The Italian School

This story is part of a series entitled Leaving Mathematics. See also:
1. The Baltimore Snowstorm; 2. The Italian School; 3. Characteristic Classes

“Of course you should go!” Professor Torino nodded honestly, smiling inexplicably. Torino hung like that for an instant, balanced in his chair. He relaxed suddenly, his smile vanishing. “Pieri is a good mathematician,” he continued. “We spent a summer together at the Institute in 1991.”

Professor Torino always seemed to like Josif, though Josif didn’t fully understand why. The professor used to invite his first-year algebra students to the board near the end of his lectures. “Do you have anything you’d like to share?” he’d ask the class in accented English, generously turning out his palms.

Josif once demonstrated to a few shifting classmates that every group of order 4n + 2 admits a subgroup of index 2. “Good!” Torino stated heartily upon the conclusion of the proof, standing near the side of the chalkboard. “Many times I’ve seen a fantastic, just a fantastic result…” Torino began, now glancing vaguely toward the back wall of the room. “When you finally see its proof, it’s just many small steps… Many small steps!” What struck Josif was not the content of the statement but Torino’s apparent conviction that in it resided the secret of all of mathematics. Torino had a way of nodding to Josif and beaming at the end of class, speechless, apparently indicating to Josif his conviction that Josif was bright. Josif would grin back and wave. He liked Torino. On the other hand, he felt that Torino’s knowledge of his mathematical mind was shallow.

It took a transatlantic flight and a bus ride to reach the small university town in Italy’s Marche province. Josif visited Pieri’s office on the afternoon he arrived. He passed through the city’s outer walls, and climbed a steep cobbled stone street toward the town’s center. On each side of the street, narrow alleyways wound yet higher, themselves lined with stucco buildings with windows decorated with shutters and flowering plants. In the empty piazza, a fountain rested, filled with water and yet not running.

The small university complex surrounded a dusty parking lot in a dilapidated development past the city’s opposite extreme. These buildings’ confused 1960s architecture, taken in alongside that of the Renaissance-era city nearby, engendered a sort of sadness. Josif entered Pieri’s office.

Pieri was old, and his smiling, squinting eyes were hardly visible. “I’m interested in algebraic geometry,” Josif began, sitting upright. “Torino’s research… in Hilbert schemes…” Pieri responded, happily. Pieri’s white hair was disarrayed, and his hands gestured vaguely. “Local rings of super-varieties, local jet-structure, local formal schemes,” he suggested. “Torino studied commutants of modular groups, and Picard groups of modular stacks.” Pieri paused, leaning forward and looking at Josif. One of his eyelids seemed to twitch. Josif hesitated. He almost spoke. “Great result!” Pieri added suddenly, then laughed and looked up toward the ceiling.

Josif left Pieri’s office after an hour. The math department’s empty halls were lined with cork bulletin boards featuring old advertisements for summer schools and post-doctoral positions. He stepped into the office designated for visiting students. A handful of large desks were arrayed in rows. At one desk sat a red-haired young woman of about Josif’s age. The name tag on her desk said “Anne Lavoisier”. She was beautiful. Josif placed his backpack in the chair in front of his desk.

Anne looked up from the notebook in which she was writing. “Hi,” Josif said. “What do you study?”

“I study number theory,” Anne replied. By her speech Josif could tell she was French. Her English was poor. “Varieties over finite fields,” she offered. She glanced back up at Josif. “I’m a masters student,” she concluded. Sun poured into the office through the broken, slanted, half-raised blinds descending over a window at one end. The office’s fluorescent lights were off. “Nice to meet you,” Josif said.

Back in the old picturesque town, the sun was declining. Josif stepped into one of the city’s twisting alleyways. He walked down the curving path, through a short covered section and past tan shuttered buildings with leafy ivy-covered walls looming above him in jagged right angles. An old man, leaning with one arm against a wall, walked slowly ahead of Josif, until Josif turned into another side street, and came upon a small garden with low stone walls. Green and yellow patched hills opened up before him as the red terra-cotta roofs of the town gave way, at the city wall, to the countryside below. Josif’s damp shirt stuck to his back, despite the breeze. The cross atop the city’s domed church was visible in the distance.

The class on algebraic geometry over arithmetic fields was taught by an Italian named Gornello. A large beard with hints of grey at the edges covered most of the young man’s face. Gornello spoke calmly, and seemed never to show emotion. Josif and Anne were among the class’s students, along with two or three slouching Italians, with dark hair and stubble, who seemed to cycle in and out of attendance and who never spoke. The class was difficult.

“Thus though a diophantine equation over the integers is in general algorithmically undecidable, the corresponding scheme has geometrical properties which can be studied,” Gornello concluded one day, at the end of class. “Have a good week.”

As Josif exited through the classroom’s doorway, he turned and saw that Gornello had sat down near Anne, and was speaking to her. Josif gathered that Gornello had perceived that Anne was struggling in his course, and had suspected that language was a factor. He was explaining some mathematical concepts to her, quietly, in near-fluent French. Anne, though flushed slightly, nodded along. Gornello had wished not to embarrass her.

Josif studied with Anne later that day. Anne had a peculiar way of saying the word “union” which made it sound like “onion”. Josif sympathized with this innocent consequence of English’s vowel sounds. Anne seemed distant to him, nervous, as if a chasm between them, generated by language and shyness, made understanding between them impossible. “We have to prove that this variety is proper over the integers,” Josif began.

“I did my PhD in France,” Gornello later told Josif, when Josif visited his office one day. Gornello was holding a small espresso cup in a dish and smoking a cigarette outside the building as Josif approached.  “I’ll see you inside,” he said, nodding meekly. “When I first got to France, I understood nothing,” Gornello explained. “’D’accord, d’accord,’ I would say, until the speaker paused expectantly. Then I would say ‘Je ne comprends pas.’ When they resumed again, I would continue saying ‘D’accord.’” Gornello paused. “This could go on for quite a while,” he concluded. As Josif laughed, a hint of joy appeared on Gornello’s still face.

Josif visited Pieri occasionally throughout his stay. “The Hodge theorem…” Pieri mumbled once, as Josif entered his office and sat down. “The quotient of the closure of the image by the image is infinite-dimensional! How do you prove this…” Josif recognized this phenomenon as an elementary detail in one of Pieri’s old papers from decades ago. “You mean the L2 case? I think this happens only in the L2 case,” Josif answered. He was surprised that Pieri had failed to recall the theorem’s proof.

Josif spent his afternoons walking alone through the small walled city. Mathematics had gradually, irreparably enervated him; it had emptied him of interests and desires, of humanity, of everything except an ability to drive deeper into a world composed purely of abstract structures. It had left him with nothing to offer to himself, and nothing to offer to someone like Anne. Josif was almost transparent, his vitality, his existence, fading.

Josif encountered Anne on one of his walks. She was sitting on a ledge smoking a hand-rolled cigarette. “Why did you choose to study mathematics?” Anne asked him.

“I just liked it a lot,” Josif stuttered. He was ashamed at the poverty of his answer. He knew, in his mind, that the reasons were rich; that mathematical objects were assembled in patterns and flavors whose complexity vastly exceeded that available in all other disciplines, and perhaps in all human experiences.

“Yet there’s more than math,” Josif suddenly declared, aware now of the bursting of a dam, of a flooding revelation that had been a long time in the making. He felt the spirit of human, visceral energy fill him, the spirit of expectation and promise.

“What do you mean?” Anne asked.

Josif paused. “I have a wire transfer… Western Union… In the nearby city,” Josif said. He spoke carefully, so as to accommodate her English. “Do you want to come?” The city was miles away, and the sun was descending.

Anne and Josif walked to the city walls, and then past them, down a road lined with trees, whose leaves glinted in the afternoon sun, which led away, to a place neither of them had been before. They had a class that evening with Gornello, and another in the morning, and homework due. They knew it, and yet they kept walking.

Josif had forgotten what life was for, but he gradually began to feel that he was going to know once again.