There are many kinds of mathematics. So many, in fact, that before we go any further, it might help to take an 11 minute video tour of the map of mathematics, which is a lovely general overview to an endeavor that only begins with counting numbers.

Within each of the fields on the theoretical side of the map, mathematicians have developed specialized tools for solving the problems they've tackled. But the tools developed to work in one theoretical field have often only proven to be useful within that field. Mathematicians working in other fields haven't been able to deploy them to solve the problems they are working upon.

Which has been disappointing because some of those tools are very powerful. What if it were possible to translate problems from other fields into ones that those powerful tools can solve?

That has been the challenge and promise of the Langlands project, which seeks to connect different fields of math by uncovering a shared language between them. Much as archaeology's Rosetta Stone made it possible to translate ancient Egyptian hieroglyphs and demotic scripts by connecting them to written Greek.

In 1967, mathematician Robert Langlands conjectured it would be possible to connect several different fields in mathematics, which if successful, would make it possible to used the tools developed in each to gain insights in the others.

From here, lets turn to Quanta Magazine's Erica Klarreich's reporting on a mathematical breakthrough:

A group of nine mathematicians has proved the geometric Langlands conjecture, a key component of one of the most sweeping paradigms in modern mathematics.

The proof represents the culmination of three decades of effort, said Peter Scholze, a prominent mathematician at the Max Planck Institute for Mathematics who was not involved in the proof. “It’s wonderful to see it resolved.”

The Langlands program, originated by Robert Langlands in the 1960s, is a vast generalization of Fourier analysis, a far-reaching framework in which complex waves are expressed in terms of smoothly oscillating sine waves. The Langlands program holds sway in three separate areas of mathematics: number theory, geometry and something called function fields. These three settings are connected by a web of analogies commonly called mathematics’ Rosetta stone.

Now, a new set of papers has settled the Langlands conjecture in the geometric column of the Rosetta stone. “In none of the [other] settings has a result as comprehensive and as powerful been proved,” said David Ben-Zvi of the University of Texas, Austin.

“It is beautiful mathematics, the best of its kind,” said Alexander Beilinson, one of the main progenitors of the geometric version of the Langlands program.

The proof involves more than 800 pages spread over five papers. It was written by a team led by Dennis Gaitsgory (Scholze’s colleague at the Max Planck Institute) and Sam Raskin of Yale University.

Please do click over to Quanta to find out more. Their coverage is an excellent entry point for understanding Langlands conjecture and how the mathematicians who published their proof of it some 57 years later tackled the challenge.

It's a massive proof and if it holds, a massive achievement. It is also a strong candidate for being the biggest math story of 2024, which is just a little over half over. What other stories might join it by the end of the year?

Gold is a hedge against inflation. Gold is sensitive to expected long-term real interest rates. Gold is regarded as protective against "bad economic times".

Each of these statements represents a factor that has been cited as a driver of the price of gold. In November 2021, four economists at the Federal Reserve Bank of Chicago tested each of these propositions, publishing their analytical results in a Chicago Fed Letter.

Their analysis found evidence to support all three statements. Here's the conclusion to their essay summarizing their results:

We have investigated several hypotheses about the determinants of gold prices—in annual levels data, quarterly data in innovations form, and daily data in differences. The negative effect of real interest rates on gold prices predicted by theory holds in all three contexts. Two of the three specifications (the quarterly innovations specification being the exception) support the notion that gold is an inflation hedge and that this effect is quantitatively larger than the real interest rate effect. The two specifications that can be used to evaluate the proposition that gold prices also reflect protection against bad economic times are highly supportive of it. In the early part of the sample, variation in inflation or inflationary expectations was the single most important consideration for the real price of gold. From 2001 on, however, long-term real interest rates and pessimism about future economic activity appear as the dominant factors. While disinflation since 2001 might have been expected to result in low gold prices, any effect of low inflation was more than compensated for by unprecedentedly low long-term real interest rates and by pessimism about future economic activity.

That was in 2021. In the years since then, it would seem at least one of the propositions isn't holding the way it used to hold. And it's the big one they found held regardless of how they parsed their data: the negative effect of real interest rates on gold prices.

What's meant by that is they found the price of gold has an inverse relationship with real interest rates. After adjusting for inflation, when interest rates rise, the price of gold falls. And vice versa: the price of gold rises when real interest rates fall.

Since 16 March 2022 however, as the Federal Reserve started increasing interest rates and succeeded in causing real long-term interest rates to rise, the price of gold hasn't behaved they way. Real interest rates are now much higher than they were at that time, but instead of falling, the price of gold has risen and is now at or near record highs. The following chart shows both the pre-March 2022 inverse relationship between gold prices and real long-term interest rates and the new relationship, or rather, the new lack of relationship, between the two in the period since.

We've previously described the behavior of gold prices since 16 March 2022 as representing a paradigm shift. We think that new paradigm is like the old one, but is resetting the relative level of the price of gold at a much higher level than it was previously. It has happened as other factors driving the price of gold have become dominant over the level of real interest rates.

The behavior of gold prices since 1 March 2024 suggests that process has not yet settled. The price of gold is now ratcheting up without any significant change in real interest rates. Which factor do you suppose is more dominant right now? Are people buying gold because they're more worried about inflation and are hedging against it or are they buying gold because they're seeking to protect themselves against bad economic times?

References

Robert B. Barsky, Craig Epstein, Adrian Lafont-Mueller, and Younggeun Yoo. What drives gold prices? Chicago Fed Letter: Essays on Issues. November 2021, No. 464. [Online article]. DOI: 10.21033/cfl-2021-464.

Image Credit: A yellow background with lots of small circles photo by Lucas K on Unsplash

Later this week, on Thursday, 25 July 2024, the Bureau of Economic Analysis will publish its first estimate of the United States' Gross Domestic Product during the second quarter of 2024.

With that date just a day away, it's a good time to check in with how 2024-Q2's GDP tracks with what a momentum-based forecasting method projected it would be for this quarter over seven months ago. That simple method, called the "Climbing Limo", uses nominal GDP data that was available back in December 2023 in its projections.

As you can see in the following chart, that method came within one percent of anticipating the final GDP estimate for the first quarter of 2024 (2024-Q!), the data for which only became available last month.

Coincidentally, the Atlanta Fed's GDPNow forecast for 2024-Q1 was correct in projecting 2024-Q1's actual GDP exceeded the climbing limo's momentum-based forecast for this period.

Looking forward, since the GDP data for 2024-Q1 has been finalized, the Climbing Limo projection of GDP built using that data point suggests the United States' nominal GDP is on track to rise through the end of 2024 at a steady pace.

Update 25 July 2024

The BEA's first estimate of nominal GDP for 2024-Q2 is $28,629.2 billion (nominal = not adjusted for inflation). That figure is 0.8% below the Climbing Limo's forecast of $28,949.4 billion for 2024-Q2, as the U.S. economy would appear to be returning to underperforming the forecasting method's projection of where it would be based on its previous momentum.

References

U.S. Bureau of Economic Analysis. National Income and Product Accounts. Table 1.1.5. Gross Domestic Product. [Online Database]. Accessed 27 June 2024.

Political Calculations. Forecasting GDP Using the Climbing Limo. [Online Tool]. 10 May 2005.

Image Credit: Microsoft Copilot Designer. Prompt: "A simple sketch of a limousine driving uphill toward the right side of a rising zig zag line chart".

One month ago, Political Calculations' monthly snapshot of the future for the S&P 500's quarterly dividend payouts dimmed for the first time in months.

This time around, we have better news. The outlook for the remaining quarters of dividends for the S&P 500 (Index: SPX) improved over the past month.

You can see how that outlook changed from one month ago in the following animated chart, which shows what the index' dividend outlook looked like on our 24 June 2024 snapshot and our new snapshot from 19 July 2024. It shows 2024-Q3's expected cash dividends increased from $18.18 to $18.29 per share, while the projected dividend payout for the final quarter of 2024 rose from $18.75 per share to $18.90 per share.

If you're accessing this article on a site that republishes our RSS news feed, you may need to click through to our site to access a working version of the animated chart.

For 2024-Q3, the increase in forecast dividends falls just a penny per share below the level they had reached in our 14 May 2024 snapshot. The 19 July 2024 projection of the S&P 500's dividend payout for 2024-Q4 however is higher than had been forecast in that earlier snapshot.

More About Dividend Futures Data

For this series, we have been taking a snapshot of the CME Group's S&P 500 quarterly dividend futures data shortly after the second or third week of each month.

Dividend futures indicate the amount of dividends per share to be paid out over the period covered by each quarter's dividend futures contracts, which start on the day after the preceding quarter's dividend futures contracts expire and end on the third Friday of the month ending the indicated quarter. So for example, as determined by dividend futures contracts, the now "current" quarter of 2024-Q3 began on Saturday, 22 June 2024 and will end on Friday, 20 September 2024.

That makes these figures different from the quarterly dividends per share figures reported by Standard and Poor. S&P reports the amount of dividends per share paid out during regular calendar quarters after the end of each quarter. This term mismatch accounts for the differences in dividends reported by both sources, with the biggest differences between the two typically seen in the first and fourth quarters of each year.

Image Credit: Microsoft Copilot Designer. Prompt: "A crystal ball with the word 'SP 500' written inside it". And 'Dividends' written above it, which we added.

S&P 500 (Index: SPX) investors were rattled in the trading week ending Friday, 19 July 2024. The index dropped two percent to close the week at 5,505.00.

That decline was triggered by the Biden-Harris administration's announcement on Wednesday, 17 July 2024 that it was planning to expand its anti-free trade restrictions against China. The new sanctions would negatively affect U.S. advanced computer chip manufacturers as well as Japanese and Dutch chipmakers that do large volumes of business with China.

The announcement sent the stock prices of U.S. chipmakers plunging, as the tech-heavy Nasdaq 100 index experienced its worst day since 2022, going on to lose 4.3% by the end of the week. The third largest company in the S&P 500, AI chipmaker Nvidia (Nasdaq: NVDA), lost $244 billion (8.75%) of its total value from the previous week as it dropped to a market capitalization of $2.9 trillion. Coincidentally, the size of that loss is about $1 billion less than the entire market cap of Advanced Micro Devices (Nasdaq: AMD) after it shrank by 16.5% from its previous week's market valuation.

The dividend futures-based model's alternative future chart shows what appears to be a new Lévy flight event, in which investors have shifted their investment time horizon from the current quarter of 2024-Q3 to the more distant future quarter of 2025-Q2, which may coincide with the timing of when the new export rules may take effect.

Trading during the week showed little sign of any impact from the Saturday, 13 July 2024 assassination attempt against former U.S. President Donald Trump. The change in stock prices for the S&P 500 on Monday, 15 July 2024 fell well below the threshold of a 2% change from the previous trading day's close that would quality as interesting.

With corporate earnings season getting underway, it's quite possible that the random onset of new information it provides may soon prompt investors to shift their focus back to the current quarter. Or not. It depends on what new information comes out in the week's ahead.

Speaking of which, here are the week's market moving headlines.

Monday, 15 July 2024

• Signs and portents for the U.S. economy:
• Oil prices ease on demand concerns in China
• Yield Curve: US30Y jumps above the US2Y for the first time since late January
• Fed minions say U.S. inflation is heading lower to their 2% target:
• Fed's Powell: Latest data 'add somewhat to confidence' inflation is returning to 2%
• Bigger trouble, stimulus, bailouts developing in China:
• China's economy falters, raises pressure for more stimulus
• China's GDP Growth Unexpectedly Tumbles As New Home Prices Plunge Most In 9 Years
• China new home prices fall at fastest pace in 9 years, more support needed
• China's Q2 GDP grows 4.7% y/y, below forecast
• Chinese Regulators Intensify Efforts To Tighten Control Over Financial Markets
• Nasdaq, S&P, Dow end higher on first trading day since assassination attempt on Trump
• Trump Media skyrockets 70% as markets reopen following assassination attempt
• Game-Changing CPI Report May Drive Big Rotation Towards Value And SmallCap

Tuesday, 16 July 2024

• Signs and portents for the U.S. economy:
• Oil prices decline over 1% on Chinese demand jitters
• IMF says Fed minions shouldn't rush to cut U.S. short term interest rates:
• No rush for US Fed to cut rates, IMF's chief economist says
• Bigger stimulus, bailouts developing in China:
• China tries to hit more birds with one stone in property rescue push
• BOJ, JapanGov minions secretly involved in propping up Japan's currency, letting failing businesses finally go under:
• Japan keeps up yen warnings; 6 trillion yen, 2-day intervention suspected
• Japan finds a 'stealth' cure for zombie businesses: Let them fail
• Possible growth signs developing in Eurozone:
• Euro zone households seek loans again after two years - ECB
• Dow jumps more than 700 points on UnitedHealth boost; rotation into small-caps continue
• Dow, small caps rally as rate-cuts bets hold firm
• UnitedHealth hits seven-month high after Q2 beat

Wednesday, 17 July 2024

• Signs and portents for the U.S. economy:
• Oil prices steady as declining US inventories offset slowing China demand
• US Floats Tougher Trade Curbs in Chip Crackdown on China
• US single-family housing starts hit eight-month low; green shoots in manufacturing
• Fed minions "optimistic" inflation will drop to their 2% target, thinking about cutting rates:
• Fed's Kugler cautiously optimistic inflation headed to 2% target
• Top Fed officials say they are 'closer' to cutting interest rates
• Fed's Williams suggests rate cut could be warranted in coming months, WSJ reports
• Fed's Waller says timing of interest rate cut 'drawing closer'
• Bigger trouble, stimulus developing in China:
• China's June jobless rate for 16-24 year olds at 13.2%, stats bureau says
• Global chip sell-off slams Nasdaq, which notches worst day since 2022; Dow tops 41K
• Chip stocks shed over $500 billion in value on China trade fears
• Gold posts second-highest close as U.S. rate cut hopes rise, Treasury yields fall

Thursday, 18 July 2024

• Signs and portents for the U.S. economy:
• Oil prices rise on bigger-than-expected drop in US crude stocks
• Key U.S. mortgage rate drops to lowest since March, Freddie Mac says
• Fed minions say they're not okay yet with inflation, pitch new way for banks to tap into bailout money:
• Fed's Daly says 'not there yet' on price stability
• Fed's Logan hails progress in prepping banks for discount window
• Bigger trouble, stimulus developing in China:
• China's stubborn savers ignore inducements to spend as growth slows
• China reaffirms lofty policy goals, offers no implementation details
• BOJ minions seeking ways to keep stimulus alive:
• BOJ wants to maintain accommodative monetary environment, Jiji reports
• ECB minions choose to sit on hands, will think about cutting rates later:
• ECB keeps rates unchanged, offers no guidance on next move
• ECB keeps rates on hold, leaves options open for Sept
• ECB hawks open to September rate cut if data supports it, sources say
• Nasdaq, S&P slip, Dow sheds 500 points as tech rotation intensifies; Netflix in focus
• Wall Street ends sharply lower as anxiety rises, earnings heat up

Friday, 19 July 2024

• Signs and portents for the U.S. economy:
• Global tech outage disrupts industries, highlights online risks
• What is CrowdStrike, the cybersecurity firm behind a global tech outage?
• Without backup plans, global IT outages will happen again
• US regional banks build stronger safety nets as CRE pain lingers
• US regional banks' Q2 profits squeezed by deposit costs, tepid loan demand
• Oil settles at one-month low on Gaza ceasefire hopes
• Fed minions looking forward to getting inconclusive data:
• Fed faces wave of data before deciding on end-of-summer rate cut
• Bigger trouble, stimulus developing in China:
• Chinese officials expect bumpy ride for economy after policy meeting
• China to boost support for equipment upgrades and consumer goods trade-ins
• BOJ minions to hold rates steady despite inflation pressure:
• BOJ to forego July rate hike, over three-quarters of economists say: Reuters poll
• Japan's capital seen facing increased inflation pressure: Reuters poll
• ECB minions thinking about cutting Eurozone interest rates after passing on cuts this month:
• ECB policymakers back more rate cuts as inflation heads back to goal
• Wall Street posts worst week in three months; focus turns to upcoming earnings deluge
• S&P 500 falls nearly 2% for worst week since mid-April as traders shun big tech
• Wall Street falters, CrowdStrike slumps on global tech outage

The CME Group's FedWatch Tool forecast is mostly unchanged this week. It continues to anticipate the Fed will hold the Federal Funds Rate steady in a target range of 5.25-5.50% until 18 September (2024-Q3), at which time, the Fed is expected to start a series of 0.25% rate cuts that will occur at 6-to-12-week intervals at least into mid-2025.

The Atlanta Fed's GDPNow tool's forecast of the annualized real GDP growth rate during 2024-Q2 continued rising to +2.7% from the +2.0% growth projected a week earlier. When the BEA's official first estimate of GDP in 2024-Q2 is released near the near of July 2024, the GDPNow tool will shift to start forecasting 2024-Q3's real GDP growth rate.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a worried Wall Street bull and a bear moving pieces around a game board".