Sandra Burton, director of the dance program at Williams College in Massachusetts, was struck immediately by Nakesha’s vibrancy and talent as a dancer. She became Nakesha’s teacher and mentor, and she began to closely track her development. Nakesha, she recalled, stood out no matter the setting: the stage, the classroom, even across a kitchen table.

But three years after graduation, Ms. Burton and other friends started getting strange phone calls and emails from Nakesha, with bizarre claims she was being followed by strangers. She abandoned people who were close to her and spurned their offers of help. In 2010, more than two decades after they met, Ms. Burton received what would be her last communication from Nakesha, an email in which her former student angrily lashed out at her.

Sandra Burton and P.J. Dearden were bookends to a promising life derailed by mental illness and homelessness. But their paths never crossed. One woman held answers to Nakesha’s past, the other to her present.

What had happened in between?

There are roughly 3,900 unsheltered homeless people in New York City, ever visible but also largely anonymous. They lie in dingy sleeping bags near buildings or construction sites, bury themselves under blankets, ponchos and cardboard boxes, or sit with propped-up signs asking for money. Their presence remains one of the most intractable challenges facing the city: Solutions are complex at best, and finding the proper balance of medical care, casework and law enforcement to ease them off the streets has proved daunting to the de Blasio administration, as it has been to many that came before.

Over the course of more than a year, I delved into Nakesha’s life, trying to understand the events and forces that put her and so many of the city’s homeless on the street. I spoke with dozens of people and benefited from having access to hundreds of emails that Nakesha sent over the years to friends and others, and scores of meticulously typed letters that she copied at libraries and handed to passers-by. In these, she offered a winding, rudimentary diary of her existence on the street.

My effort revealed a deeply complicated, at times contradictory, journey — a life of spectacular promise undone by demons. No simple answer to the puzzle of Nakesha emerged. But at the same time, another narrative revealed itself: a story of New Yorkers and others who went to extraordinary lengths to try to help her, only to be left frustrated.

On her grate at East 46th Street and Park Avenue, Nakesha was surrounded by a collection of strangers who offered their company, care and generosity. Sidewalk vendors watched over her and her belongings. Restaurant owners let her use their restrooms. People like P.J. brought her food and warm clothing.

And yet Nakesha continued to spiral downward. She rejected P.J.’s suggestion that she enter a shelter or seek help. “I was like, ‘I have a friend that you can talk to,’” P.J. recalled telling her, referring to a social worker she knew. “She’s like, ‘Oh, no, I’m fine.’” P.J. tried again. “‘Do you need any medication?’” The answer was no.

So P.J. tried to address her other needs. She brought her sandwiches and toiletries. She went online and ordered her a raincoat, a duffel bag, two pairs of black pants, shirts, socks and underwear. She bought a pair of black leather boots that she helped Nakesha pull over her swollen feet. She gave her presents on her birthday and at Christmas.

Then one day in August 2016, P.J. realized that she had not seen Nakesha for weeks. With a sense of foreboding, she began to search for her friend.

As a student at Woodrow Wilson High School in Camden, N.J., Nakesha had excelled. She was class president and a member of the National Honor Society; she sang in two choirs and worked on the school newspaper.

The yearbook described her as one of two students who “did the most” for their class and school. She graduated in 1988 third in her class of more than 200 seniors, and her college tuition was covered by scholarships, some from local churches.

To Nakesha, Williams College, a competitive private liberal arts school in pastoral Berkshire County, must have seemed a world away from home.

Yet she adjusted easily. A former classmate, Valda Clark Christian, now a health care executive in Maryland, recalled how Nakesha had warmly introduced her to the relatives who accompanied her to campus. “I can’t think of anybody who met her and didn’t fall right into some comfort and familiarity,” Ms. Clark Christian said.

Part of a close-knit group of students of color, Nakesha joined the college’s gospel choir and a traditional African dance troupe called Kusika, which Ms. Burton directed.

Nakesha, Ms. Clark Christian and two other students were seen together so often that they became known on campus as the Four Musketeers.

“They were like the women you looked up to, that you admired,” said Tanya Nicholson Miller, another former classmate and now an educational consultant in New Jersey. “It was just like the way they carried themselves on campus, their sense of self-esteem.”

Nakesha was also a source of support for other students — a “really bright light to all of the faculty and her peers,” Ms. Burton recalled.

Becky Dickinson, who had felt self-conscious about performing, said she joined Kusika only after Nakesha encouraged her.

“I do remember her saying like, ‘You can do this,’” Ms. Dickinson said. “I admired her. She was inspiring.”

In her dorm room, Nakesha kept a photograph of her mother, Geraldine Williams, and she had saved her mother’s cards and letters, which often bore an imprint of a lipstick kiss.

In the fall of 1990, several weeks into her junior year, Nakesha learned that her mother, who had breast cancer, had become gravely ill. Nakesha withdrew from school and returned to New Jersey, moving into the Willingboro home of her aunt, Jacqueline Fattore, who was caring for Geraldine.

Nakesha slept in a recliner at the foot of her mother’s bed in the den. “I spent every moment I could at her side,” Nakesha later wrote in an essay about her mother’s final days.

Geraldine was an unwed teenager when Nakesha was born in 1970. Although she soon married Nakesha’s father, Marvin Rothmiller, they divorced several years later. Geraldine became involved with an abusive man who repeatedly molested Nakesha when she was just a child, as Nakesha eventually confided to Ms. Burton and others. (The man was later convicted in a drug case and has since died.)

Nakesha was sent to live with her maternal grandmother, with whom she stayed until leaving for college. But she remained close to her mother, and was devastated by her death in February 1991, at the age of 37.

“I looked at her face,” Nakesha wrote. “For the first time in my life, I saw a calm that wasn’t clouded by bitterness, hurt or anger.”

Nakesha enrolled for the spring semester at Rutgers University’s Camden campus, and in the fall, returned to Williams College, with a renewed sense of dedication and seriousness, Ms. Burton recalled.

Over the next two years, Nakesha focused on researching, writing, choreographing and directing an ambitious senior project — a panoramic performance of black dance, from its origins in Africa through contemporary hip-hop.

Nakesha, lithe and petite, also performed in the two-hour show in March 1993 which involved some 20 dancers and musicians. It was called “If You Can Walk, You Can Dance, If You Can Talk, You Can Sing.”

Navin Girishankar, a drummer, remembered the hours they had spent rehearsing in a campus studio.

“Everything I would throw at her, with these little drumming moves, she’d come up with something even more brilliant,” he said.

The show was a sensation.

“The audience was immediately aware of the great care and love that went into the production,” the college newspaper, The Williams Record, said in a review.

Nakesha was awarded a prize established in memory of the college’s first black graduate for the “best scholarly work submitted by a Williams undergraduate in the field of Africana studies.”

Reginald Hildebrand, who taught Nakesha African-American history, said he had “expected her to be one of the leading individuals in whatever she was going to do in her generation.”

‘A Wandering, Unstable State’

After graduation, Nakesha went to work at Sewickley Academy, a private school near Pittsburgh that at the time enrolled about 700 students. One former student, Lisa Bevevino, to whom Nakesha taught sixth-grade history, remembered Nakesha as beloved by students, often sitting on the floor with them, talking through their problems and assigning projects that provoked creativity.

Once, she said, Nakesha had each student invent a holiday and write about how it would be celebrated, the values it promoted and what artifacts would be involved.

“She was one of the foundational people,” said Ms. Bevevino, now an assistant professor of French and Latin at the University of Minnesota, Morris, “that really cared about how we were doing and how we thought about the world.”

In 1996, after three years, Nakesha abruptly left Sewickley. Neither Ms. Bevevino nor anyone else could recall her giving a clear reason for her departure.

Nakesha drifted over the next decade, moving among family members and friends and holding odd jobs, working as a sales clerk at Victoria’s Secret and as a bridge toll collector. She took computer software courses, and a writing class at Temple University in Philadelphia, where she wrote the essay about her mother’s death. Hints of her mental illness began to surface.

In 1997, Ms. Burton and her husband invited Nakesha to live with them in Hancock, Mass., a short distance from Williams College, where Nakesha found administrative work in the dance department. The couple had been concerned that Nakesha had quit her teaching job, and they sensed that something was not right. One day, Ms. Burton said, she arrived home to find all the curtains drawn. Nakesha said she believed someone had followed her home.

After a few months, Nakesha decided to move to New York, against the advice of Ms. Burton and her husband. “What could we do?” Ms. Burton said. “She was an adult.”

By the summer of 1998, Nakesha was living in Fort Greene, Brooklyn, with Anim Steel, a former college classmate. Mr. Steel had been captivated by Nakesha when they first met at Williams, writing in his journal, “I’ve never met anyone like her.”

In Brooklyn, they shopped together, played kickball and visited the Statue of Liberty for Nakesha’s birthday. But Nakesha was having bouts of anxiety, Mr. Steel said, and he found a therapist who agreed to see her on a pro bono basis.

The therapy did not last; Mr. Steel moved to Boston at the end of the summer, and Nakesha stopped seeing the doctor. She moved in with another friend, Greg Holland, a paralegal and friend from high school, who she once described as being “like a brother.”

After about two years, Mr. Holland recalled, Nakesha began claiming that the C.I.A. was looking for her. She stopped working and paying her share of the rent. She moved back to New Jersey, first staying with her father and then with other relatives. Her erratic behavior continued, and she began to cut off family members.

In the summer of 2001, a former college classmate, Joan Rocklin, invited Nakesha to dinner in Manhattan. As first-year students at Williams, they lived across the hall from each other, and had remained friends. Ms. Rocklin was planning to move to teach at the University of Oregon School of Law and wanted to see Nakesha before leaving.

As they relaxed in Ms. Rocklin’s apartment, Nakesha mentioned that she had found messages in books she was reading. Ms. Rocklin assumed Nakesha meant certain books had seemed meaningful to her; she was disturbed to learn that Nakesha meant actual written messages.

Nakesha moved a few years later to Philadelphia, where she taught at an adult literacy center. Her supervisor recalled her as a talented teacher but said she began talking to herself and complaining that people were stealing from her, and she eventually left the job without collecting her last paycheck.

It was in November 2006 that Ms. Rocklin received the first in a stream of emails from Nakesha.

“Hey Joan, What’s going on?” Nakesha wrote. “I’m stuck in California right now.”

Nakesha said she was “stranded” with no address and could be reached through the Los Angeles library system.

Ms. Rocklin responded that she cared about Nakesha and was worried “whether you have shelter at night and whether you are getting enough food.”

Nakesha, then 36, replied that she was “not exactly homeless,” but was in “a wandering, unstable state that is sad at best.”

‘I’m Not Homeless’

Ms. Rocklin continued to receive Nakesha’s emails, which were sometimes copied to politicians and others. She complained that the Los Angeles police had been harassing her and that she needed clean clothes, and money for food. She described picking up pennies on the sidewalk. “I think I now have a whole 35 cents at my disposal,” she wrote.

She said that her eating habits depended on whether she could find free samples somewhere, or a church that served food after services. She said that her feet were blistered, sore and bleeding, and that walking was painful.

“The socks are disgusting, the tights have holes and runs, and I have no winter jacket and it’s cold here at night now,” she wrote.

Ms. Rocklin consulted with a psychiatrist in Los Angeles, collecting the names and numbers of shelters, clinics and a hospital, which she forwarded to Nakesha.

“When we were in college together,” Ms. Rocklin wrote, “I was always impressed by how bright, creative and resourceful you were. There are resources in West Hollywood and people who can help you. Please reach out to them.”

Nakesha thanked her, saying the information seemed “meant for the homeless.”

“I’m not homeless,” she said.

Her emails indicated otherwise. Nakesha wrote of sleeping in a Los Angeles motel lobby where the clerks knew her and sometimes got her a room, and of sitting in a doughnut shop in Hollywood until she fell asleep.

A week before Christmas, Nakesha wrote to say that she had collected enough money to buy a bus ticket to the East Coast. A week later, she emailed to say she had arrived in Asheville, N.C., where the police had taken her to a Salvation Army homeless shelter.

Three days later, Nakesha emailed again, this time from Washington, where, she later noted, she was writing from the Martin Luther King Jr. Memorial Library. In messages that followed, she described sleeping in the lobby of one apartment building and the laundry room of another, and outdoors on the National Mall.

“I still have no money to make sure I’m fed & barely eat, if I eat at all,” she wrote on Jan. 14, 2007. A week later, she claimed that she had been “rudely asked” to leave the Greyhound bus terminal and Union Station, and was “desperate as to what to do.”

Ms. Rocklin tried to arrange for a Washington-based ministry to have a photograph of Nakesha distributed to homeless outreach teams and the city’s mental health services department.

That July, Ms. Rocklin was traveling to Washington and emailed Nakesha, suggesting they have lunch together.

They met at the library, then dined at a nearby seafood restaurant, where Ms. Rocklin said the conversation was pleasant if awkward; she felt she could not raise Nakesha’s homelessness, since Nakesha had always denied it. She also took Nakesha shopping, buying her CVS gift cards and a fare card for the subway.

Nakesha’s emails eventually stopped, Ms. Rocklin said. Nearly a year later, in September 2008, Ms. Rocklin was in Washington again, for a wedding, and visited the library. She searched for Nakesha on each floor, without finding her.

Rejecting Offers of Help

By then, Nakesha had been in New York for nearly a year, arriving in December 2007, according to her letters. Around that time, Ms. Burton’s husband, Don Quinn Kelley, a professor at the New School, came upon Nakesha at Grand Central Terminal. He called his wife, saying he was shocked at Nakesha’s appearance: frail, underweight and apparently homeless.

“I said, ‘Sweetie, go look for her,’” Ms. Burton recalled. “He was really clear that she needed our help — and she needed help beyond us.”

The couple invited Nakesha to move into an apartment they kept in Washington Heights, clearing space for her in the living room. But after they broached the idea of her getting treatment, Nakesha moved out.

It would not be the last time Nakesha rejected offers of help, often from former classmates who had chance meetings with her on Manhattan streets.

Abby Dobson, a singer and songwriter who had known Nakesha from the Williams College gospel choir, recalled bumping into her repeatedly in Manhattan, including several times in Penn Station, where she saw Nakesha standing by the Seventh Avenue escalators, appearing homeless. Their conversations were friendly and they exchanged email addresses, but Ms. Dobson could see the deterioration in her friend, who mumbled to herself and claimed people were stealing her identity.

During one encounter, Ms. Dobson gave Nakesha some cash.

“Your twenty-five dollar ‘donation’ got me a very necessary new pair of pants,” Nakesha emailed later, “and for once in a long time I was lucky enough to get a shower.”

Ms. Dobson said that when she suggested delicately that Nakesha might benefit from seeking help, Nakesha rejected the idea. When they next saw each other, on the street, Nakesha snapped at her and accused her of spreading lies.

Angelique Feaster, Nakesha’s former college roommate and now a bank vice president, learned of Ms. Dobson’s encounters with Nakesha and began a search for her. “You are never alone,” she emailed Nakesha. “I love you, my sister, for life.” She never caught up with her.

Wole Coaxum, a banker and former Williams classmate, said he was at Grand Central one night when he heard someone call his name. He was stunned to see Nakesha, bundled on a bench. They had not seen each other since college, where they had been friends. He greeted her, silently trying to process the scene. He asked himself: How did you get here? Why are you sitting here homeless?

He gave Nakesha his card, and later emailed her. “How wonderful it was to see you the other day,” he wrote. “Clearly, you’ve been working through a lot.” He offered to help “in whatever way I can.”

Mr. Coaxum told Nakesha that he was on the board of Phoenix House New York, a nonprofit with a network of resources that could “start the process of helping you.” Nakesha ultimately rejected the offer, saying the agency was not for her, it was for substance abusers.

After his offices moved to Midtown, Mr. Coaxum continued to see Nakesha, on the grate at 46th Street and Park Avenue, sometimes on cold days when the rising steam kept her warm. He understood that homelessness was a reality, he said, “but it doesn’t necessarily happen to people that you know.”

Nakesha’s college friends communicated with one another about her predicament. But her resistance strained their relationships with her, and one after another they fell out of touch with Nakesha.

“There was just a deep sense of helplessness I think we all shared,” said Robyn Carter, a former nonprofit executive in Boston and one of the Four Musketeers on the Williams campus.

Laboring Alone

In January 2010, Ms. Burton received an email from a colleague in the school’s alumni relations department.

“Sandra, F.Y.I.,” the email said. “Looks like Nakesha had a baby.”

A New York City child protective specialist, Charles Castro, had emailed the college, saying he was seeking any of Nakesha’s “family or friends who can be considered as resources in a time of crisis for her.”

Nakesha, now 39, had apparently hidden her pregnancy under clothing and blankets. She was discovered going into labor at Grand Central and was rushed to the hospital. “It’s a miracle,” Nakesha wrote in an email several months later, that the baby “was born at all.”

Ms. Burton recalled Mr. Castro saying that he was trying to persuade Nakesha to accept mental health treatment, to improve her chances of maintaining a relationship with her child while his fate was decided in family court. “Otherwise,” Ms. Burton recalled, “the child was going into foster care.”

Nakesha was not receptive. When Ms. Burton emailed her in July 2010 to see how she was doing, she responded angrily, threatening to sue her, adding that if Ms. Burton intended to discuss social services or doctors, “don’t bother.”

The baby was placed in foster care, and although Nakesha took some parenting classes and visited him at the nonprofit social services agency that was supervising the case, her parental rights were ultimately terminated by the family court. The boy was adopted by his foster mother.

In 2012, Nakesha gave birth to another son, who was allowed to remain with his biological father. The agency’s social worker who oversaw the children’s cases said in an interview that both boys were in loving households, and each was thriving and happy.

The older boy, now about 8 years old, is autistic and gifted, his adoptive mother said in a separate interview. She said the two boys had exchanged photographs and messages, and she added, “They look identical.”

‘This Is Not Me’

By the time P.J. Dearden first encountered Nakesha on the sidewalk in 2014, she had been homeless in Manhattan for more than six years, with only occasional breaks, her writing suggests. She described living for several months in Long Branch, N.J., a short “home stay” in the Bronx and spending some nights in a Brooklyn motel. Some periods are not documented. For the most part, she stayed on sidewalks near Bryant Park, Herald Square, Grand Central, Penn Station, a heated grate on 32nd Street near Seventh Avenue, and the sidewalk at 46th and Park. On some cold nights, she rode the subway.

P.J. and several others had saved copies of the letters that Nakesha had handed them on the sidewalk. Nakesha typically wrote the letters on laptops she borrowed at the Grand Central branch of the New York Public Library, a short distance from the grate, and then printed copies. The letters, like her emails, were a mixture of delusional statements, literary references, tart commentary and harsh attacks on other people. Some were addressed to public figures, like President Obama — “Dear Barack” — and others to “Children All Over the World.”

“I am still stuck in New York City without housing,” she wrote on April 10, 2014. “I don’t care for ‘bag lady’ as a profession any more than ‘homeless lady.’”

She later described spending a rainy Christmas under a makeshift tent. “I don’t seem to be welcome in places where I actually should be,” she wrote, “though I keep hearing my appearance and cart & bags are deterrents.”

She noted that her bags and property had been stolen “by an F.B.I. team,” and that her sons, too, had been “stolen.”

She listed the books she was reading, and described her favorite hangouts: a Cosi eatery on 45th Street where she would “buy my dinner or get a hot beverage and use the bathroom”; a passageway under the Helmsley Building, where she would “thaw and stay out of the cold”; and an atrium where she went at night after a late-January snowstorm.

In her letters, she described rejecting offers of temporary housing by a homeless services outreach worker named “Luis.”

“Still a ‘no’ and will remain that way,” Nakesha wrote.

For many in the neighborhood, Nakesha was a fixture on the grate at 46th and Park. Daniel Ferrigno, who managed the Grand Café, a lunch counter beneath the Helmsley Building, said Nakesha was a frequent customer. They talked, he said, though not about her past.

“She never needed me to open that door, so why go there?” Mr. Ferrigno said.

Nakesha was also a regular at the Grand Central library branch, according to Genoveve Stowell, a former managing librarian.

Ms. Stowell said that the branch welcomed the homeless as long as they respected library rules that applied to everyone — no napping, no loud noises, no oversized bags.

Ronnie Hicks, another former employee, said that although she had seen Nakesha outside the library talking to herself, “she never did that” inside the branch, and had “perfect manners.”

Occasionally, patrons complained about Nakesha’s body odor, and Ms. Stowell took the matter up privately with Nakesha. “I had to address it,” she said.

Others who engaged with Nakesha were often left with more questions than answers.

Carolee Hildenbrandt, who passed Nakesha on her way to work, had once been a nurse at Bellevue Hospital Center. “They’re wonderful there,” she recalled telling Nakesha. “They can help you.” Nakesha demurred.

Myrna Cruz, a senior executive assistant at JPMorgan, who bought Nakesha a cart, said Nakesha had seemed appreciative, yet within a week, the cart was gone and Nakesha was using an older one.

Nakesha’s closest observers were two street vendors. One, Hamid Elhiri, a Moroccan immigrant who each morning parked his coffee cart a few feet from the grate, said he typically made Nakesha a breakfast of eggs and a roll, with iced tea or cranberry juice. When he would ask how she was doing, she would respond, “Alive.”

Sometimes, Nakesha stood in line and placed a few dollars at his window, he said. If she did not have money, she would approach the rear of his cart. “She has — how we say — pride?” he recalled.

Mr. Elhiri said one day a man began taunting her, and Mr. Elhiri shouted at him to stop. Another time, he blocked a homeless man from stealing Nakesha’s purse while she slept. Mr. Elhiri said he joked with her, “I’m your bodyguard.”

But he regretted that one day he failed to stop a security officer from confiscating Nakesha’s belongings after she had asked Mr. Elhiri to watch them. “They throw everything away,” he said; Nakesha was enraged.

At lunchtime, Nakesha often crossed Park Avenue to sit beside another vendor, Magdy Eltantawy, an Egyptian immigrant, who operated a sandwich cart with his wife and brother. They had known Nakesha for several years, and enjoyed bantering with her and preparing her favorite lunch — chicken over rice, with raw onion.

Mr. Eltantawy said he remembered Nakesha saying she had children, and once showing him a photograph of her holding a baby.

Mr. Eltantawy said he saw all sides of Nakesha’s personality. One day, he recalled, she began screaming uncontrollably, and he asked her, “What’s going on, Ms. Williams?”

“‘This is not me,’” she said. “‘This is other people in me.’”

Mr. Eltantawy’s suggestions that she seek help were rebuffed: “I’m functioning very well,” she told him.

An Avid Reader Loses Ground

For Nakesha, Christmas 2015 was a relatively happy time, at least by one measure. On Dec. 28, an outreach worker, Luis Alfredo Garcia, found her on her grate, surrounded by a pile of presents she had received from passers-by.

“Lots of gifts, food,” Mr. Garcia jotted down in his journal.

Nakesha’s holiday bounty did not last long, Mr. Garcia noted several days later. “Sanitation took gifts” that she had left unattended, he wrote: “food, blanket, vase.”

Mr. Garcia — the “Luis” cited by Nakesha in one of her letters — worked for Breaking Ground, a nonprofit contractor for the Department of Homeless Services, which sends two-member outreach teams on daily walks to check on street dwellers in Midtown and to try to patiently encourage them to accept housing, treatment and supportive services. Occasionally, psychiatrists accompany the teams.

At first, Nakesha was dismissive of the workers, calling them the “hi and goodbye” people, said Bill Hughes, a Breaking Ground supervisor. Mr. Garcia said he drew out Nakesha by discussing the books she was reading; a co-worker, Shanna Knotts, said she used the letters Nakesha handed out for hints about what she was feeling and thinking.

Mr. Garcia’s journal served as a running account of his interactions with the street people he checked on; one entry shows that he observed Nakesha as early as August 2013, at Fifth Avenue and 42nd Street.

“Nakesha: Reading. Looked up smiled, waved,” Mr. Garcia wrote.

The next month, Nakesha was at 46th Street between Park and Lexington Avenues, sitting, writing and surrounded by bags. “Spoke but waved off,” he noted.

His journal entries offered vivid snapshots of Nakesha’s daily activities:

“Counting $.”

“Reading War of the Worlds.”

“Reading Huck Finn.”

“Seated w/ umbrella, rain coat. Reading Anna Karenina.”

“Still reading Anna K; about 200 pages left.”

“Plans to stay here to keep wind from blowing away her things.”

“Standing. Arguing w/ herself.”

Breaking Ground’s records show that from April 2014 on, its workers saw Nakesha at 46th Street and Park Avenue 319 times, Mr. Hughes said. Three-quarters of those times, they stopped to speak with her. On 10 occasions, they tried unsuccessfully to persuade her to take a room reserved for the homeless at the Harlem Y.M.C.A.

”I can arrange for a ride there so you can see the place before deciding,” Mr. Garcia once told Nakesha in June 2015, his journal shows.

“Not today,” Nakesha responded.

Nakesha resisted meeting formally with psychiatrists, although several spoke with her briefly on different occasions on the street and read some of her letters, Mr. Garcia recalled. He said that one doctor had suspected that Nakesha might have schizophrenia.

Mr. Garcia said he had hoped Nakesha would agree to treatment so she could return to the “level of functionality” she had once enjoyed, allowing her to reconnect with her family, become employable and even write in a more coherent way.

“That’s really what I wanted for Nakesha,” Mr. Garcia said. “She’d have her life back.”

He said that he had also wondered whether Nakesha’s refusal to seek shelter on the coldest nights was itself sufficient evidence that she posed a danger to herself, and thus could be committed for evaluation.

Mr. Garcia said he got a new perspective one day in January 2016 when he was surprised to see that Nakesha had been interviewed by a local news station.

New York’s governor, Andrew M. Cuomo, had just signed an executive order compelling the authorities to move the homeless — forcibly if necessary — into shelters when the temperature dropped to 32 degrees or below.

The governor’s order spurred widespread debate, and the television station, WPIX, seeking reaction from the street, found Nakesha.

“Nakesha Williams braces for the first really cold night of this winter, hunkering down on the corner of 46th and Third Avenue,” a reporter narrated over an image of Nakesha seated on a bench.

Asked what she thought of Mr. Cuomo’s order, Nakesha responded that it “makes sense,” but noted that there were risks associated with shelters.

“Your choice is between the cold and an unsafe situation,” Nakesha explained. “Is it sanitary enough and is it safe enough? Those are the issues.”

Mr. Garcia was impressed, he said, that Nakesha, despite her delusions, had cogently analyzed the choices as she saw them: a potentially dangerous shelter, or the street, where she had proved self-sufficient.

“So, which is saner?” Mr. Garcia said. “It’s such a moment of extreme clarity.”

A Last Encounter

On the morning of July 22, 2016, Scott Erickson, an optician who worked on Fifth Avenue near 45th Street, smiled at Nakesha as he passed her on his way to buy a lottery ticket; she nodded back, he recalled. He had seen her frequently in recent weeks, introducing himself at one point, and had given her small gifts: hand lotion, fresh socks and a pair of unused sneakers from his closet.

That day, Nakesha was seated on a bench outside the Barnes & Noble store, surrounded by shopping bags, sandwiches wrapped in cellophane and an empty bottle of Minute Maid lemonade, a green straw inside. A blue utility cart, stuffed with clothing, papers and a floral-patterned sack, stood a few feet away.

When Mr. Erickson returned from his errand a few minutes later, he found Nakesha sitting motionless, her head tilted backward, her eyes open but empty.

He shook her arm and called 911. He began CPR, until the police and paramedics arrived and took over.

“Come on!” he called to Nakesha.

“You know her?” a paramedic asked.

“Just from seeing her out here,” Mr. Erickson replied.

Nakesha was rushed to Bellevue, where she was pronounced dead at a minute past noon. She was transported to the medical examiner’s office, where an employee wrote, “Unknown woman of unknown age found dead in a park.”

Her photograph was emailed to the Department of Homeless Services. Outreach workers identified her and reported that she was mentally ill and had spurned offers of shelter or safe haven.

“The team is not aware of any next-of-kin information,” the workers said.

Nakesha had died of a pulmonary embolism, a complication of her obesity, an autopsy showed. She was 5 feet tall and weighed 255 pounds, the report said. There was no evidence of alcohol or drugs in her system. She was 46 years old.

Her remains were placed in a white body bag and kept in a basement mortuary, where she lay unclaimed.

A Call to a Reporter

P.J. Dearden had been searching for Nakesha on the streets near the grate, and in and around Grand Central. Finally, she approached Mr. Elhiri, the coffee vendor, who said a customer had told him that Nakesha had died.

P.J. sought a confirmation. Her wife, Dawn Dearden, a federal employee in Manhattan whom I knew, called me. How could she find out if Nakesha was alive? I suggested she call the medical examiner’s office.

Dawn did, and she learned that Nakesha had died and her body had been in the morgue for nearly six weeks. If no relative claimed her remains, Dawn was told, Nakesha would most likely be buried on Hart Island, where the city inters unclaimed bodies in mass graves.

P.J. resolved to not let that happen. She claimed Nakesha’s remains, had her cremated at a Greenwich Village funeral home and had her ashes placed in an urn made with mother-of-pearl with flecks of gold. She left her phone number with Mr. Elhiri, in case anyone asked about the woman on the grate.

An office worker, Karen Davis, who had known Nakesha and learned of P.J.’s efforts through Mr. Elhiri, emailed the secretaries in her office, collecting money to help defray the funeral expenses. Ms. Davis gave P.J. about $300 in an envelope that listed 21 donors.

I, meanwhile, used copies of Nakesha’s letters, which P.J. and others had provided to me, as a guide to track down her relatives and former classmates.

Nakesha’s aunt, Ms. Fattore, sobbed at the news of her death. But she also seemed to have found a sense of relief. She said the family had not heard from Nakesha in years.

In the living room of her house in Willingboro, N.J., Ms. Fattore, along with Nakesha’s uncle, Danel Williams, and the younger of her two brothers, Cherron Rothmiller, proudly recounted her academic achievements, strong friendships and devotion to family members. “Everybody called her Kesha,” Danel Williams said.

Displayed on Ms. Fattore’s wall was Nakesha’s high school portrait: She is wearing a blue-and-white-striped dress and resting her chin on her arms. She looks radiant, relaxed and confident.

The family also described Nakesha’s difficult childhood, her molestation and how she had kept it a secret for years. They also talked about her mental deterioration.

“We should just be happy that she’s at peace,” Ms. Fattore said, weeping. “Because she wasn’t at peace as long as she walked around this earth.”

Nakesha’s former college classmates were brokenhearted to learn of her death. Scores of tributes were shared through private group messages on Facebook. One appreciation came from Krystal Williams, who had danced in Nakesha’s senior project but had not seen her since college.

“I am trying to wrap my mind and heart around Nakesha’s final years,” she wrote. “In my mind, I still see her hair in a ponytail with her rocking dance tights, and a slightly oversized dance jersey that slides off one shoulder. That is how I will remember her — in light, dancing and leaping over the drum beat.”

Ms. Burton, sitting in her campus office at Williams College, read through letters Nakesha had written in her final months, including some in which Nakesha had bitterly attacked her.

“It’s really powerful for me to know that I was in her mind until her death,” Ms. Burton said. She said that as Nakesha had struggled with her illness, she must have asked herself, “How do I anchor myself, right now, in this storm?”

“Her discipline was writing every day,” Ms. Burton continued. “The names, the places, the jobs, the memories of relationships, the snatches of memories of relationships; the books, the discipline to read all the time and to talk about what she was reading and to have a book list, right? To me, that goes back to the joy that she had as a learner, as a scholar, as a creative person. And those skills remained with her throughout her life.”

“Somehow, she was also holding on to all of us,” Ms. Burton added, “but she wouldn’t let us hold on to her.”

On Oct. 15, 2016, Nakesha’s father, Marvin Rothmiller, then 66, a retired postal worker, arrived at Penn Station with several relatives, including two of his sisters, Norlyn Garlic and Valerie Dillard, who had been close to Nakesha. I accompanied them on the walk to 46th Street, where P.J. and Dawn Dearden were waiting by Nakesha’s grate, with a suitcase containing Nakesha’s urn.

P.J. spoke for a few minutes about Nakesha, and the people who had watched over her. She showed the family the passageway under the Helmsley Building where Nakesha had sought refuge from the cold and rain, and the Grand Central library branch, where she had spent so many hours. The family returned to New Jersey, taking the suitcase with them.

P.J. still asks what else she could have done for Nakesha. “Everyone let her down on some level,” she said. “She died on a street corner.”

A Reckoning, by Candlelight

On Nov. 1, 2016, 10 people gathered on Nakesha’s grate to commemorate her life. Nicole Moore, a friend of Nakesha’s from college, had organized the event. Each guest lit a candle and reminisced.

Many were Nakesha’s former college classmates, including Becky Dickinson, the once-insecure dance performer; Anim Steel, who opened his apartment to Nakesha in 1998; and Abby Dobson, the singer and songwriter. Many spoke of their regret at not being able to do more for Nakesha.

P.J. and Dawn Dearden were also there, as was Karen Davis, the office worker who had taken up a collection, and who happened on the gathering as she was heading to the subway.

Ms. Feaster, the former college roommate who had been unable to find Nakesha in New York, said she was devastated that she never saw her.

“Because I would have grabbed her and just took care of her, crazy or not,” Ms. Feaster said. “I would be like, ‘O.K., they’re coming after you? Well, we’ll run!’”

Ms. Dickinson, now a social worker, said that every day she walked by people “who are going through trauma and adversity.”

“You never know — you never know anybody’s story,” she said.

P.J., who was meeting Nakesha’s college friends for the first time, could barely hold back tears. She said that she hoped the group was comforted knowing that “there were people who really cared for her, and looked out for her.”

When it was Ms. Moore’s turn, she spoke about how mental illness did not discriminate. “It could be us tomorrow,” she said. She turned to P.J. “We are grateful for you, for giving her whatever time you had,” she said.

With candles burning and a few pedestrians looking on, Ms. Dobson began singing a modern-day spiritual that she had written years earlier with her cousin. Inspired by the Toni Morrison novel “Beloved,” it was called “Lonesome Child.”