How Will Fashion Megabrands Handle the Next Phase of the Coronavirus Crisis?This week, everyone will be talking about LVMH's assessment of the pandemic's impact, the prospect of extended lockdowns in the US and UK, and China's economic recovery. Get your BoF Professional Cheat Sheet.LVMH reports first-quarter financial results on April 16The owner of Louis Vuitton, Celine and other brands said in late March that it expects Covid-19 to cut Q1 sales by up to 20 percentThe luxury conglomerate initially downplayed the coronavirus threat but has since taken drastic steps to address the crisisLVMH, the luxury conglomerate turned hand sanitiser and face mask manufacturer, reports results this week. The company has undergone a remarkable transformation since its last financial update in January, when executives downplayed the risk of Covid-19. ("Do you know how many people die from the flu every year?" one executive said on an analyst call before floating the idea that the then-nascent crisis "won't be that bad" if it passed in a couple months.)LVMH had changed its tune by late March, predicting a 10 percent to 20 percent drop in Q1 revenue compared with the same period in 2019. Last week, Louis Vuitton started making hospital gowns at its ready-to-wear atelier, an admission that life won't be returning to normal anytime soon (while also making it easier to restart regular production once the pandemic subsides).The tone LVMH strikes on Thursday will be closely watched, as will any insight into how quickly China is rebounding. (Louis Vuitton's stores in that country had all reopened by last week, though it's not clear if sales have fully rebounded to pre-outbreak levels.) In an interview with BoF, Louis Vuitton Chief Executive Michael Burke declined to offer a date for normal production to resume, though he did indicate work is underway on future collections.The Bottom Line: All brands can do is wait out the pandemic. LVMH, with its enormous scale and pile of cash, is better equipped to do this than virtually any other company in fashion.
How Will Fashion Megabrands Handle the Next Phase of the Coronavirus Crisis?
This week, everyone will be talking about LVMH's assessment of the pandemic's impact, the prospect of extended lockdowns in the US and UK, and China's economic recovery. Get your BoF Professional Cheat Sheet.
LVMH reports first-quarter financial results on April 16
The owner of Louis Vuitton, Celine and other brands said in late March that it expects Covid-19 to cut Q1 sales by up to 20 percent
The luxury conglomerate initially downplayed the coronavirus threat but has since taken drastic steps to address the crisis
LVMH, the luxury conglomerate turned hand sanitiser and face mask manufacturer, reports results this week. The company has undergone a remarkable transformation since its last financial update in January, when executives downplayed the risk of Covid-19. ("Do you know how many people die from the flu every year?" one executive said on an analyst call before floating the idea that the then-nascent crisis "won't be that bad" if it passed in a couple months.)
LVMH had changed its tune by late March, predicting a 10 percent to 20 percent drop in Q1 revenue compared with the same period in 2019. Last week, Louis Vuitton started making hospital gowns at its ready-to-wear atelier, an admission that life won't be returning to normal anytime soon (while also making it easier to restart regular production once the pandemic subsides).
The tone LVMH strikes on Thursday will be closely watched, as will any insight into how quickly China is rebounding. (Louis Vuitton's stores in that country had all reopened by last week, though it's not clear if sales have fully rebounded to pre-outbreak levels.) In an interview with BoF, Louis Vuitton Chief Executive Michael Burke declined to offer a date for normal production to resume, though he did indicate work is underway on future collections.
The Bottom Line: All brands can do is wait out the pandemic. LVMH, with its enormous scale and pile of cash, is better equipped to do this than virtually any other company in fashion.
China Has Become a Testing Ground for Luxury Brands
Product launches are being moved to China and high sales targets are set, hoping to offset the losses from the rest of the world.
With much of the world still in lockdown to slow the coronavirus outbreak, China has become the best hope for luxury and fashion brands eager to get tills ringing again, making it a key launchpad for products and a testing ground for innovative marketing approaches.
Dior, for example, introduced Gem Clutch, a bespoke hybrid between fine jewelry and fashion handbag in China on WeChat on Friday before the brand begins to roll it out in the West. A day before that, Dior released a video featuring its China ambassador Angelababy touring its Shanghai flagship at Plaza 66 as a virtual shopping trip to promote the new Book Tote and Lady Dior bags.
The video gained 297,000 likes, 97,000 reposts and 43,000 comments on Angelababy’s Weibo post. It’s also understood that Dior will follow the path of Louis Vuitton, Burberry and Longchamp and host livestreaming events. But instead of engaging with China’s expansive online audience, Dior’s session is expected to be invitation-only for its VIPs.
Louis Vuitton worked with Xiaohongshu, a popular social commerce platform to promote its summer 2020 collection using livestreaming. Some 15,000 users watched the streaming with 6.25 million comments and likes.
Burberry recorded 1.38 million in engagement with its livestreaming debut with Tmall. Six out of the 10 products introduced via the streaming, including handbags and earrings, promptly sold out.
Josie Zhang, president of Burberry China, said, “In the past two months, we have seen a visible change in the behavior of Chinese consumers. We are very happy to be able to connect with so many consumers with livestreaming on Tmall. Consumers are eager for new things, and for us, when some consumers are unable to experience off-line boutiques because of the epidemic, livestreaming happens to be a safe and exciting way to bring them fresh experiences.”
Besides the trend toward livestreaming, WeChat Mini Programs, which were at the top of luxury brands’ minds in 2019, continue. These — along with Tmall stores and an official presence on various social media channels including Weibo, TikTok, Xiaohongshu and Shipinhao, which is WeChat’s answer to Instagram — are helping brands build a 360-degree digital ecosystem that brings them closer to customers, making direct-to-consumer sales easier than in most other regions of the world.
Luxury brands are banking on China, which is quickly recovering from the outbreak, to get revenues flowing again. Brunello Cuccinelli said sales in China saw improvement in the first 10 days of March and over the past few days, after significant drops since the last week of January and throughout the month of February.
It’s not yet clear whether signs of improvement will become major gains. Chinese luxury consumer confidence is down, according to consultancy firm China Luxury Advisors. Bubble tea and hot pot are two inexpensive treats experiencing an instant surge in demand after lockdown measures were removed, as reported by local news. But brands are working hard to create desirability to move items off the shelves as fast as they can.
Richard Lu, founder of communication firm Instar — which works with 30-plus luxury clients including Gucci, Loewe, Prada, Valentino, Moynat, Bottega Veneta, Dunhill, Alexander McQueen, Tom Ford and Fendi — said brands have been introducing items to stimulate the market.
“People haven’t been shopping for three months. There is definitely a resurgence, but I am worried about what’s to come in two to three months’ time. We are supposed to receive samples for pre-fall and fall-winter collections now, but since factories stopped working from March in Europe, I hope we will be able to get them by May, or it will be difficult for us to help promote them,” Lu said.
Meanwhile, an account director at a global fashion agency’s Beijing office, who requested anonymity, said its sample room has been empty since late March. And two other agencies confirmed to WWD that sample requests are reaching a historic high.
“I can feel that China is the last straw for many of our clients. All the new product launches are being moved to China, as it’s the only place where shops are open. But store openings, previews and fashion shows are still being canceled or postponed before Q3. It’s very risky because the government hasn’t signaled a clear attitude on hosting events. You still need to self-quarantine for 14 days if you are coming to Beijing and no one wants a COVID-19 carrier showing up at their events,” the director added.
Vanessa Wu, director of Europe at Reuter Communications, a Shanghai-based agency representing Lanvin, Maison Margiela, Alaïa, Vivienne Westwood, Harrods, Farfetch and Mytheresa, believes brands should reexamine their go-to market strategies to rebuild the momentum with China leading the way of recovery post-COVID-19.
“Instead of being overly commercial to leverage the pandemic, it’s important to offer products or services to help people ease into a new norm, whether through digital migration or domestic consumption,” she said.
While online innovation is helping brands reach a wider audience, physical retail still brings in the majority of revenues.
Hermès is aiming to hit the 40 million renminbi, or $5.7 million, sales goal for its renovated store for the rest of April in Guangzhou’s luxury landmark Taikoo Hui, which is set to open on Saturday, according to a source working in luxury retail.
The amount is a bold monthly target. Only the flagships in China of the top luxury brands, such as Chanel and Louis Vuitton, are able to reach that number, the source said, adding that the rest of Hermès’ stores in the country will be supporting the Guangzhou unit to hit the target within 20 days.
In Shanghai, 80 percent of the business had recovered by early April, said a sales associate at Louis Vuitton over WeChat. Loyal customers have returned, exotic leather bags sell just as well as iconic monograms, but walk-in sales haven’t rebounded. The brand’s spring runway collection is nearly sold out, and now customers are snatching up the summer collection.
A China sales manager at Porsche said its headquarter have asked the dealer to hit its 2020 target despite a visible drop in January and February.
“Our goal for this year is to sell 90,000 luxury cars, which is more than the entire Europe[an continent] or the U.S. We have had 10 percent annual growth for years. The performance of the second half will be crucial. But I am not too worried, our clients come from a wide range of backgrounds and if they don’t buy it now, the car will no longer be produced as our factories in Germany are shut,” the manager said.
Hermès Hauled in $2.7 Million in One China Store on Saturday: Sources
Its China's second-largest Guangzhou flagship, reopened last Saturday, saw a strong rebound of luxury consumption in post-coronavirus China.
Hermès is said to have brought in at least 19 million renminbi, or $2.7 million, in sales on the reopening day of its flagship store in Guangzhou’s Taikoo Hui last Saturday, according to multiple sources.
Rare bags, including a diamond-studded Himalayan Birkin, were shipped to the location. VIPs from across the Guangdong province, the wealthiest area in China with Guangzhouits capital, descended on the store to purchase tableware, shoes, furniture and leather goods.
The single-day tally, believed to be the highest for a single boutique in China, offers a confidence boost for luxury brands who are eager to get tills ringing again after the coronavirus outbreak.
Hermès could not immediately be reached to verify the sales figure, but its VIPs documented their shopping foray across Chinese social media, such as Weibo and Xiaohongshu.
Atomniu, a user on social commerce platform Xiahongshu, posted several pictures of herself in the two-level store with interiors designed by the Parisian architecture agency RDAI, and said she spent nearly one million renminbi, or $142,124 at current exchange, on the reopening day. She snatched a black crocodile Birkin 30, as well as some clothes and shoes.
The new location, spanning some 5,500 square feet and previously occupied by Prada, boasts an extensive facade that “features a wall with a modern yet minimalist aesthetic, cleverly combining the local tradition of brick making and enamel craftsmanship.” It offered for sale two exclusive Birkin designs inspired by the jacket worn by the Imperial Guard of Napoleon III, according to a statement seen by WWD.
“This reopening affirms the house’s commitment to Southern China and marks a new chapter for the Parisian house in Guangzhou, where it has been present since 2004,” the statement added.
Before the reopening, Hermès closed its first store in Guangzhou, located in what was once the luxury hub of the city, La Perle Plaza, and merged that team with its Taikoo Hui store, which first opened in 2011.
Boeing’s Problems Predate the Virus. Should the U.S. Come to Its Rescue?
The nation’s largest aerospace company is in discussions about three different federal aid programs amid confusion created by its new chief executive about its intentions.
Two deadly crashes that left its 737 Max airliner fleet grounded. An aborted mission of its new spacecraft. Problems with a tanker it makes for the Air Force, a depressed stock price and the abrupt dismissal of its chief executive.
Boeing faced serious problems largely of its own making long before the coronavirus crisis wounded the United States economy and led Congress to put hundreds of billions of dollars of corporate aid on the table. But that has not kept it from putting its hand out.
If the Trump administration obliges, the company — whose prospects have been further clouded by the virus-induced travails of the airline industry — could become one of the top recipients of federal aid in the most sweeping economic program in American history. That could give it a new chance to overcome its recent struggles, even as businesses without such flaws are fighting for assistance to survive.
Boeing is now engaged in a delicate dance with an administration that wants to keep the American industrial base strong but aims to avoid charges of handing out corporate welfare. Only weeks ago, the company signaled a reluctance to accept the conditions that might come with aid, including the possibility of handing over an ownership stake to the government or limiting layoffs. But it is still trying to work out a deal.
Boeing, the nation’s largest aerospace company and its second-largest military contractor in the United States, is considering seeking assistance from as many as three of the federal programs established by the $2 trillion coronavirus relief package signed into law last month by Mr. Trump.
After initially saying it had plenty of options and would not accept taxpayer help if it came with certain strings attached, the company is now working closely with the administration to see if it can work out a deal.
Inquiries into the crashes of the 737 Max led to revelations of corner-cutting and distrust, leaving Boeing mired in the worst crisis of its history. In December, the company dismissed Dennis A. Muilenburg as chief executive and later replaced him with David Calhoun, who continues to grapple with a wide array of operational and financial problems — not least the drying up of demand from the airline industry for new jets in a period when air travel has been severely depressed.
Yet even its critics acknowledge that Boeing is important to the nation’s security and economy, and that if its problems deepen, they could have wide-reaching ripple effects.
Boeing employs more than 150,000 people in the United States, while supporting more than a million additional workers through a supply chain that includes thousands of businesses. Those operations span the country, giving Boeing clout in the communities and the Capitol Hill offices of many influential lawmakers, as well as in the White House.
“Making sure that Boeing is strong again is very, very powerful and very important, and we’ll do whatever is necessary to do,” Mr. Trump said on Friday at a briefing on the government’s coronavirus response. He noted that the company “has not asked for aid yet, but I think they probably will.”
Yet as negotiations over the stimulus heated up late last month — with Boeing officials working closely with the White House and key members of Congress to communicate the company’s needs and preferences — Mr. Calhoun gave an interview that provided fodder to critics, left supporters puzzled and created confusion among government officials involved in the negotiations.
Mr. Calhoun suggested in an interview with the Fox Business host Maria Bartiromo last month that Boeing would not accept taxpayer money if it meant giving the government a stake in the company, a condition that was being debated at the time as part of the federal bailout legislation working its way through Congress. The final legislation included conditions for some of the aid, including the possibility of the government taking an equity stake and limits on layoffs and stock buybacks.
“I don’t have a need for an equity stake,” Mr. Calhoun said in the interview. “If they force it, we just look at all the other options, and we’ve got plenty of them.”
He added, “If they attach too many things to it, of course you take a different course.”
The comments prompted a flurry of phone calls from members of Congress to Boeing executives and lobbyists, according to people familiar with the exchanges.
Was Mr. Calhoun saying that the company didn’t want to be included in the bailout, or that its internal prognosis was better than the company’s debt-laden balance sheet suggested?
The responses from the company’s lobbyists to lawmakers were emphatic, if at odds with Mr. Calhoun’s pronouncement: Yes, Boeing needed the federal money, and no, it was not in better shape than it looked. Internally, Mr. Calhoun’s colleagues informed him that he had sent the wrong message, according to two people familiar with the matter.
Three days after Mr. Calhoun’s interview, Mr. Trump signed the bailout bill. It included several programs to be administered by the Treasury Department and the Federal Reserve, for which Boeing could be eligible.
The bill authorized the Federal Reserve to make credit available to companies, and it allowed the government to make direct loans to big companies through a program to be administered by Treasury Secretary Steven Mnuchin. The legislation specifically allotted $17 billion for loans and loan guarantees to “businesses critical to maintaining national security” — a pot of money believed to be intended to benefit primarily Boeing.
Yet on the day Mr. Trump signed the bill, Mr. Mnuchin suggested in an appearance of his own on Fox Business that he was still operating under the premise that Boeing was not looking for federal help.
“Boeing has said that they have no intention of using the program,” he told Ms. Bartiromo, adding: “I appreciate the fact that Boeing is saying they can operate on their own. That’s what we want them to do.”
Mr. Mnuchin and the Treasury Department will have wide latitude over the assistance funds, and the agency is currently drafting guidelines for doling out the money — a process that Boeing is closely monitoring.
Hours after Mr. Mnuchin’s interview, Greg Smith, Boeing’s chief financial officer, called him to try to clarify Mr. Calhoun’s comments. Mr. Smith explained that the company was just keeping its options open and studying the strings attached to each pool of money.
While some Boeing allies cringed at Mr. Calhoun’s Fox Business interview, others viewed the defiant stance as shrewd.
The company’s stock, which has plunged this year, briefly ticked upward as investors seemed buoyed by the idea that its value would not be diluted by possible government ownership. Within the company, there was a sense that it might prove useful to take an initial hard line, if only as a negotiating tactic to signal that the company had leverage and was not going to roll over.
In a statement, Gordon Johndroe, a Boeing spokesman, said that assistance from the government would not benefit the company alone, but would also help prop up companies in its supplier network. “We are continuing to pay our suppliers,” he said, adding that “70 cents of every dollar we take in goes directly to maintain the supply chain.”
Boeing is devoting considerable resources to analyzing and shaping the government assistance programs, as well as avenues for raising money from Wall Street. It retained a pair of New York investment banks — Lazard and Evercore — to advise on its options, The Wall Street Journal reported on Friday.
Mr. Smith has been consulting Mr. Mnuchin on how the conditions attached to the bailout money will affect the company. Boeing’s lobbyists have been regularly consulting with Treasury staff members.
In addition to the potential equity to be provided to the government in exchange for assistance, people familiar with the company’s lobbying effort say it is monitoring the maximum amounts of financing available under the various programs and whether anything in the guidelines would prevent it from seeking assistance from multiple programs at once.
For its defense contracting business, Boeing could seek loans from the $17 billion pool for national security companies. And for its commercial airplane manufacturing business, it could seek credit from the Federal Reserve, or direct loans or loan guarantees from the more general program administered by the Treasury Department.
The Treasury Department has yet to issue guidelines for the large company loans. It did issue late last month preliminary guidance for the national security company loans, indicating that in order to borrow from that $17 billion pool, companies would be required to identify equity interest that the government could take in the company.
The companies borrowing from that pool would also be required to maintain their employment levels “to the extent practicable” through September, according to the legislation and subsequent guidance from Treasury. And they would face limits on compensation increases for executives, along with prohibitions on using the loans to buy back stock.
But details on how those restrictions will be applied are still being hammered out. The Treasury Department indicated last week that it was developing more specific guidance and application materials regarding the “eligibility, mechanics and timing” of the national security loans.
Once the conditions of the various loan programs become clear, Boeing will balance them against its need for financial help to decide which of the options makes the most sense, according to the people familiar with the company’s lobbying effort.
Boeing is loath to agree to any restrictions on cutting employment, since the aerospace industry has been hit hard by the coronavirus crisis and the company has no idea what levels of aircraft production will make sense in one month, let alone in one year.
Mr. Trump on Friday seemed sympathetic to Boeing’s plight. He said a “determination hasn’t been made” about whether Boeing or other companies could be penalized for laying off workers after accepting federal assistance. But he suggested that he was considering the prospect of compelling the company to “keep people that they absolutely don’t need.”
“They do have to run a company,” the president said. “You know this isn’t a great time to sell airplanes; let’s not kid ourselves.”
Boeing is in fact considering involuntary layoffs, people familiar with its deliberations said. It has suspended production at some factories amid concerns about the spread of the virus and has already begun offering buyout packages to employees.
The moves come despite the release this month of $882 million in payments that the U.S. Air Force had been withholding from Boeing because of technical problems with a tanker jet the company had sold to the government.
On the day Mr. Trump signed the stimulus bill, he spoke by phone with Mr. Calhoun, who offered to use the Boeing Dreamlifter, an enlarged 747, to ferry medical supplies to states in need, according to four people briefed on the call. Mr. Calhoun also mentioned that Mr. Smith and Mr. Mnuchin were in close touch over the stimulus terms, one of the people said.
Later that evening, Mr. Trump held up a photo of the Dreamlifter at his coronavirus news briefing.
“They’re letting us use that for the distribution of product all over the country,” Mr. Trump said. “Each plane can carry 63,000 pounds of cargo per flight. That’s a lot of cargo.”
The aircraft has not yet made any humanitarian flights. Boeing says it expects to deploy the plane in “the next couple of weeks.”
The New Math Bridge Beyond Fermat’s Last Theorem
Two papers, representing the work of more than a dozen mathematicians, have finally figured out how to connect two realms that were once seen as distant.
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Pierre Fermat (1601-1665) PHOTOGRAPH: GETTY IMAGES
WHEN ANDREW WILES proved Fermat’s Last Theorem in the early 1990s, his proof was hailed as a monumental step forward not just for mathematicians but for all of humanity. The theorem is simplicity itself—it posits that x n + yn = zn has no positive whole-number solutions when n is greater than 2. Yet this simple claim tantalized legions of would-be provers for more than 350 years, ever since the French mathematician Pierre de Fermat jotted it down in 1637 in the margin of a copy of Diophantus’ Arithmetica. Fermat, notoriously, wrote that he had discovered “a truly marvelous proof, which this margin is too narrow to contain.” For centuries, professional mathematicians and amateur enthusiasts sought Fermat’s proof—or any proof at all.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
The proof Wiles finally came up with (helped by Richard Taylor) was something Fermat would never have dreamed up. It tackled the theorem indirectly, by means of an enormous bridge that mathematicians had conjectured should exist between two distant continents, so to speak, in the mathematical world. Wiles’ proof of Fermat’s Last Theorem boiled down to establishing this bridge between just two little plots of land on the two continents. The proof, which was full of deep new ideas, set off a cascade of further results about the two sides of this bridge.
From this perspective, Wiles’ awe-inspiring proof solved just a minuscule piece of a much larger puzzle. His proof was “one of the best things in 20th-century mathematics,” said Toby Gee of Imperial College London. Yet “it was still only a tiny corner” of the conjectured bridge, known as the Langlands correspondence.
The full bridge would offer mathematicians the hope of illuminating vast swaths of mathematics by passing concepts back and forth across it. Many problems, including Fermat’s Last Theorem, seem difficult on one side of the bridge, only to transform into easier problems when shifted to the other side.
After Wiles came up with his proof, other mathematicians eagerly extended his bridge to slightly larger portions of the two continents. But then they hit a wall. There are two natural next directions for extending the bridge further, but for both, the Taylor-Wiles method faced what seemed like an insuperable barrier.
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Andrew Wiles, the mathematician who proved Fermat’s Last Theorem, was awarded the Abel Prize in 2016.PHOTOGRAPH: ALAIN GORIELY/MATHEMATICAL INSTITUTE AT UNIVERSITY OF OXFORD
“People wanted to do this for a long time,” said Ana Caraiani of Imperial College London. But “we pretty much didn’t think it was possible.”
Now, two papers—representing the culmination of the efforts of more than a dozen mathematicians—have overcome this barrier, essentially solving both problems. Eventually, these findings may help mathematicians prove Fermat’s Last Theorem for some number systems beyond the positive whole numbers.
They are “pivotal results,” said Matthew Emerton of the University of Chicago. “There are some fundamental number-theoretic phenomena that are being revealed, and we’re just starting to understand what they are.”
Needle in a Vacuum
One side of the Langlands bridge focuses on some of the least complicated equations you can write down: “Diophantine” equations, which are combinations of variables, exponents, and coefficients such as y = x2 + 6x + 8, or x3 + y3 = z3. For millennia, mathematicians have tried to figure out which combinations of whole numbers satisfy a given Diophantine equation. They’re motivated primarily by how simple and natural this question is, although some of their work has recently had unforeseen applications in areas such as cryptography.
Since the time of the ancient Greeks, mathematicians have known how to find the whole-number solutions to Diophantine equations that have just two variables and no exponents higher than 2. But searching for whole-number solutions is anything but straightforward with equations that have larger exponents, starting with elliptic curves. These are equations that have y2 on the left and a combination of terms whose highest power is 3, like x3 + 4x + 7, on the right. They’re a “massively harder problem” than equations with lower exponents, Gee said.
On the other side of the bridge live objects called automorphic forms, which are akin to highly symmetric colorings of certain tilings. In the cases Wiles studied, the tiling might be something along the lines of M.C. Escher’s famous tessellations of a disk with fish or angels and devils that get smaller near the boundary. In the broader Langlands universe, the tiling might instead fill a three-dimensional ball or some other higher-dimensional space.
These two types of mathematical objects have completely different flavors. Yet in the middle of the 20th century, mathematicians started uncovering deep relationships between them, and by the early 1970s, Robert Langlands of the Institute for Advanced Study had conjectured that Diophantine equations and automorphic forms match up in a very specific manner.
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Robert Langlands, who conjectured the influential Langlands correspondence about 50 years ago, giving a talk at the Institute for Advanced Study in Princeton, New Jersey, in 2016.PHOTOGRAPH: DAN KOMODA/INSTITUTE FOR ADVANCED STUDY
Namely, for both Diophantine equations and automorphic forms, there’s a natural way to generate an infinite sequence of numbers. For a Diophantine equation, you can count how many solutions the equation has in each clock-style arithmetic system (for example, in the usual 12-hour clock, 10 + 4 = 2). And for the kind of automorphic form that appears in the Langlands correspondence, you can compute an infinite list of numbers analogous to quantum energy levels.
If you include only the clock arithmetics that have a prime number of hours, Langlands conjectured that these two number sequences match up in an astonishingly broad array of circumstances. In other words, given an automorphic form, its energy levels govern the clock sequence of some Diophantine equation, and vice versa.
This connection is “weirder than telepathy,” Emerton said. “How these two sides communicate with each other … for me it seems incredible and amazing, even though I have been studying it for over 20 years.”
In the 1950s and 1960s, mathematicians figured out the beginnings of this bridge in one direction: how to go from certain automorphic forms to elliptic curves with coefficients that are rational numbers (ratios of whole numbers). Then in the 1990s, Wiles, with contributions from Taylor, figured out the opposite direction for a certain family of elliptic curves. Their result gave an instant proof of Fermat’s Last Theorem, since mathematicians had already shown that if Fermat’s Last Theorem were false, at least one of those elliptic curves would not have a matching automorphic form.
Fermat’s Last Theorem was far from the only discovery to emerge from the construction of this bridge. Mathematicians have used it, for instance, to prove the Sato-Tate conjecture, a decades-old problem about the statistical distribution of the number of clock solutions to an elliptic curve, as well as a conjecture about the energy levels of automorphic forms that originated with the legendary early 20th-century mathematician Srinivasa Ramanujan.
After Wiles and Taylor published their findings, it became clear that their method still had some juice. Soon mathematicians figured out how to extend the method to all elliptic curves with rational coefficients. More recently, mathematicians figured out how to cover coefficients that include simple irrational numbers, such as 3 + √2.
These papers, right now, are kind of the pinnacle of achievement.
What they couldn’t do, however, was extend the Taylor-Wiles method to elliptic curves whose coefficients include complex numbers such as i (the square root of −1) or 3 + i or √2i. Nor could they handle Diophantine equations with exponents much higher than those in elliptic curves. Equations where the highest exponent on the right-hand side is 4 instead of 3 come along for free with the Taylor-Wiles method, but as soon as the exponent rises to 5, the method no longer works.
Mathematicians gradually realized that for these two next natural extensions of the Langlands bridge, it wasn’t simply a matter of finding some small adjustment to the Taylor-Wiles method. Instead, there seemed to be a fundamental obstruction.
They’re “the next examples you’d think of,” Gee said. “But you’re told, ‘No, these things are hopelessly out of reach.’”
The problem was that the Taylor-Wiles method finds the matching automorphic form for a Diophantine equation by successively approximating it with other automorphic forms. But in the situations where the equation’s coefficients include complex numbers or the exponent is 5 or higher, automorphic forms become exceedingly rare—so rare that a given automorphic form will usually have no nearby automorphic forms to use for approximation purposes.
In Wiles’ setting, the automorphic form you’re seeking “is like a needle in a haystack, but the haystack exists,” Emerton said. “And it’s almost as if it’s like a haystack of iron filings, and you’re putting in this magnet so it lines them up to point to your needle.”
But when it comes to complex-number coefficients or higher exponents, he said, “it’s like a needle in a vacuum.”
Going to the Moon
Many of today’s number theorists came of age in the era of Wiles’ proof. “It was the only piece of mathematics I ever saw on the front page of a newspaper,” recalled Gee, who was 13 at the time. “For many people, it’s something that seemed exciting, that they wanted to understand, and then they ended up working in this area because of that.”
So when in 2012, two mathematicians—Frank Calegari of the University of Chicago and David Geraghty (now a research scientist at Facebook)—proposed a way to overcome the obstruction to extending the Taylor-Wiles method, their idea sent ripples of excitement through the new generation of number theorists.
Their work showed that “this fundamental obstruction to going any further is not really an obstruction at all,” Gee said. Instead, he said, the seeming limitations of the Taylor-Wiles method are telling you “that in fact you’ve only got the shadow of the actual, more general method that [Calegari and Geraghty] introduced.”
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David Geraghty at Boston College in 2015.PHOTOGRAPH: GARY WAYNE GILBERT/BOSTON COLLEGE
In the cases where the obstruction crops up, the automorphic forms live on higher-dimensional tilings than the two-dimensional Escher-style tilings Wiles studied. In these higher-dimensional worlds, automorphic forms are inconveniently rare. But on the plus side, higher-dimensional tilings often have a much richer structure than two-dimensional tilings do. Calegari and Geraghty’s insight was to tap into this rich structure to make up for the shortage of automorphic forms.
More specifically, whenever you have an automorphic form, you can use its “coloring” of the tiling as a sort of measuring tool that can calculate the average color on any chunk of the tiling you choose. In the two-dimensional setting, automorphic forms are essentially the only such measuring tools available. But for higher-dimensional tilings, new measuring tools crop up called torsion classes, which assign to each chunk of the tiling not an average color but a number from a clock arithmetic. There’s an abundance of these torsion classes.
For some Diophantine equations, Calegari and Geraghty proposed, it might be possible to find the matching automorphic form by approximating it not with other automorphic forms but with torsion classes. “The insight they had was fantastic,” Caraiani said.
Calegari and Geraghty provided the blueprint for a much broader bridge from Diophantine equations to automorphic forms than the one Wiles and Taylor built. Yet their idea was far from a complete bridge. For it to work, mathematicians would first have to prove three major conjectures. It was, Calegari said, as if his paper with Geraghty explained how you could get to the moon—provided someone would obligingly whip up a spaceship, rocket fuel, and spacesuits. The three conjectures “were completely beyond us,” Calegari said.
In particular, Calegari and Geraghty’s method required that there already be a bridge going in the other direction, from automorphic forms to the Diophantine equations side. And that bridge would have to transport not just automorphic forms but also torsion classes. “I think a lot of people thought this was a hopeless problem when Calegari and Geraghty first outlined their program,” said Taylor, who is now at Stanford University.
Yet less than a year after Calegari and Geraghty posted their paper online, Peter Scholze—a mathematician at the University of Bonn who went on to win the Fields Medal, mathematics’ highest honor—astonished number theorists by figuring out how to go from torsion classes to the Diophantine equations side in the case of elliptic curves whose coefficients are simple complex numbers such as 3 + 2i or 4 − √5i. “He’s done a lot of exciting things, but that’s perhaps his most exciting achievement,” Taylor said.
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Peter Scholze won the prestigious Fields Medal in 2018, partly for his work in extending the Langlands bridge.PHOTOGRAPH: BARBARA FROMMANN/HAUSDORFF CENTER FOR MATHEMATICS
Scholze had proved the first of Calegari and Geraghty’s three conjectures. And a pair of subsequent papers by Scholze and Caraiani came close to proving the second conjecture, which involves showing that Scholze’s bridge has the right properties.
It started to feel as if the program was within reach, so in the fall of 2016, to try to make further progress, Caraiani and Taylor organized what Calegari called a “secret” workshop at the Institute for Advanced Study. “We took over the lecture room—no one else was allowed in,” Calegari said.
After a couple of days of expository talks, the workshop participants started realizing how to both polish off the second conjecture and sidestep the third conjecture. “Maybe within a day of having actually stated all the problems, they were all solved,” said Gee, another participant.
The participants spent the rest of the week elaborating various aspects of the proof, and over the next two years they wrote up their findings into a 10-author paper—an almost unheard of number of authors for a number theory paper. Their paper essentially establishes the Langlands bridge for elliptic curves with coefficients drawn from any number system made up of rational numbers plus simple irrational and complex numbers.
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In the fall of 2016, Ana Caraiani and Richard Taylor convened a “secret” workshop at the Institute for Advanced Study in 2016 that quickly resolved two outstanding problems and led to a major 10-author paper.PHOTOGRAPH: HAUSDORFF CENTER FOR MATHEMATICS; ROD SEARCEY
“The plan in advance [of the workshop] was just to see how close one could get to proving things,” Gee said. “I don’t think anyone really expected to prove the result.”
Extending the Bridge
Meanwhile, a parallel story was unfolding for extending the bridge beyond elliptic curves. Calegari and Gee had been working with George Boxer (now at the École Normale Supérieure in Lyon, France) to tackle the case where the highest exponent in the Diophantine equation is 5 or 6 (instead of 3 or 4, the cases that were already known). But the three mathematicians were stuck on a key part of their argument.
Then, the very weekend after the “secret” workshop, Vincent Pilloni of the École Normale Supérieure put out a paper that showed how to circumvent that very obstacle. “We have to stop what we’re doing now and work with Pilloni!” the other three researchers immediately told each other, according to Calegari.
Within a few weeks, the four mathematicians had solved this problem too, though it took a couple of years and nearly 300 pages for them to fully flesh out their ideas. Their paper and the 10-author paper were both posted online in late December 2018, within four days of each other.
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Soon after the secret workshop at the IAS, Frank Calegari (left), Toby Gee (center) and Vincent Pilloni, working with George Boxer (not pictured), found a way to extend the Langlands bridge beyond elliptic curves.COURTESY OF FRANK
“I think they’re pretty huge,” Emerton said of the two papers. Those papers and the preceding building blocks are all “state of the art,” he said.
While these two papers essentially prove that the mysterious telepathy between Diophantine equations and automorphic forms carries over to these new settings, there’s one caveat: They don’t quite build a perfect bridge between the two sides. Instead, both papers establish “potential automorphy.” This means that each Diophantine equation has a matching automorphic form, but we don’t know for sure that the automorphic form lives in the patch of its continent that mathematicians would expect. But potential automorphy is enough for many applications—for instance, the Sato-Tate conjecture about the statistics of clock solutions to Diophantine equations, which the 10-author paper succeeded in proving in much broader contexts than before.
And mathematicians are already starting to figure out how to improve on these potential automorphy results. In October, for instance, three mathematicians—Patrick Allen of the University of Illinois, Urbana-Champaign; Chandrashekhar Khare of the University of California, Los Angeles; and Jack Thorne of the University of Cambridge—proved that a substantial proportion of the elliptic curves studied in the 10-author paper do have bridges that land in exactly the right place.
Bridges with this higher level of precision may eventually allow mathematicians to prove a host of new theorems, including a century-old generalization of Fermat’s Last Theorem. This conjectures that the equation at the heart of the theorem continues to have no solutions even when x, y and z are drawn not just from whole numbers but from combinations of whole numbers and the imaginary number i.
The two papers carrying out the Calegari-Geraghty program form an important proof of principle, said Michael Harris of Columbia University. They’re “a demonstration that the method does have wide scope,” he said.
While the new papers connect much wider regions of the two Langlands continents than before, they still leave vast territories uncharted. On the Diophantine equations side, there are still all the equations with exponents higher than 6, as well as equations with more than two variables. On the other side are automorphic forms that live on more complicated symmetric spaces than the ones that have been studied so far.
“These papers, right now, are kind of the pinnacle of achievement,” Emerton said. But “at some point, they will just be looked back at as one more step on the way.”
Langlands himself never considered torsion when he thought about automorphic forms, so one challenge for mathematicians is to come up with a unifying vision of these different threads. “The envelope is being expanded,” Taylor said. “We’ve to some degree left the path laid out by Langlands, and we don’t quite know where we’re going.”
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
Looming Earnings Season Offers Next Test for Rebounding Stock Market
‘We haven’t really seen markets reflect the full extent of the damage that coronavirus is having to corporate profitability,’ one strategist says
The kickoff of earnings season this week will give investors a first glimpse of the impact of the coronavirus shutdown on corporate profits—and potentially clues about the outlook for the rest of the year.
Those results will offer a test for a stock market that is attempting to rebound after a bruising selloff. The pandemic is expected to cause a severe economic contraction and a sharp decline in corporate earnings in 2020. What remains unknown is the extent of the damage.
Companies from General Electric Co. to FedEx Corp. and Starbucks Corp. have warned they can no longer forecast their own results in a period of such uncertainty. Businesses across the country say revenue has evaporated following stay-at-home orders and the closure of nonessential businesses, leading them to furlough employees and drastically cut spending as they try to stay afloat.
Despite the turmoil, stocks have rallied over the past three weeks on early indications that social-distancing practices are helping to slow the spread of the virus. The S&P 500 climbed 12% last week, its best weekly performance since 1974, and it has rallied 25% from its March 23 low. The index is still down 14% for the year.
“It’s been remarkable to watch markets just climb higher and higher,” said Emily Roland, co-chief investment strategist at John Hancock Investment Management. “We haven’t really seen markets reflect the full extent of the damage that coronavirus is having to corporate profitability.”
For that reason, some analysts worry the stock market is on the cusp of a reckoning and another painful selloff could be in store if corporate profits plunge. Others fear Wall Street’s current earnings estimates don’t fully reflect the extent of the expected damage. Because projections for earnings are a key part of how stocks are valued, the opaque view into corporate profits suggests major indexes could see more volatility ahead.
Big banks including JPMorgan Chase & Co. and Bank of America Corp., along with health-insurance giant UnitedHealth Group Inc., transportation bellwether J.B. Hunt Transport Services Inc. and health-products company Johnson & Johnson, will be among the first big companies to open their books this week. While those results will be of great interest, investors will more carefully scrutinize comments from executives for indications of what will come later this year.
“There is more uncertainty for this quarter than almost any quarter I can remember,” said Bob Doll, chief equity strategist and senior portfolio manager at Nuveen. “My guess is somewhere between a half and three-quarters of analysts have yet to take a knife to their earnings [estimates] because they don’t know what knife to take to them and how deep to cut.”
The range of estimates reflects the deep uncertainty about the path ahead. FactSet projects a 9% year-over-year decline in earnings for all of 2020, based on analysts’ expectations for individual companies in the S&P 500, a sharp reversal from the 9.2% growth anticipated as last year ended.
Such a decline pales in comparison with the profit collapse forecast by big banks. BofA Global Research projects a 29% drop in per-share earnings this year, an estimate that incorporates “cataclysmic losses in travel, restaurants and other industries directly impacted by social distancing.” Goldman Sachs GS 4.13% has predicted profits will tumble 33%, while cautioning that in a more painful slowdown, the decline could be 57%.
The second quarter is expected to see the brunt of the damage based on the current scale of the shutdown. Profits among companies in the S&P 500 are projected to drop 21% in the current quarter after sinking 11% in the first three months of the year, according to FactSet estimates. In the second half of the year, profits are expected to continue shrinking, but at a slower pace, falling 9.6% in the third quarter and 1.6% in the fourth.
“We’ve never, ever, ever seen a sudden stop of the economy,” said Lisa Shalett, chief investment officer at Morgan Stanley Wealth Management. “That’s why this is going to be such a scrutinized petri dish, if you will, of an experiment for people to understand, really, what are the companies that have a true resilient and recurring revenue stream?”
First-quarter earnings forecasts have dropped for all 11 sectors in the S&P 500. The pain is projected to be particularly acute among companies in the consumer-discretionary group—a category that includes hotels, cruise lines and restaurants—where profits are expected to sink 32% from a year earlier, according to FactSet.
Marriott International Inc., for one, has begun furloughing what it expects will be tens of thousands of employees, while temporarily closing properties and curbing other spending. Shares of the company, which has withdrawn its financial guidance, are down 46% this year.
Meanwhile, earnings among energy companies, which have been hit both by an unprecedented drop in demand and the price war between Saudi Arabia and Russia, are expected to plummet 52%. Both Exxon Mobil Corp. and Chevron Corp. have slashed their capital spending plans in response to the crash in oil prices. Those stocks are off more than 30% in 2020.
Marriott International Inc., for one, has begun furloughing what it expects will be tens of thousands of employees, while temporarily closing properties and curbing other spending. Shares of the company, which has withdrawn its financial guidance, are down 46% this year.
Meanwhile, earnings among energy companies, which have been hit both by an unprecedented drop in demand and the price war between Saudi Arabia and Russia, are expected to plummet 52%. Both Exxon Mobil Corp. and Chevron Corp. have slashed their capital spending plans in response to the crash in oil prices. Those stocks are off more than 30% in 2020.
"We won't see a tourism comeback before 2021"
Assaf Goren, CEO of Israel tourism company Issta, sees the coronavirus changing customers' behavior patterns.
"Globes" spoke to Assaf Goren, CEO of tourism company Issta Lines (TASE: ISTA), about the impact of the coronavirus pandemic on his company and the tourism industry in general, and his plans for the day after.
Tell us about the current situation in the company and what actions you have taken in the face of the coronavirus crisis.
Goren: "At Issta, we found ourselves dealing within a short time with tens of thousands of requests for changes and cancellations, whether at the customer's initiative or because of flight cancellations by the airlines. All this happened in a short period of time, in which we were already working with limited manpower, with people working from home and having to become accustomed to a new and complicated way of working, which mainly involved following up cancellations with customers and with the airlines.
"There were calls from Israelis who were already overseas, and countless situations that had to be resolved. We built an automatic system to monitor changes by the airlines and notify customers of flight cancellations. This was dealt with in such a way that there was no need to contact us. We introduced an Internet system for rescue flights, and we set up an emergency team to deal with each of these channels."
Goren says that dealing with the airlines over returning money to customers whose flights had been cancelled was not easy. Many of them refused to refund fares and offered a credit against future travel instead. For many customers, however, the situation had changed, and money allocated to a vacation was now needed to live on. "The task is to refund the money by any means that will succeed, including lawsuits," Goren says.
How are you preparing for the day after the crisis ends?
"We have shareholders' equity, we will be capable of getting through the crisis, and we will adapt and change. We will try to identify opportunities and expand our activity. In the past two years we have invested resources in the "Issta 2020" program, and it will adapt itself to change and to the need to expand the technological tools available to travel agents.
"We will have to adapt ourselves to the new reality. There will be more people working from home, fewer people will come to the branches, and work methods will be more advanced. These processes make us more efficient.
"The crisis will speed up processes, and our customers too will be more exposed to technology. Every grandparent now knows how to use Zoom. Issta is has a presence all over the country, with 40 branches. I don't know whether all of them will remain, but we will maintain deployment from Kiryat Shemona to Eilat. There are people who like to physically meet an agent who already knows them and their preferences, and that's a service that can't be provided by a call center.
"We will also see changes in customer behavior patterns. The low-cost airlines taught us to make bookings a year in advance, but that won't happen anymore. There's a crisis of confidence between customers and retailers in every sector. Under the cloud of uncertainty, people will prefer the money to stay with them. In tourism, this will mean last-minute bookings." <pHow long do you think it will take before activity in the sector gets back to what it was before the crisis?
"In the throes of the crisis, it's hard to say when it will end. It depends on the degree to which people feel a sense of personal security, and on the state, as restrictions are slowly lifted. The first to be able to get back to work in the industry will be the hotels, which will be dependent on internal tourism. Tourism will return like a pair of compasses: first internal, after that the Mediterranean basin, then Europe, and then more distant destinations. It could be that instead of tourist visas there will be health visas. Other notions will become important to the consumer, such as standards of cleanliness.
"We estimate that fares will be higher, and airlines that survive will try to recompense themselves until they reach profitability. In internal tourism, the price will perhaps make it possible to bring vacationers back. I assume that we will start to see bookings as we get closer to the Jewish holiday season in the fall, but a return to full activity - I don't believe we'll see that before 2021. It depends on the Ministry of Finance and the government getting the economy moving forward. With high unemployment and a worsening recession, we know that tourism is a luxury."