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tilting

A project by Mateus Domings and Hipolito Treffinger as part of MmATT at University of Leicester

This version of the page is almost as it originally appeared. Some styling has been removed and images are behind clicks, to match the rest of bruise.in.

/ Foreword

/ Tilting (story)

/ Aiming for meaning (essay)

/ Images

/ Game

⬑ Foreword

In the year 2020, at the University of Leicester took place an event in which mathematicians and artists would meet and work together towards the dissemination of Mathematics in a new, original way.

The main idea of this event was that artists would patiently listen to the mathematicians as they try to give some explanation of their work, often filled with farfetched, inconsistent analogies, hoping to get some inspiration. Afterwards, the artist would work based on the mental images they retained and would present that work to the mathematicians, whom in turn would see the bits and pieces that were lost in translation and go again with another imperfect explanation of their subject of expertise. This process would repeat itself several times until everyone involved in the collaboration is satisfied with the final product: a piece of art inspired by and reflecting a piece of contemporary cutting edge mathematics.

This booklet is one such piece, product of the collaboration of the writer and artist Mateus Domingos and the representation theorist Hipolito Treffinger. In these pages you can find two short complementary texts that can be read independently from each other.

The first is titled “Tilting”, written by Mateus, is a short story accounting a fantastical journey of Carlo Luiz to the village Tau, whose people have a mythology and culture based on the mathematical knowledge of their ancestors, even if real meaning of the rituals are now lost to them.

The second, entitled “Aiming for meaning: a brief  account of quivers in representation theory” and written by Hipolito Treffinger, is a highly condensed account of the developments in representation theory of finite dimensional algebras. The aim of this text is to present and explain briefly the mathematical objects that live behind the culture of the people of tau.

Occupying some space between the purely fictional and the mathematical, you will also find  a set of rules for a game. It was the development and working out of this game that allowed a common language to be found between the collaborators.

We hope you like these stories.

Mateus & Hipolito

⬑ Tilting

by Mateus Domingos

By this time Carlo Luiz was an old man. He had correctly assumed that he would pass from our world to the next before the month was over. Fifty years had passed since he had returned to Buenos Aires and said goodbye to the dreams he had once held. It was already late in the evening when he decided to visit one of the old university bars.

Over the years he had often cursed himself for ever leaving.

Some might doubt the story that is set down here - for it has been told, like many, by a desperate man. It deals with a place, the existence of which had long passed from memory to myth. And if this man had not seen this place, then all his research was for naught. In fact his position at the esteemed University of Buenos Aires depended on it. Some had said the place was other places given a new name, Shambhala, Babylon, Doggerland. But these places each had their own devotees, followers, mystics and experts.

The place here in question was Tau, or more accurately 𝛕.

The world had changed many times over since the first mathematicians had entered exile and began building 𝛕. Their disappearance could be traced in the artefacts of war. The exponential curve that seemed destined to pass through the nuclear discovery had stumbled. Their terrain was that of new ways of seeing the world as it really was - more accurately, and in turn this created new opportunities to more effectively yield reality to the will of man. In the changing interests of those holding the power, the truth of the mathematicians had been seen as a threat and as such banned - except for within the confines of closely monitored government labs. In the years since, this position had changed - as it always had in fact, between control, relaxation, freedom and secrets - like a tide, making and unmaking the slope of a beach; each time some things lost, others found.

Have you ever stood in the shallows as the waves move back and forth - a glint of silver, a fish, or the dark grey of a stone, or round green of a bottle shard spotted near your feet. Treasure that as the surf washes over disappear, and once the water has cleared again, can never be found? Difference and similarity washed over.

Carlo had found 𝛕 in a sheltered limestone valley, far enough within the borders of some small European state, never to see the military marching and shifting of borders, imaginary and physical.

As if to give credit to his story, he says that many others have found the place over the years, only they chose to stay. They had gone there, seeking refuge. However, for all of these, he was the first mathematician to set foot in the valley for generations. However, the vision that greeted him was instantly familiar, he said it was as if the limestone bed was a great chalkboard and the place, the people and even himself now stepping there, were just the traces of chalk, to which some had argued this is the way of all life, since mathematics describes it so, and to which he countered, the difference here is an intention and specifically the desire to reduce life to one unifying algebra. Usually we imagine a unifying theory as encompassing everything as it exists, but if you reverse this and describe everything contained within a single theory what do you build? “Well that is simply a set,” one retorts.

Carlo could remember his brief summer there more clearly than anything before or since. He remembered these things so clearly that he could sometimes convince himself he was simply inventing it anew.

He had no way of knowing if 𝛕 still existed as he had seen it. Perhaps by now it had changed once more - the valley reclaimed by forest or meadow, the community itself dispersed and forgetful, or maybe a town had been built and the village rites quashed with the customs of the state. Traces of 𝛕 would remain. The old cross, that had been an origin marker, now anointing instead a small shrine to Our Lady of Fatima.

He had never again tasted bread that was quite like that made by the baker in 𝛕. She had been following the principles taught to her by a grandmother, who had been a young girl at the time 𝛕 was founded. She had told her granddaughter stories of their passage from the city university campus, via trains, passing cars, and eventually by foot, passing through smaller and smaller villages, the distance between them always increasing. The granddaughter liked to claim that the dough was still made from the same yeast culture that had been brought along on that founding journey and that it was therefore the oldest part of 𝛕. Whether or not this was true, the culture had a name, that all inhabitants of 𝛕 knew, and that was Casimir.

In the years before his journey, if asked by someone who was not an academic, what the purpose of his studies was, or what their application was, to them, he would have said something along the lines of, ‘to further understand the system of mathematics by which we can describe existence.’ Later, Carlo would have answered instead that it was  ‘mostly to mask reality, and occasionally to reveal something that surprises everyone.’

The vocabulary of maths is a language of myth and magic. Looking across the valley at the exposed roofs and fenced yards of 𝛕, Carlo was struck by its appearance as familiar to him from diagrams. The buildings are mostly equidistant apart, and from this vantage he can see the worn paths that thread between them, almost uniformly straight with several exceptional cycles, but each path moving between buildings. No signs of the usual arterial paths or streets that so often approximate a line of best fit.

Carlo had waited on that ridge as night fell, making a fireless camp and sleeping out the night under a star filled sky, mostly awake and restless thinking of the village below. After daybreak he waited for signs of life from the village. Threads of smoke rising from chimneys, children yipping and the tumbling bells of goats being driven out to pasture, over the far side of the valley. He made his way from the ridge, through old forest to a stream. It was too wide to cross easily and instead he walked slowly upstream, closer to the village, until he found a rough bridge, made from a single tree. He made it across, and climbed the shallow hillside to the village, finally being sighted in the late morning sun.

He was welcomed in without question, and it was in fact some time before either Carlo or the villagers who had met him spoke a word to each other. When they did, Carlo found that conversation was easy enough. The people of 𝛕 spoke in a, pidgin, mostly French and German -hardly any of the Spanish that he knew so well, but at its core built around the kind of mathematical terms that usually float between languages interchangeably. Carlo would learn quickly that this followed with their use of writing, which appeared mostly as algebra.

In the notebooks that had remained with him over the years, he would sometimes marvel at the entry written on that first day. Only a simple translation of a few mundane words, and which he would note wryly to himself were not actually translations as much as equations.

𝒙 = eggs

0 = 𝛕

𝒊 = 1

𝒛 = bread

They were playful and quietly magical in the way that all of 𝛕 was. It charmed him.

By the time Carlo visited 𝛕, the founders had all passed on long ago. The villagers were surprised to learn that Carlo had been looking for them specifically. They believed the nearest villages were probably not so different, and in founding this place of refuge they had gained a kind of anonymity or forgetfulness from the rest of the world - and the forces they had been running from. It was revealed soon that they stayed in the village also to protect what they had taken there. That knowledge existed now, externally realised, through rites, building, song and arts.

Carlo was invited to tour the village. He met those who were there, and later would meet those returning from foraging or play. He was invited into the homes and different buildings. Indeed, moving around the village, during the day, at least, it was customary to pass through each building - as truly through a vertex. At night, or after a dispute it was permitted to move around the building perimeter. This movement was directed also - it was customary to walk towards the north of the valley. Journeys in the opposite direction could be made along paths marked out with stepping stones. These appeared to run in parallel tracks between certain buildings. The stepping stones made use of these paths less convenient and often the villages would instead walk the Northward paths backwards.

He was told that most of these buildings were old, and built by the founders. Occasionally new buildings would be constructed. However, their placement would involve reference to The Graph. Blind lots are cast until the acceptable place is found. These are rules they inhabit blindly. They are customs and tradition. They are known whilst remaining undefined. The orientation of the directed paths, Carlo would later come to believe, were placed against the current of the stream, that traversed the valley north to south.

Carlo had long subscribed to the belief that every action has an inverse. The practical implications of this belief had evidently eased its importance on 𝛕 over the years. However there was still one dwelling on the opposite side of the village to the ceramicist and kiln where people would take old or broken pots and plates and for each item produced by the ceramicist the ceramicist-inverse would smash a corresponding item. Therefore the production of new things was kept in check by the ability to destroy the old. Making and unmaking. And of course remaking would follow the unmaking. The broken vessel in pieces stacked and carried away to a third place (the convergence of two equidistant vectors from the first and second position). Here the pieces would carefully be placed back together. It was held that it should be possible occasionally, to create two objects from the single object. Identical instances of the same. This had never been done - but still for each item the method was tried.

Carlo stayed in the village for several weeks (not that days were counted in 7 anymore) and after he left if the villagers had thought of this time, would have given it the measure of one moon. He spent this time learning about their lives. He would reflect that it was mostly the mundane requirements of sustaining life as foragers and diligent farmers. Tasks attended to collectively, without hierarchy and a generosity that Carlo had never known. There was also much time for play and socialising. Buildings could quickly overflow with villagers converged, chatting and joking.

There was a kind of ghost story among the adults of sightings of the last of the founders, who still roamed the nearby mountains. Several claimed to have seen the man on the trails. A wiry figure, tangled white hair and beard, jogging, topless through the trees. This seemed to be told in admiration for the founders and as a reminder to honour their intentions.

Carlo learned that it had been an issue of contention for many years, but recently was proven, by the use of knotted ropes, that where the houses were grouped more closely, those furthest from the centre were slightly larger. This was not an issue of differing layouts and spaces, for the houses were built simply and although most comprised of three rooms, the size of these rooms and their function followed no logic that had ever, in recent times, been noted. They scaled up, around each measurable axis. The walls were thicker.

Carlo was given use of a small house near the centre of 𝛕. He was told that this was the House of Translation. It was a space in which the occupant could move from one space to another. It was an old building and despite its centrality within the village only two of the dotted, stepping stone paths connected it. Therefore it remained more private and secluded than he had expected. Carlo was of particular interest to the children. Some of them could not remember the last time an outsider had entered their world. At first, they would run, scared, as a dare, through his rooms. But soon he was invited to go with them to the meadow.

It was here that Carlo had learned the game. The meadow was scored with the tracks of the learning games. The boards or gamespaces were large - one had to move physically between points. The game in some ways resembled baseball. Players occupying bases and tracing vectors - albeit the diamond shrunk somewhat, and no bats or balls - but often pebbles that were cast about - placed in piles. Closing in on 𝛕, Carlo had noticed a change in the stone cairns that resided along all trails that had been made by humans, seemingly irrespective of age or time since their last usage. The monuments had been rebuilt as diagrams, dismantled stacks that, at first, one might have attributed to wind or perhaps a curious boar. And Carlo saw again these diagrams, in the placement of pebbles around the meadow. Elsewhere in 𝛕 they would draw in the dirt or make marks on the stone, but it seemed the placement of pebbles was reserved as the domain of the young. These were the words in the songs they would sing. They had laughed at Carlo when he tried to place the stones himself. Instead they let him join the first drawing games. These were conducted on the bank of the stream gurgling from the cave next to the meadow.

Old Carlo, half-drunk paused for a moment, looking into the faces of those gathered around him. He tried to draw the things of which he was speaking. A napkin was slowly filling with marks. They were listening to him, for now.

This version of the game was the basis of much activity in 𝛕. Trade and bartering were conducted in this way, poems were written and names were given. The basic forms were known by all and made a performative space. The further reaches of their common understanding were explored communally at times of feast and celebration.

It seemed that the children were not formally taught the rules. They were inferred from the society around them. Still, they managed to teach them, in a fashion, to Carlo. It was easier to correct a misplaced mark than explain beforehand the possible permutations of placement. First, the vertices are placed. The children begin with ‘settlements’ of 3,4 or 5 vertices. Games played with 1 or 2 vertices are often used in specific ritualised parts of 𝛕 life, and the adults mostly play within realms of 9 or more vertices.

Next, the counting begins as numbers are assigned in turn to the vertices. The outcome of the game is based around these values but they need not be numeral. Indeed this is often where the first levels of metaphor or substitution are involved. All that is required is that the inherent value of the vertices is agreed by the players, or known to them - even if it is a point of disagreement.

After this, the players take turns to place lines between the vertices. It is often in this placing, if the game is being played to prove a winner, that the loser is known.

When no more lines may be placed, the shape is drawn again. To Carlo this closely resembled the Auslander-Reiten Quiver, but to the people of 𝛕 it was simply known as The Knitting.

A thought that had always puzzled him, was the proximity of the mathematic and the poetic. The quivers he could understand. The logic he followed. But they used it with a lightness. For example, their clothes were mostly like his own, woven closely. However, they also wore interchangeably, loose tunics that were woven as intricate quivers with beads at each vertex.

There was a spirituality present in 𝛕 that Carlo wished to pour himself into. As if he could be absolved by it. Walking one afternoon up along the north ridge from 𝛕, Carlo resting on the limestone, twisted and bent the silver crucifix he was wearing into the shape of 𝛕. Once, many years later, his mother would notice this and scold him.

One morning a village elder had entered Carlo’s shelter. He carried a flask of hot tea and they shared a cup before the elder imparted his message, “these are the rules for us. It has worked for some time and we are happy, but we know, and the drawings show, that simply in our being, we are connected to the wider world. The vectors are there, but their direction and values are unknown. It is true that many people have joined us, seeking refuge, just as our founders asked refuge of this valley. But they are accepted, as we all were, under a condition of ephemerality. Don’t mistake our community for utopia.” After this they had drank again, and then he left.

Next to the meadow there was a spring. Here the villagers gathered their water, sometimes caught fish, and swam and washed clothing. The spring was cold and clear bursting out from time-smoothed rock. Not far from the source of the spring the limestone climbed to a modest shelf. Within this wall of rock there was an entrance to a cave, large enough for an adult to pass through.

The mathematics Carlo knew - had demonstrated the alignment of different representations. The Auslander-Reiten quiver overlaid perfectly with the ordinary quiver which together provided a third reading, or like a seeing pool- the ability to move between different spaces. He had sometimes noted that he imagined these techniques all amounted to that of an increasing number of Gauchos arriving to lasso a mythical steer. Each rope slowing the beast a little, but always the gaucho tiring before the steer. In 𝛕, Carlo could see the quiver laid over everything. He believed, then at least, in the university, academia - now he wasn’t so sure - but for 𝛕, life experience could be understood through the principles of drawing quivers. In this act, meaning was made. Without it, there was no truth.

The cave near the meadow was a sacred space. The children never played close to it’s entrance. They called it The Place of All Trails. They were clearly afraid of it. A deeper fear than the kind Carlo could remember from his own childhood, of abandoned buildings and bank-owned factories, all places that they would inevitably break into and explore, to conquer that fear and flirt with the invincibility of youth. It seemed that these children had other caves in which they could do such things. The karst was always hiding more portals to its interior. The forests too were dangerous. Wolves could often be heard, and on his journey to 𝛕, Carlo had spent several nights unable to sleep for fear of letting some beast approach unseen from the darkness. One child had said that within The Place of All Trails there was a terrible darkness, but on the stone, vectors and vertices that could be felt. But they were dangerous. Too complex to follow.

One afternoon, whilst helping to collect water from the stream a woman had appeared on the meadow. She was wearing only a few of large knitted shawls, that Carlo had seen elsewhere in the village, beneath which she looked emaciated, grey skin, smeared and dark with mud. She walked toward the stream. The people there moved away, quietly. Someone whispered to Carlo, “she has come from The Place of All Trails.”

She stepped into the icy stream, letting the mud and silt wash off. After a while she began to smile, and her body loosened a tension it had been holding. She looked around at the people standing nearby, smiling and nodded to them. Her gaze pausing a moment on Carlo, before she left the stream and walked back toward the village.

A few days later there was a celebration for the woman’s return. A feast, dancing and music.

By this time Carlo had pieced together from the fears of the children and overheard fragments of gossip between adults, the ritual of passage into the cave was undertaken periodically by adults. It appeared to be a personal choice, not a requirement of life in 𝛕 but respected and celebrated when it was accomplished. After the scene at the stream he had asked some question, to be met only with, “The Place of All Trails = 0 place of Carlo.” It was spoken quietly and with the musical lilt and smile that seemed to accompany all 𝛕-speech, and yet it was clearly a warning.

A great fire was built, a pyre burning atop a dais of broken ceramics. It was a time of cleansing and the broken pieces that had been collected were added to by the breaking of perfectly good tools and pots.

Large fires were surely rare, as they would sooner or later attract attention from the outside world, and yet if this was a fear for the villagers, they didn’t show it. When the fire had started there was a large feast. Food, which was usually cooked and consumed by the families individually within their homes was brought out and shared around. Several people took charge in distributing the food across a quiver, that the children had been tasked with drawing earlier in the day. It was a form of performance in which the food was shared out seemingly according to the random balance of the quiver and yet it was manipulated such that each person was fully satisfied, irrespective of their portion size. Most of the villagers were wearing specially woven shawls. One had been given to Carlo to wear for the day. It was heavy and beautifully complicated. The festivities had been accompanied all day by the grappa that was distilled there in the village, and by the time evening had fallen, Carlo was absorbed in the play of the firelight on the various beads and markers that adorned each knot of his shawl, and following the various paths created by ribbons of material.

There was, of course, music and dancing also. The songs he had started to learn from working alongside the farmers or foragers. They worked like much of the speech as a movement around a quiver of words. They were clearly practiced often, and everybody knew their part and place. They seemed to be using a form based around perhaps, four vertices, the vectors between them changing and shifting, constructing the next verse.

The woman who had been in the cave, whose name was Emmy, had been absent until the sun had long vanished from the sky and it was only stars and indigo, except for the black smudge of smoke that danced against it. A space was made for Emmy within the singers. There, she began a dance. With this she was drawing a map, of a new quiver. The singers were singing the movement, the musicians playing their instruments faster, faster. It was a new thought - a gift from the cave.

In a sense the traditional thought of an expanding attention, a focus always toward the infinite, macro or micro, which struck Carlo now as something horizontal, was, in 𝛕, ignored in favour of the vertical. The stacking or mining for correlating diagrams, the relationships between these tracings and the palimpsest reading of everything together at once.

Of course no one had known that 𝛕 would be as it was. Carlo had expected to find mathematicians. They had all died, but the maths survived. He hadn’t foreseen that their break from society would even allow such a radical rebuilding of community and shared belief. Perhaps they were fortunate to share a belief in algebra - something more concrete than love, freedom, happiness, which it seems they mostly found as well. Perhaps it had not happened overnight. Perhaps the 𝛕 cross really was a crucifix - the forgotten grave of a shepherd, or something forged by someone who held both maths and Jesus in their heart. The buildings and large scale quivers were always aligned, projectives eastward, saluting the rising sun or in reverence to Mecca. Maybe they had abandoned Gods - recognising them as they overseers of the endless wars and weaponisation of their research. Or perhaps it was whilst fishing in the stream, making nets for their first time, and finding comfort and joy in their proximity to a language they could understand.

Carlo had started to wonder about the implications of him staying in the village. In the life he had left, he would have been missed, presumably, for a while, but no search parties would be sent out. At the most he would just become another line in the fable of 𝛕. In 𝛕 itself, he was beginning to sense that things would be more changed. The process of passing through the House of Translation, was a buffer, as the community conferred carefully on Carlo’s placement within their careful fabric of quivers. He could be placed, like a counter in the meadow games. He was undefined still, his weight on the vertix unknown - they would have to estimate this, or force it. He was concerned that his acceptance within the village could force the dismissal of someone else. He had no evidence of this, and it was only in certain glances, or empty midnight moments, that his mind would take him there. Perhaps this was his burden to understand and solve to make sense of the translation and leave that small house.

Carlo’s curiosity of the zero-place as he’d started to think of it, only grew over the following weeks. This wasn’t helped by the silence around it that permeated village life. After the celebration the most he could elicit from folks around the village was that it had been “une bonne fête”. The woman who had been in the cave had returned to regular life alongside the other villagers.

Increasingly it seemed that the villagers were expecting something of Carlo. His time at the house of translation must come to an end. Often there would be storms that broke the pressure trapped in the valley. For several days the storm had failed. Clouds had gathered, darker and darker each day, yet nothing sparked, no rain fell and thunder did not roll around the ridges or tremble through the potholed karst. The villagers accepted this and knew that it would pass. Carlo however felt it was 𝛕 itself, that valley, speaking to him- conveying the meaning hidden behind glances and smiles from the villagers. On the third day of this, exhausted and confused, Carlo decided that the next day he would go to the zero-place. The storm broke before dawn, waking Carlo and likely the whole village. The rain was falling in heavy sheets. He dressed quickly and gathered a few things together in a small pack. It was still dark when he left, but lightening would burst in fits that filled the village with the brightness of day, but flattened - like photographic exposure the graph being burned into image. Vectors and Vertices. Values clear from this heavenly perspective. Carlo fled the village quickly by ignoring the vectors and moving instead to the closest edge. Then, giving wide berth to the outlying houses he moved around to the path that led to the meadow. The sky was beginning to lighten now and he moved quickly along the trail.

Soon he reached the meadow and paused by the stream to fill his canteen. The cave eyed him darkly. A void within the dim rock face. He checked again behind him. Nobody there. The sun would rise soon, the storm was almost passed, and the village would stir. His absence could go unremarked until evening perhaps. With that, he climbed across the scree to the cave and within a moment was inside of it. He felt his way forwards in the darkness as far as he dared, before switching on a small flashlight. Fearing the light might escape, he kept it close to the ground and illuminating only what he needed to move forward. The cave was sloping gently down, and seemed to twist and turn at every moment. Twice he heard the thunder that snaked through the system like a locomotive in a tunnel- approaching, faster and louder, overtaking itself somehow- but like a ghost passing unseen. After a time, (Carlo could not say how long) he reached a point where the tunnel split into two. Each pass as inviting as the other. They were near perfect reflections of each other. He sat against a damp wall of rock. The path would be directed, it had to be. He realised he had not yet eaten anything and foolishly devoured the supplies he had brought. His stomach satisfied, and nerves calmed a little from the descent thus far Carlo considered the paths. Soon he noticed a small mark on the lip of the right-hand path. It was only slight, but similar to markings he had seen used in writing around the village. He switched off the light. His eyes not used to the darkness were completely blind. He felt for the mark, and by touch it was more noticeable than it had been in light. It felt made and unnatural against the geological time of the stone wall. He groped around to find the other tunnel, but felt no marks. He continued along the marked path. He decided to move in darkness now, one hand skimming the wall for markings, as his eyes relaxed to the darkness and began to reveal the occasional suggestions of form. Some loose excitement of light.

There were more markings and guiding points. The tunnel split several times and forced him through crawlspaces and over narrow shelves before it began to change. The space had become a series of halls and chambers. He trampled the remains of a fire in one and decided to switch on the flashlight. Revealed now were seemingly endless scratches and marks. It was the professor’s chalkboard, a thought captured in script, and overwritten repeatedly. A convoluted skein of meaning. He passed to the next chamber, and the next, finding the same each time. He turned off the light once more and moved slowly between the chambers. He continued on for a long time.

Finally, from the darkness, something yields. Suddenly he can see it, that somehow each chamber is a description of the next. The wall, indeed also the floors and ceilings describe the space of the next chamber. The light falls on moments of rock which together render primitive polyhedra. He thought back to Emmy’s dance - could it have been a map to a new chamber. How far must she have travelled to do so? Perhaps it was a more accurate description of some earlier chamber. He felt weakened and giddy at the scale of this endeavour. He laid down in the chamber he had reached and studied the light. He wondered if the people of 𝛕 could have made such cathedrals wherever they settled. He must have fallen asleep for a while, as he awoke, suddenly, cold, and already hungry once more. He moved back to the previous chamber and attempted to follow the markings. He could see the shape building slowly, faces emerging but always it would reach a complexity that he could no longer hold in his mind. He laughed and the chambers echoed, laughing with him. He began to return to the surface now. They had been right he supposed, this was a zero place. Beyond him. He would never realise his time in the House of Translation. He knew also, that he would not tell folks of this place when he returned home. He would abandon his life at the university and the future it held, tell people he was wrong, it was just a myth.

Eventually he reached the surface. The sky was bright but without sun, the ground and those close to it, lost in hazy darkness. He could hear voices though. Near the stream and back towards the village, shouting. His eyes attuned to the darkness of the cave could see a few figures now, moving silently over the meadow, crouched, approaching the cave. He was still hidden and unseen in the dark mouth of the cave. Had he really spent the whole day there? He crouched down and crawled out around the rocks, he moved quickly, toward the tree line. Once there, he turned and could see the figures still approaching the cave as before.

Carlo climbed quickly through the forest towards the ridge. He was tired and his body ached all over. The darkness had almost overtaken him, by the time he reached the ridge. From there the sun was visible once more, sinking quickly over the horizon. He was able to move further now along the ridge until eventually he picked up the trail he’d used to approach the village. A quiver-cairn greeted him. He followed this a way, into the darkness, trees again, moving slowly now, carefully, until he reached the place he’d camped the night before entering 𝛕. There were a few leafy branches piled against a rock and he pulled these aside to reveal a pack he’d left there. It contained some small rations of food, a blanket and tarp. He ate and drank. He wrapped up in the blanket, against the cold of the night, and to ease the chill of the cave that seemed to have settled in his bones. He watched the distant lights of 𝛕, flickering. After a while some more returned from the meadow side of the village. A central fire was lit. Carlo fell asleep watching the torch-bearers moving in their paths, vertex to vertex.

He awoke before sunrise and packed his things blindly. He gazed across the valley a final time, but a mist had settled and he could see only the pale ghosts of a quiver.

The bar had emptied out by now. Old Carlo happily drunk, believing he had finally told some young students about what he’d seen all those years ago. Of course they’d not recall, in the morning the sad old man, who’d stood them several rounds before they’d left in search of dope and pizza, a notion stirred by vague thoughts of the Auslander-Reiten quiver of a type-A, Dynkin diagram.

⬑ Aiming for meaning: a brief account of quivers in representation theory

Oftentimes, our first encounter with this subject happens in school, when we are presented with equations that we try to solve (not very successfully at beginning) by finding 𝒙. If we are lucky[1], we have a nice teacher that explains us the underlying mechanics of the subject and guides us to construct an algorithm which will allow us to solve every equation of a given type. Once we have mastered the rudiments of this subject, we become more ambitious and we try to solve several equations at once. By doing so, we realise quite quickly that this new task is much more difficult, sometimes simply out of our reach.

But, as we are resilient learners, we keep going and after working for a while with multiple equations at once, we start to see that some equations are superfluous. For instance, if we have three equations, call them e1, e2 and e3, and we see that e3 is equal to the sum of e1+e2, then any solution of e1 and e2 is a solution of e3. So we can forget about e3 and work only with the other two. The same happens if e3 is the multiplication of e1xe2 or if it is the multiplication of r*e1, where r is any given number. Basically, we realise that are different ways to mix different equations to find new ones without losing or adding any solution:

(+)  We can add two equations to get a third equation.

(x) We can multiply two equations to get a third equation.

(*) We can multiply any equation by any number to get another equation.

It turns out that there are many other things that can be added and multiplied in their own special way. Since mathematicians are creatures that like to work in a framework as general as possible, they decided that it was a good idea to put a common name to all these things and try to study them, if possible, all at once. After some consideration and to make things a bit more confusing for anyone who is not an expert, they decide to call any set that has a notion of addition and two multiplications and that respect some basic properties an algebra (with a lowercase “a”).

Note that I just said before that an algebra is a set that has some notion of sum and multiplication. Believe me that these notions can be really complicated, to the point that our intuition is completely lost and we have no grasp of what is going on (Later on I will give an example of this). One way to think about this is that any new algebra you want to understand is a foreign language that you want to learn. For certain languages you might pick up some words in a conversation while for others you will be completely lost. When you work with algebras, something similar happens. There are algebras that are nice and simple, while others are extremely complex.

Pushing this analogy further, a good way to learn a new language is by comparing a text in the language we want to understand with one or several translations of this text to our mother tongue. After some work, we first start to understand the meaning of some words. Then we are able to see how the grammar works. And finally, after a deep study, we start to understand the true meaning of the phrases in the text, recovering the information that was lost in translation.

In Mathematics, there is one language that we understand pretty well, and that is the language of linear algebra. In some sense, linear algebra studies how different points are distributed in a given space, regardless of your perspective. Let me explain this with a small thought experiment.

We are in a dystopian future where you and I are the prisoners of a malefic dictatorship, waiting in a cell for our executioner to come get his job done. However, at the last moment, the Dictator decides to spare our life if we are able to solve the following puzzle. One at a time, we will be guided into a dark room and after we are left alone, four small LED-like lights are lit up: one white, one blue, one red and one green. Moreover, next to each light there is a vector with the coordinates of each of them. After both of us went into the room and came out with this information, we are told that none of the colour lights has moved between the times when each of us was inside the room and we need to decide if the white light has moved or not. It seems like a straightforward problem, until we realise that none of the coordinates you have coincides with the coordinates that I have. However, not everything is lost. We know that the green, blue and red lights did not move. So, even if these vectors are different, they represent the same point in space. Then, if we work out how to translate one set of vectors to the other, we will be able to translate the coordinates of the white light as well, which will allow us to decide if it moved or not.

The theory of linear algebra revolves around these kinds of problems, studying how we can transform one set of coordinates into another and even allowing us to decide if the room we were in was the same or not.

As I said before, for mathematicians linear algebra is fairly well understood, in the sense that virtually every question we can ask about vector spaces can be answered. Then, we can use linear algebra to answer questions in Algebra. Because, if we are able to translate an algebraic question into the language of linear algebra, then we will get an answer for the original one. This is the basic idea behind one important branch of mathematics known under the name of Representation Theory.

You may say: “OK, why you say that Algebra is so complicated if you can answer any algebraic question using linear algebra?” and you will be making a very good point. The main issue with this approach is that there is a lot of information that gets lost in translation, because linear algebra can only express only one aspect of the question at a time. This means that the answer we get is partial, at best. But this is not something new for any of us.

In some sense, the approach of Representation Theory to the study of algebras is like trying to explain the concept of love to a robot which is incapable to feel. In order to do so, we would need to give to the robot a definition of the concept of love. Now, it is certain that any given definition of love is imperfect and will never be able to describe every aspect of the concept of love. However, if we give many different definitions of this concept, maybe the robot would be able to reconstruct the concept by fitting together all the definitions we fed it. Using this analogy, Representation Theory tries to understand a given algebra (the concept) by means of its representations (the definitions of the concept).

If you want to study an algebra via Representation Theory, it would be helpful to know how to construct all its representations. In general this is quite difficult to do, even considered to be an impossible task. However, for algebras satisfying a certain technical condition^[2], a mathematician, Gabriel, proved that one can construct a quiver to which we impose some relations such that every single representation of the original algebra can be realised as a representation of the quiver satisfying these relations.^[3]

What Gabriel has noted back in the 1970’s is that each algebra can be thought of as a bunch of blocks influencing each other in a coherent way. Then we can take a piece of paper and draw this phenomenon by putting a point for each block of the algebra and an arrow going from the point 1 to the point 2 if the block 1 has an influence on the block 2. Now, algebras can be really complicated. A given block might influence another in multiple ways, so we need to put many arrows going from the first vertex to the second. Or a block can influence itself, so we might even need one or many arrows starting and ending in the same vertex. This schematic model of the algebra is called the quiver ^[4]of the algebra.

However, the quiver is not a faithful picture of algebra just yet, because it might contain too much information. For instance, suppose that in our algebra we have three blocks, call them 1, 2 and 3, and suppose that 1 has an influence on 2, while 2 has an influence on 3. Then, in the quiver of this algebra we will have an arrow going from 1 to 2 and another going from 2 to 3. But then the quiver is telling us that the block 1 actually has an influence over the block 3 via the influence it has over the block 2. This is perfectly fine in some cases, but if in our algebra the block 1 had no influence whatsoever over 3, then we need to say it. So, we add to the quiver a set of relations, encoding all the information that we should ignore to recover the behavior of the original algebra.

Moreover, as I mentioned before, the quiver also give us a way to see all the representations of our algebra. This is because we can separate in blocks any given representation, as we did for the algebra when we were constructing our quiver. And the relation there is between the different blocks of the representation is determined by the relations between the blocks of the algebra. So, every representation of the algebra can be taught as a set of blocks, one for each vertex of the algebra, that are related to each other in any way we want which is allowed by the relations of our quiver.

One important way in which representation theory has benefited from the introduction of quivers is the fact that they allow to build many examples easily, by simply drawing dots and arrows in a piece of paper. Also, the use of quivers has helped to build bridges between Representation Theory and other branches of Mathematics and Physics, because as soon as you have some phenomenon that can be represented with a quiver, you have an underlying algebra and its representations can tell you a lot about the phenomenon you want to study.

As every chemical compound in this world is made out of a set of chemical elements, every representation of an algebra can be seen as a combination of a set of fundamental representations, the so-called indecomposable representations of the algebra, whose name indicates clearly that they can not be decomposed into smaller representations. An incredible result proved by Drozd told us in the mid 1970’s is that all algebras can be separated in two quite different families: tame and wild. As is suggested by their name we can only hope to have some control over tame algebras. This is because if an algebra is wild, then every  representation of any algebra can be seen as a representation of our wild algebra. In other words,  if we completely understand the representation theory of only one wild algebra, we understand the representation theory of every single algebra there is.

Thanks to the work of Mendeleev, we know that  the chemical elements can be nicely arranged into a pristine table. In Representation Theory some similar happens, since all indecomposable representations of an algebra can be arranged diagrammatically in what is called the Auslander-Reiten quiver of the algebra.

This object originated with the work of Auslander and Reiten on the so-called almost split sequences back in the mid 1970’s. They proved that for almost every^[5] indecomposable module 𝑴 we can associate a sequence of three objects with very particular characteristics finishing in 𝑴. Moreover they showed that this sequence is unique and always start with another indecomposable module, which is known as the Auslander-Reiten translation of 𝑴 and is denoted by 𝛕𝑴.

In general, the second element of an almost split sequence finishing is not indecomposable. However, as we briefly mentioned before, it can be built out of indecomposable modules. These indecomposable modules have almost split sequences of their own. It turns out that all the almost split sequences can be arranged coherently, in a big picture that we call the Auslander-Reiten quiver of the algebra.

A priori, if you want to build the Auslander-Reiten quiver of your algebra, you need to know all the indecomposable representations of the algebra, calculate all their almost split sequences and then fit them together like a giant jigsaw. This can be very challenging, especially because we need to have a lot of information beforehand. However, the almost split sequences are nicely balanced, in the sense that we can attach some numbers to each of the three representations in the sequence in such a way that the sum of the numbers in the extremes is equal to the number of the representation which is in the middle of the sequence. Then if we know how some sequences fit together, then we can calculate neighbouring sequences via a process known as knitting and, if the algebra is nice enough, find every single indecomposable representation of the algebra in the process.

An amazing fact about the representations of quivers without relations is that the number of indecomposable representations of each quiver depends only on the shape of the quiver and not on the orientations of its arrows. In other words, if you and I construct two different quivers having the same shape, the total number of indecomposable representations that we will find is exactly the same, even if all of the arrows in your quiver point towards a given vertex while none of them point to that vertex in mine.

This phenomenon has been (and still is) greatly studied and has motivated the introduction of many important tools in Representation Theory. Arguably the most important consequence of this phenomenon is the introduction of the so-called tilting theory by Brenner and Butler in the beginning of the 1980’s. Their explanation they found can be described as follows.

Suppose that you have two quivers without relations which have the same shape. Take the first of these quivers and construct its Auslander-Reiten quiver as we described before, which for the purpose of this explanation we can imagine as a piece of paper. Then they showed that one can draw a vertical line in the paper is such a way that, if we cut along this line and we glue the two resulting pieces of paper by the original borders we obtain the Auslander-Reiten quiver of the second quiver. As a consequence we have that both quivers have the same number of indecomposable representations.

As a matter of fact, it has been shown that tilting theory is a powerful tool that allows mathematicians if working in different areas of this science to compare the representations of a multitude of different mathematical objects. Allowing tilting theory to join the notion of quivers as another great export of Representation Theory to other branches of mathematics.

To understand an algebra from its representations we need to know the representations of the algebra, of course, but that is not enough. We also need to understand how the different representations interact with each other. In Representation Theory, these interactions are known under the name of morphisms. It turns out that also the morphisms between representations are encoded in the Auslander-Reiten quiver of the algebra. For the algebras in our previous example, when we were considering algebras whose quivers don’t have relations^[6], it is known that all morphisms are going from left to right in the Auslander-Reiten quiver of the algebra.

Then, in the same example, we cut the Auslander-Reiten quiver following a vertical line. This implies in particular that we are separating the indecomposable representations in two different categories, those to the left and those to the right of the line. By  what we said in the last paragraph there are no morphisms from category on the left to the category on the right. When two categories have this property, we say that they form a torsion pair.

The notion of torsion pair has been central in Representation Theory and that is why different mathematicians have been trying to find ways to construct them. We have seen before that tilting theory generate some torsion pairs, but not all torsion pairs are generated using tilting theory. The breakthrough for this problem was made by Adachi, Iyama and Reiten (yes, the same Reiten as before) at the beginning of the 2010’s, when they introduced& 𝛕-tilting theory, a powerful combination of tilting theory and Auslander-Reiten theory. Note that Auslander-Reiten theory is represented in the nam 𝛕-tilting theory by the greek letter 𝛕, which is usual notation for the Auslander-Reiten translation. With the introductio 𝛕-tilting theory, Adachi, Iyama and Reiten were able to show 𝛕-tilting theory generates every single torsion pair that can be generated^[7].

This relation between torsion pairs 𝛕-tilting theory has allowed to understand torsion classes much better. For instance, it has been shown very recently that 𝛕-tilting theory of  each algebra creates walls in a n-dimensional space, creating chambers as many chambers as torsion pairs has the algebra. This structure is known as the wall and chamber structure of the algebra.

This space happens to appear in different forms in many other areas of Mathematics and Physics and many scientists are working hard to unveil some of its mysteries. For instance, suppose that you are inside one of the chambers of this structure. Since there are no doors here (the architect in charge wasn’t really good it seems), if you want to go to a neighbouring chamber you need to cross a wall.

As I am writing these lines, little we know on what would happen to us when we cross through this Looking-Glass. But whatever happens, I am sure that an adventure as incredible as Alice’s is waiting for us on the other side.

~gg 02/23

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