew existed.
It seemed odd we plowed away in this seminar on some stuff related to what control engineers called system's theory, something about input-output systems more related to operators than to martingales on manifolds. I'd asked him once and he'd only told me that the papers he'd given me were all relevant, but he might not get far enough to cover their utility. They did assume more machinery in probability and geometry than many here could deal with, particularly the relationship to manifolds. But it didn't seem new stuff, either, since McKean did it in his almost decade-old book, a principal work on my reading list.
This day he talked more about transfer functions and Fourier transforms, working towards some sort of decomposition theorem based on an operator representation. He made forays into Hp spaces as places where transfer functions lived, spent an inordinate time getting conditions for positivity of a kernel using interpolation and now was working to get a reproducing kernel representation for "realizations" of causal, linear, dissipative systems. He hinted at some kind of contractivity, but I think we were all clueless on where he was going; I followed every argument he made but saw no shape looming at the end of it all. I'd felt the same way about two semesters of theorems regarding modules, no hint of the algebraic geometry or whatever had been the original impetus.
The Frenchman's work had been important enough to someone to get him tenure at Stanford, so plenty of people attended but so far not many had any questions for him, a bad sign. I guessed his real goal was to teach at the École Polytechnique, recently relocated to the outskirts of Paris, or more likely the École Normale Supérieure, though I'd heard the place to be for his stuff was really the Université de Strasbourg. None of it made that much difference to me; I'd be lucky to get a teaching position at some third rate US university with a sickly graduate program.
First I had to pass my orals.
After seminar I attended my advisor's class, then bolted for the gym to do the excruciating leg workout. Then home on the streetcar, not quite jammed up with commuters though plenty of tourists. I got a window seat anyway, starting out at nothing until we made the final straight leg to Canal where I dumped out and walked home, working off the stiffness from the exercises, wondering why I'd not heard from Lori.
Dinner, music, sleep. I forwent jazz for Stravinsky: Sir Colin Davis's Rite and Petrouchka. Then Shostakovich's Seventh, promising myself something more abstract later.
Stumbling awake through late morning already muggy, bright and shining, many joints with coffee and then a forced march down Bourbon to the streetcar and a swaying ride uptown, Audubon Park and the ugly old stone monstrosities, approaching from the front. Expecting ambush. Knowing of no reactionary force in readiness.
I smoked a heavily hashish laced cigarette, marched up the stone steps, up the giant's stairs inside, and trudged down the hallway towards the room where my inquisitors awaited me. Deserted like an abandoned firebase. Not a soul in sight.
I opened the closed door and entered a room full of them. Spread across the front row the PDE group: my advisor, beside him the group's brash young genius, then the old maestro, and then Oberst, the topological algebra maestro. The next row: Momus against the wall behind Oberst, the younger PDE guys strung out beside him, one an up and coming star, and then scattered beyond them the remnants of the topological algebra crew.
A sudden hush; I'd surprised them at something.
The young PDE group genius stood and said, "You're a few minutes early. If you don't mind waiting outside, we'll call you."
I stood outside the door like a troublesome brat expelled from the classroom, pushing my nose into the wall, butterflies like going on my first patrol knowing armed men wishing me harm awaited. My advisor fetched me.
The PDE group genius began with a softball, doubtless planned to set me at ease.
"Tell us about your favorite PDE and its solution method."
I launched into a spiel on the Dirichlet problem in a bounded region of Euclidean space with a smooth boundary, conflicted between two favorite approaches. Taking first the elegant proof from Folland using Dirichlet's principle, interrupted by a random popping of irrelevant question from the back of the room, "What's smooth?," "What's the normal field?," "What's a harmonic function?," "What's a vector field?," "What's a norm?," "What's a bounded domain?" which I at first attempted to answer but soon realized led to an essentially infinite regress. Smiles on all the faces except Oberst's, whose visage bespoke boredom at foolishness, told me it was sport and I pretended not to hear any longer, proceeding to the set-up with the Dirichlet integral I said was a kind of potential energy on connected components. I showed that adding the standard norm and taking a square root gave a proper norm obtained from an inner product so the completion became a subspace of L2 and also a Hilbert space in its own right.
No one interrupted so I plowed ahead, showing that the restriction map to the boundary could be extended to all of L2, corollary of a bound on the L2 norm by the potential norm, an estimate I proved with a hand wave. I mentioned in passing that in essence we'd restricted to functions in the Hilbert subspace possessing L2 derivatives of order one half on the boundary.
I stepped back and examined what I had wrought so far, waiting for real questions or objections or a call for more detail. The board contained a goofy drawing of a domain, the norm, an inequality, and an integral along the boundary crucial to the hand wave, the merest hint of what I had spoken.
They allowed me to pass from the unproven result and I sailed on into the last stages, sketching a proof of a beauty proclaiming equivalence between functions harmonic in the region and functions orthogonal with respect to the potential on the subspace of compact support in the region. Home free, I showed that since the functions with zero potential were locally constant, hence harmonic, they could be written as direct sums, their orthogonal projection onto the harmonic space being the solution to the Dirichlet problem. I then erased what little remembrance I'd left on the board and stated Dirichlet's principle of equivalent conditions, leaving only the drawing.
Professor Momus broke the silence with his characteristic throat clearing.
"Is this a good solution, I mean in a classical sense?"
I knew he would be aware of the history of this principle, stated by Dirichlet, used by Riemann and discredited by Weierstrass, rehabilitated by Hilbert. In the century or so since this rolling controversy arose mathematicians had built heavy machinery to give logical substance to these statements, equipment those giants could not even imagine: Hilbert spaces for functions to live in with right angles defined in infinite dimensions, Lebesgue integration to complete those spaces of integrable functions into Hilbert spaces where one could rest assured the orthogonal projection onto a subspace existed. Stuff like that.
"Not in some sense," I said. "Its weaker. We used that weak solutions of the Laplacian are harmonic. We also cheated a bit on the boundary functions, but I don't think we have time to show the boundary functions are continuous given continuous data. But we can go there if you want…"
"No, it isn't necessary," the PDE genius said. I noticed the random interference from the peanut gallery had ceased.
"Anyway," I continued, "it seems remarkable that one can also solve this problem by using a random process, the Brownian motion." I drew a wiggle inside to the boundary where I stopped it with a stopping time, evaluated it with the initial data on the boundary and took the conditional expectation given the starting point. "That, I said, was a harmonic function that solved the Dirichlet problem, essentially the same one we had already found by deterministic means.
I added one more interior squiggle heading to the boundary, talked about the infinitesimal generator of the process for extending the method to more general PDEs. A realization dawned on me and I let it out into the room: perhaps the mystery of randomness providing the same solution could in part be explained by considering that the conditional expectation operator was really an orthogonal projection onto a subspace of functions, just as before and likely the same space.
Standing back, the sparsity of marks on the board surprised me. A nebulous shape with some squiggles like lonesome spermatozoa or maybe spirochete emanating from a couple points to the boundary, a formula for the conditional expectation on the boundary, nothing else. Yet I had talked a good stretch, leaving only hints.
My advisor's turn. "Give the defining properties of Brownian motion."
Another softball. I decided to expand this as far as I could.
I started by saying I found it helpful to think of Brownian motion as a drunk walking down the street, changing direction by coin toss at each step, the steps decreasing to infinitesimal length at delta times going to zero, a limit of a random walk on the real line.
"Constrained to a line," someone said and with the word random the chorus in back resurrected the inane hooting. "What's random?" "What's a random variable?" and more, all of which I ignored.
I gave the standard four characterizations of Brownian motion: almost sure starting point at zero, almost surely continuous sample paths, independent increments and normally distributed increments with mean zero and variance the size the delta of the time step. Then I added the two conditions equivalent to the last two, namely normally distributed linear combinations at finite times and covariance at times s and t equal to the minimum of s and t. I considered proving the equivalence but thought better of it. Instead I plowed into Einstein's argument that a particle beginning at some fixed point and subjected to random bombardment in a medium allowing its motion to be rotationally symmetric would exhibit such a normal distribution. I wrote the density integral on the board, then talked about transition probability functions. From this, before anyone could stop me, I obtained the finite-dimensional distributions, then hurried on to give the basic idea of how it could be considered the solution to the heat equation on an infinite rod with a Dirac delta at the origin at time zero, diffusing to a stationary distribution uniform on the real line after infinite time, assuming one believed in infinitesimals. I considered veering off into a discussion of such improper distributions as priors for Bayesian statistics, a technique I thought begun by Laplace for silly arguments Christians repeat for the existence of God, but realized it would be stupid and suicidal.
The hooting from the gallery had stopped but it wouldn't have mattered, so involved was I in laying out this landscape of the mind. I tore on to sketch two proofs for the existence of such a process, zipping along the top of the Wiener proof using the sample space for the probability as the set of paths themselves, mentioning that the finite-dimensional distributions could be extended to a real measure on a sigma-algebra of functions concentrated on the continuous functions. Beside the integral for the finite-dimensional distributions, I wrote an example: the set of all sample paths b such that b at time 1 is between .1 and .3 and at time 3 is between 1 and 1.1.
By this time, I'd begun writing BM for Brownian motion. Before anyone could say anything I launched into the complete proof of the existence of this measure on the function space constructed around the Haar functions, the approach due to Ciesielski lifted out of McKean. That killed some time and also led to a proof for the expression of the covariance of the process. This left us square in the midst of a space of continuous functions with Wiener measure as model for the Brownian motion process, righteously called the Wiener process. From this vantage point I drew out a sketch of the fact that the measure was concentrated on continuous functions that were almost surely not differentiable, albeit it rather hurriedly with a lot of vague but authoritative arm-waving. I waded into the conditional expectation properties: Markov property, stopping times, strong Markov property, martingales and semi-martingales, ending it all with a proof that BM was a martingale.
I stepped back to view my handiwork and again found little on the board, a sloppy integral fdb and several large BMs. I shrugged.
"Should I prove Blumenthal's zero-one law or Khincin's law of iterated logarithm?"
"No, that's more than I expected," my advisor said.
"Just let me bring out the semigroup connection to the heat equation and the abstract Cauchy problem," and I stepped to the board, gave a short spiel on semigroups of operators, infinitesimal generators, and a bogus explanation for the semigroup property of BM on bounded uniformly continuous functions on the real line based on the Markov property, ending with a derivation of the fact that the flow of the semigroup satisfied the heat equation. In the middle of the board a giant BUC(R) sprawled amidst BMs.
As I stood back basking in the glow of ideas lingering in the room, my advisor asked me to sketch the construction of the Ito integral and derivation of Ito's lemma.
I started by talking about Wiener's integral with a nonrandom integrand. The idea of using BM for an integrand in the Stieljes sense was cute, but of course the sample paths were so badly behaved, of unbounded variation in every interval, that it was not possible in a standard Riemann sense to do this path by path, and Lebesque's method didn't apply. Then I gave Wiener's trick for this particular case, starting with functions of compact support so it was possible to use integration by parts and reverse the path differentials. I stated that the map defined by this integral was an isometry of the subspace of continuous functions with compact support that overlapped L2 with L2 itself; added that this so-called white noise integral was itself normal with mean zero and variance the L2 norm of the integrand.
Dramatically pausing, I said such an approach could not work if the integrand was a function of the sample paths.
I detected eyes glazing over in the gallery, most of whom had long since lost interest. Their glazed expressions spurred me to greater heights as I waxed eloquently on the irregularity of the functionals depending on both time and the BM sample paths which were themselves now considered the random elements. I gave conditions for a functional to be an integrand: jointly measureable with respect to the Borel sets on the product of the extended real line and the sigma-algebra of all sample paths, non-anticipating in that the functional as a function of a path at time t depended only the path up to time t, reviving the connection to the Brownian sigma-fields generated by BM paths b(s) for times t less than s, and finally that the square of the functional with respect to time was finite almost surely. I said a few words about the causality implication of non-anticipation, then defined simple functionals and, taking care to emphasize that the simple functionals be evaluated at the left endpoint to keep from sticking out into the future, defined the integral as the sum, showed that general non-anticipating Brownian functionals could be approximated in probability by simple functionals, and defined the general integral. I finished by showing that the integral itself was continuous and defined simultaneously for almost all sample paths. I proved the integral a martingale, and mentioned in closing Stratonovitch's stochastic integral, obtained by evaluating the integrand at interior points of the intervals, leading to a more intuitive integral but without the martingale property, a serious defect.
By now I wrote more but erased it almost as soon as it hit the board, chalk in one hand and eraser in the other, modeled on the professor who had dragged me through a semester of multilinear algebra, tensors and exterior algebras and modules. On one end of his classroom high above the ground stood a window from floor to ceiling at the end of the chalkboard runway down which he had sped with chalk and eraser, wiping as he wrote, bouncing off the bars across the window and starting back the opposite direction. One day I stopped him with the comment that if we were to cut through the bars he would plunge to his death almost surely. He paused to laugh, saying the rumor was it had been a math professor falling to his death that had prompted the bars in the first place, then picked up where he'd left off.
I wore down one soft yellow chalk, the dust settling around me in a dull haze, and picked up a white one that screeched when I wrote with it, someone in back likely awakened by the rush up his spine hollering to use another piece of chalk.
I ignored him and sped into a discussion of what might happen if one did integrate with respect to paths not of bounded variation, presenting an intuitive and wildly incoherent sketch based on Taylor series that perhaps instead of going to zero might wear a second order term, a finite remainder of the wildly jumping path. I gave Paul Levy's result regarding the quadratic variation of BM and came up with the Ito lemma for a pure BM process, with the second derivative terms of the integrand appearing for the quadratic variation in the integral. I used the formula to show that the integral of BM with respect to BM from time zero to time t is BM squared divided by two minus t divided by two, the extra term the quadratic variation.
At this point my advisor, laughing softly, stopped me.
"Don't prove the general Ito formula," he said. "Just state it."
And so I did, leaving as evidence on the board the mnemonic two-by-two table of stochastic differentials with dtdt, dtdb and dbdt equal zero, dbdb equal dt.
I finished with the fact that the Stratonovich integral doesn't have the extra term, a dividend of knowing the future, and then wrote the conversion formula, warning that the Stratonovich integral doesn't have dominated convergence properties and is really less a stochastic integral than integro-differential operator.
I stepped back from the board, stick of chalk in one hand, eraser in the other, chalk streaking my jeans and t-shirt. I put both down and wiped my hands on my butt.
"Should I derive the exponential martingale?" I asked and someone at the back of the room, one of the topological algebra types, said, "No, please. We surrender." My advisor chuckled and gave the floor to Oberst.
He said he would appreciate it if the room would remain quiet while he asked me a few questions. Then he looked at the board and said, "You don't leave much evidence, do you, Mr. Butcher?"
The quieted room let out a joint grin.
"No sir, I think it's dangerous. No tape recorders, either."
Some laughter, not long, not loud.
"I have a few questions I'd like to ask." Precise, clipped, authoritarian, when Oberst spoke it came across like a Gestapo colonel giving orders. "Can you define Lie groups and Lie algebras and explain their relationship."
I decided to go halfway between Miller's specific approach and the general, heavy equipment approach of Kobayashi and Nomizu, starting with Lie groups as differentiable manifolds with differentiable group product, diving into left translation by an element of the group, left-invariant vector fields and then defining the Lie algebra to be the set of all left-invariant vector fields with respect to the usual vector field bracket and of course addition and scalar multiplication. I generalized to define Lie algebras as vector spaces with a bilinear product called the Lie bracket satisfying the two standard conditions and showed that the bracket operation on vector fields provided a Lie algebra isomorphic to the tangent space of the Lie group at the identity.
Instead of pressing on with the exponential mapping, I stopped, letting Oberst ask for it.
He did. I flew into it, everything I had gone over in the previous week in the abstract and in the concrete rushing into my brain faster than I could speak, like walking through hallways of hallways leading to yet more hallways stretching away to infinity in static perfection, vector fields and integral curves and differential equations, flowing one-parameter groups the local version of global orbits of vector fields from whence they began their flow on an arbitrary differentiable manifold. I specialized it all to the Lie group, local one parameter groups commuting with left translation, unique solutions to differential equations defined by a Lie algebra element A giving rise to a unique one-parameter group flowing out from the origin defined as exp(At), evaluated at t equal one giving the map exp from the algebra to the Lie group.
I decided to zero in on the matrix case and as I wrote a general time-invariant differential equation over the general linear group I realized I'd been limping, more like dragging my right leg along, and people watched that, not what I did on the board. My noticing brought on the pain.
I leaned against the board to catch my balance, then wobbled to the first row and pulled over one of the small vacant desks and plopped down in it.
"Excuse me," I said. "My bum leg is attacking me."
From my chair I wrote the exponential of the matrix to give the solution as a series, showing the flow and the exponential map into the group from the algebra, talking about the general linear group and its Lie algebra of matrices, bringing up the classical examples.
"I can go into the Campbell-Baker-Hausdorff Theorem if you want," I said. "I just need to give my leg a minute."
I needed a drink and a joint. Pain rippled like hot spikes flowing from some nexus deep in the shattered remains of my pinned and pasted femur.
"No, that's fine," Oberst said. "I don't have more questions."
"I have one, if no one objects." It was Momus. No one said a word, so he cleared his throat and I expected the next words to be about Daniell integrals in this stochastic sense. "It seems that these topics are relatively disjoint. I don't see any connection," he said in his officious tone, as if he were beginning a speech. I waited for more, but he seemed finished.
"Of course, the Lie groups we are concerned with are the symmetries of the PDEs, the ones that transform the solution manifolds to new solution manifolds."
"Yes," he interrupted, "I am aware of those."
"The stochastic integrals correspond to PDEs via infinitesimal generators, and there is a tie there with symmetry groups to evaluate them in closed form. That is on one of the papers in the list."
Back in the hallway of hallways I stumbled on a new crossover lying ready at hand and stood up wobbly to head back to the board, leaning against the chalk rack for support.
"Ito and his school did some work on diffusion processes on manifolds and McKean has specialized it to a few sections in his book with Brownian motion on Lie groups, mostly the orthogonal group, and also on the space of symmetric matrices. It was amusing but not all that interesting. But it struck me just now that one can tie quadratic variation to connections on manifolds, since you can think of connections as a Hessian, mapping from smooth functions to symmetric bilinear forms."
I stopped, thinking perhaps I'd gone slap happy with the pain streaking up into my groin and radiated out as background waves pulsing my thigh, amplitude from dull ache to excruciating pierce and back, a periodic misery. I forgot about even considering Noether's theorem and the relation of conservation laws to symmetry. And maybe cohomology.
"I think we're done here," my advisor said.
I knew the drill. I had to leave while they decided my fate. I gripped the chair and raised myself, pitched forward and made the door, grabbed the knob for support and hobbled down the deserted hallway to the lounge and the comfort of the couch, hugging the walls for support.
As I passed the secretarial office next to the lounge Joelle looked up.
"You've been in there three hours," she said. "I thought they were only supposed to have you for two— My God, are you all right?"
She jumped up from her desk and came to help me over to the couch. "It's your leg, isn't it?"
I didn't say anything. I hadn't talked about my leg to anyone. Just the thought of being branded a baby-killer in Vietnam kept me mum. Now it seemed one of the damned secretaries knew about my leg. I wondered who else knew.
"I'm all right," I said. "Happens all the time when I'm on my feet too long."
My advisor appeared at the door.
"You did fine," he said. "You passed."
"Good," I said, clammy with sweat, almost cold, probably pasty pale.
"They were only a little unhappy with the way you used the board. They would have liked more written out."
"It isn't like I was lecturing first year grad students," I replied. "If I'd written it out with all the gory details we'd still be there." I paused to a mingling of burning cramp and nausea bouncing through some part of me I couldn't find. "Fuck 'em if they can't take a joke," I mumbled.
"What's wrong with your leg?" he asked. "Are you all right?"
Joelle sent him a withering stare. "It's a war injury."
I shook my head. Now everyone would know. Son of a bitch.
"You were in Vietnam?"
"I didn't know. What happened?"
"He was shot."
I decided to let it all out. "That's true. Six or seven times, in fact. Not sure exactly how many or if it was a machine gun or AK-47 or both, but at least six. The bullets shattered my femur, turned it to a kind of pulp. I'm lucky I kept it."
By now a small group stood around watching. Mrs. Dupre appeared and Joelle told her she was going to drive me home.
"No," I said. "I can grab the streetcar right out front here, take it to the quarter and catch the shuttle bus. It stops in front of my apartment."
"No sir. You wait here. You better be here when I get back. I'll get my car and get you home."
"You don't have a class to cancel today do you?" Mrs. Dupre asked.
I don't remember what happened. Joelle reappeared in a deserted room. She and Mrs. Dupre helped me down the endless stairs and wide stone steps in front of the building and into her car. She knew I lived on Bourbon Street and I stopped her at my building. She pulled up onto the sidewalk and left her flashers blinking, flagged down a massive queen cruising between the bars. Together they helped me up the winding narrow stairway to my apartment. He left to watch over her car to keep it from being towed, and she asked me what else I needed as I lay on the bed, near to passing out from the pain.
"In the medicine cabinet, a big bottle of pills and some water, please."
She brought both and I fished a thirty milligram morphine tablet from the menagerie of illegal narcotics.
"You better go," I said. "That big queen won't be able to hold off the city's tow truck force for long. They're always circling like vultures. And thanks."
"Should I get a doctor?"
"No. I've lived with it for a while now. It's not the first time. I'll be fine."
She hovered, uncertain about leaving me.
"Really, you're an angel. I'll be fine. Don't worry."
She hesitated in the doorway, staring at me, and then was gone. I pulled the box of good reefer from under the bed and rolled a joint, passing into sleep while smoking.
© Jim Chaffee 2011