There are quite a few famous unsolved problems in mathematics. Some are so famous they have million dollar bounties on them that will be paid out to the first mathematicians who definitively either prove or disprove them.
And then there are other important conjectures out there that, in addition to academic recognition, might pay smaller rewards to the people who crack them. But which if they are cracked, can be even more valuable if they can successfully be put to use.
One of those smaller conjectures is the Talagrand Convexity Conjecture. Proposed by French mathematician Michael Talagrand in 1995, Talagrand famously offered a $2,000 prize to anyone who solved it. Here is Jules Aknin's simplified description of the conjecture:
Talagrand’s convexity conjecture is a statement about high‑dimensional geometry: even in enormous, messy clouds of points, simple convex shapes are guaranteed to appear. In other words, no matter how chaotic the configuration looks, there is unavoidable structure hiding inside it.
If such structures are truly unavoidable, they can open the door to practical applications, including in financial markets. Aknin describes how they might contribute to finding order within chaos:
At a deeper level, results like this sit exactly at the intersection of geometry and probability. They don’t just create “pretty shapes” in higher dimensions, they show how structure emerges inside high‑dimensional random systems, which is the same conceptual problem we face when building models for complex datasets. In fact, as the original article notes, this kind of unification between geometric and probabilistic thinking could eventually influence how machines process high‑dimensional data and how we design algorithms that operate in those spaces.
[...]
On the surface, a high‑dimensional cloud of points in geometry and the Australian equity market do not look related. But in practice, we deal with a very similar object every day: hundreds of stocks, multiple time horizons, momentum profiles, risk factors, macro shocks and behavioural flows all interacting at once.
From a distance, that system may appear as pure noise. Talagrand’s result reinforces an idea that systematic managers have believed for a long time: structure is not optional, it is inevitable; the real question is whether you have the tools and discipline to find it.
Even though it doesn't carry the million-dollar prize of mathematics most famous unsolved problems, the stakes for those who can successfully build on the proven conjecture are still very high. That work is quite possibly worth a lot more than Talagrand's two-thousand dollar prize for the proof itself.
A team of three mathematicians stand to collect Talagrand's $2,000. Merrick Hua, Antoine Song, and Stefan Tudose posted a preprint paper of their proof on 11 May 2026, in which they converted the problem from one involving geometry to one involving probability and combinatorics. Like many math stories this year, AI is involved, but unlike those other stories, it's only a bit player.
The new proof was worked out by Dongming Hua and Antoine Song from the California Institute of Technology, and Stefan Tudose from Princeton University, who joined the other authors after hearing about their work. Together, the mathematicians reformulated Talagrand's geometric conjecture to a problem of probability theory and random vectors. In their paper published on the arXiv preprint server, they proved an equivalent conjecture for probability, showing that any 1-subgaussian random vector in n dimensions can be expressed as the sum of three standard Gaussian random vectors.
This result solves Talagrand's convexity problem, proving that for any large enough set in Gaussian space, a convex set of significant measures can be found inside a triple sum of the original set. The solution also confirms a combinatorial analog of the problem, which is important for discrete mathematics.
Initially, Song and Hua say they attempted to work out a solution with the help of ChatGPT. However, while the LLM helped to answer some of their questions and move them closer to a solution, it was Tudose who provided the final proof. Ultimately, the team did not use the work done with ChatGPT. In their paper, the team writes that Tudose's proof was "more general and conceptual."
It's a remarkable achievement. It is also a harbinger of the kind of role AI will find as just another tool used by people.
Image credit: Texture of an iced leaf image by Daniela deGol on Wikimedia Commons. Creative Commons CC By 4.0 Attribution 4.0 International Deed.
The U.S. new home market has largely recovered from the disruption of January 2026's blizzards. Unfortunately, rising mortgage rates combined with an uptick in the average sale price of new homes to reduce the quantity of sales. This combination of factors resulted in the total valuation of new homes sold in April 2026 to decline below the levels recorded a month earlier.
Political Calculations' initial estimate of the total value of new home sales in the United States during April 2026 is $28.30 billion. This value is slightly higher than the initial estimate of $28.24 billion for March 2026, but has declined from a revised value of $28.43 billion for the month.
The number of new home sales continues to hold relatively steady. The initial estimate of the annualized trailing twelve month average of the total number of new home sales for April 2026 is 665,000. This value falls below the range of 671,000 and 684,000 that had held since January 2024.
The initial estimate of the trailing twelve month average of a new home sold in April 2026 is $521,300. New home prices have generally rising since bottoming at $502,525 in September 2024. The average remains below the peak of $529,692 recorded for June 2022 at the height of the high inflation unleashed by the Biden administration.
All these figures represent time-shifted, partial trailing twelve month averages for each data series, which will be subject to revision for the next ten months before being finalized. The following charts present the U.S. new home market capitalization, the number of new home sales, and their average sale prices as measured by their time-shifted, trailing twelve month averages from January 1976 through April 2026.
Declining trend for new home sales:
Rising trend for new home prices:
New home sales were reported to have surged in March as prices fell to a five-year low, but much of this boost in sales may represent a springback from the impact of blizzards in much of the U.S. in January 2026 that shrank sales far below expectations.
Bloomberg confirms the April 2026 sales slump for new homes was not expected:
Sales of new US homes declined in April by more than forecast as builder incentives failed to motivate potential buyers at the start of the spring selling season.
Purchases of new single-family homes decreased 6.2% from March to a 622,000 annualized pace, according to government data released Thursday. Economists expected a 660,000 rate, based on the median estimate in a Bloomberg survey.
It would seem the springback in sales from January 2026's blizzards was truly that and not the start of an upward trend.
References
U.S. Census Bureau. New Residential Sales Historical Data. Houses Sold. [Excel Spreadsheet]. Accessed 28 May 2026.
U.S. Census Bureau. New Residential Sales Historical Data. Median and Average Sale Price of Houses Sold. [Excel Spreadsheet]. Accessed 28 May 2026.
Image Credit: High angle shot of suburban neighborhood photo by David McBee on Pexels.
Labels: real estate
Six months and two earnings seasons have come and gone since Thanksgiving 2025 when we were introduced to the worst performing stocks of the S&P 500 (Index: INX). How many of those 10 stocks have seen their fortunes improve and how many are proving to be an even bigger investment turkey than they appeared on the day after last Thanksgiving?
Let's cut to the chase! Here are the relative winners as measured by the percentage of their stock price recorded value on 28 November 2025:
- Dow Inc. (NYSE: DOW) - 148.1%.
- Deckers Outdoor (NYSE: DECK) - 126.6%.
- Molina Healthcare (NYSE: MOH) - 118.9%.
The stock price of these S&P 500 component companies are all higher than they were on 28 November 2026 and are also beating the S&P 500's growth, which has risen to 109.8% of its day-after-Thanksgiving-Day-2025 level. What each of these companies have in common is improved business performance combined with an improved outlook for their earnings.
Meanwhile, all seven of the other Thanksgiving Leftover stocks have experienced continuing declines in their stock prices. Here they are, ranked from best-to-worst performing over the past six months:
- Chipotle Mexican Grill (NYSE: CMG) - 93.6%
- Fiserv (NASDAQ: FISV) - 90.5%
- Alexandria Real Estate Equities (NYSE: ARE) - 90.3%
- Factset Research Systems (NYSE: FDS) - 83.3%
- Lululemon Athletica (NASDAQ: LULU) - 69.1%
- Gartner (NYSE: IT) - 67.8%
- Trade Desk (NASDAQ: TTD) - 56.1%
The following spaghetti chart shows how each performed throughout the last six months:
Since our last update, Lululemon Athletica (NASDAQ: LULU) has taken the most negative turn for the worse. The athletic apparel company is struggling to sell its mostly foreign-made clothing line in the U.S. after hiking prices to cover the cost of new tariffs. But higher costs are not the "athleisure" clothing company's biggest problem. Its latest products have been on the wrong side of fashion trends as it faces increased competition.
If that weren't enough, the company's top management is involved in war of words with Chip Wilson, the company's founder, who criticized them for losing the company's "cool" factor.
TLDR: Poorly managed Lululemon has become costly and unfashionable with few indications that will change anytime soon, sending its stock price even lower.
Breaking away from the ongoing drama of a failing business, the next chart reveals how the performance of the Thanksgiving 2025 Leftover stocks compares as a group with the S&P 500 index, both as a market-cap weighted index and as an equal-weighted index.
By both grouping methods, the Thanksgiving Leftover stocks are substantially underperforming the S&P 500 index.
Compared to a month ago, the equal-weighted group is close to the same, but the market-cap weighted group is worse off. If you went double-or-nothing in betting whether the Thanksgiving Leftover stocks were going to be doing better or worse than they were a month ago, we'd have to give the edge to worse this month.
Labels: SP 500, stock prices
The S&P 500 (Index: SPX) rose 0.8% above its previous week's close to end the trading week at 7,473.45 on Friday, 22 May 2026 as investors went into the Memorial Day holiday weekend.
In doing so, the index confirmed that stock prices have fully recovered from the Iran war impact. We find the trajectory of the S&P 500 falls very close to the central trend line of the redzone forecast range we added to the alternative futures chart back on 23 February 2026, several days ahead of when the geopolitical event began. For us, that timing has been fortunate because the redzone forecast range has been able to function as a counterfactual projection of how the S&P 500 would have changed if the Iran war geopolitical event had never happened.
So for the trajectory of stock prices to fall so neatly near the middle of that forecast range now that we're coming to its end is a strong confirmation the negative shock of the geopolitical event upon them has fully waned. Here is the latest update of the alternative futures chart that shows that outcome:
The redzone forecast range is based on the assumption investors would primarily focus on 2026-Q2 in making the decisions that set the trajectory of stock prices throughout its run. The Iran War geopolitical event didn't alter that focus throughout this period, but rather, added noise on top of the signal provided by the dividend futures-based model we use to forecast the S&P 500's future.
There are other factors that affect both the signal and noise investors consider in their decision making, which is provided from the random onset of new information. Here are the past week's market-moving headlines.
- Monday, 18 May 2026
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- Signs and portents for the U.S. economy:
- China to buy at least $17 billion in US agricultural products annually, White House says
- Explainer: What do China's new US farm purchases mean for global trade?
- White House gets small rare earth win, but China's export regime is here to stay
- Oil prices rise 3% to two-week high on Iran war supply concerns
- Trump says he’s postponing ‘scheduled attack of Iran tomorrow’ at Middle East leaders’ request
- Former Fed minions claim size doesn't matter for some reason:
- Bigger trouble developing in China:
- Wall Street finished mixed as tech slid ahead of Nvidia earnings
- Tuesday, 19 May 2026
-
- Signs and portents for the U.S. economy:
- Fed minions told to stop bitterly clinging onto assets they bought, expected to avoid rate hikes in 2026:
- Warsh details first round of asset divestments
- U.S. Fed to avoid cutting rates this year; economists still say war-driven inflation is transitory
- Bigger trouble developing in China:
- BOJ minions rethinking plan to reduce Japanese bond holdings as JapanGov minions gear up to borrow more:
- Japan's solid first-quarter GDP faces reality check as Iran war threatens economy
- Exclusive: Japan's extra budget to include funding from fresh debt
- BOJ could temper its taper as bond wobbles rattle markets
- Bigger trouble avoided in Eurozone:
- Wall Street closed lower as Treasury yields jumped amid inflation fears
- Wednesday, 20 May 2026
-
- Signs and portents for the U.S. economy:
- The AI bots are coming and the young are booing, not applauding
- As cocoa prices melt down, real chocolate is making a comeback
- Oil prices slide after Trump says US-Iran negotiations in 'final stages'
- Iran deal looks like a 30-day ceasefire extension - Eurasia Group
- Tankers exit Strait of Hormuz with 6 million barrels of crude oil
- Low unemployment streak revives debate over job market's new floor
- Fed minions outed for hiding their growing excitement for rate hikes:
- Fed minutes show more policymakers open to a rate hike
- Fed's Paulson says current policy appropriate, but 'healthy' to consider extended hold or hikes
- Bigger trouble, no stimulus developing in China:
- Exclusive: US not in a hurry to extend China trade truce, Bessent says
- China leaves lending benchmarks unchanged for 12th month in May
- BOJ minions coordinating with US on plan to hike Japan's interest rates:
- Bigger trouble developing for the Eurozone, ECB minions gearing up to hike Eurozone interest rates:
- Europe must break China's grip on rare earths pricing to spur investment, sector body says
- ECB June rate hike case is nearly sealed but July is fully open, sources say
- Wall Street ended higher as investors eye possible U.S.-Iran deal and Nvidia earnings
- Thursday, 21 May 2026
-
- Signs and portents for the U.S. economy:
- US manufacturing activity rises to four-year high in May, S&P Global survey shows
- US energy storage additions hit a first-quarter record, report shows
- Oil prices close 2% lower on uncertain prospects for US-Iran deal
- Fed minions finding other things to worry about:
- Some Fed officials and staff are fretting about state of financial markets
- Fed proposes limited payment accounts for fintechs, others
- US Fed officials propose prolonging dollar swap lines to underpin stability
- Fed's Barkin: Policy in good place to respond to ongoing shocks
- BOJ minions / Bigger trouble developing in Japan:
- Japan exports rise for eighth month despite supply disruptions
- Japanese auto exports to Middle East plunge in April as war disrupts shipping
- Bigger trouble developing in Eurozone, ECB minions getting excited to hike Eurozone interest rates:
- Europe's economy hit hard as energy price shock stifles demand
- ECB's Rehn sees few signs yet of high inflation taking root
- S&P 500, Nasdaq eke out gains in volatile trading session
- Friday, 22 May 2026
-
- Signs and portents for the U.S. economy:
- US labor market shows resilience in face of war; housing still subdued
- Oil prices rise as investors doubt US-Iran peace talks breakthrough
- Fed minions backing off rate cut plans as new chief minion starts work:
- Fed's Waller's ready to axe 'easing bias,' says rate-cut talk now is 'crazy'
- Kevin Warsh is sworn in as Federal Reserve chair
- Bigger trouble developing in China:
- Chinese Refining Rates Unexpectedly Plunge To All-Time Lows As Economy Falls Off A Cliff
- China shipped a little fuel to Southeast Asia in April but curbs still bite
- China kept building its crude stockpile in April despite Iran crisis
- "Huge Shock": China Launches Crackdown On Cross-Border Stock Selling To Block Capital Outflows
- BOJ minions get core inflation data that's too low, look to energy shock to justify rate hike plans:
- Eurozone minions admit Eurozone economy not heading to good place:
- ECB minions thinking harder about ditching their perfectly positioned monetary policy to hike Eurozone interest rates:
- S&P 500 finishes week higher with nine sectors close in green
The CME Group's FedWatch Tool moved up the expected timing of a quarter point increase in the Federal Funds Rate to 28 October (2026-Q4). The FedWatch tool also now anticipates another quarter point rate hike will come on 28 April (2027-Q2), with a strong probability its timing could also move earlier.
The Atlanta Fed's GDPNow toolestimate of real GDP growth for the U.S. economy in the current quarter of 2026-Q2 increased to +4.3%, up from the +4.0% it projected a week earlier.
Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull and bear who are at Indianapolis watching cars labeled 'Dow', 'SP 500', and 'NASDAQ' race around the track".
Which states are the most and least affordable places for American families to live after paying taxes and their essential expenses?
The Common Sense Institute tallied up the numbers and ranked each state after subtracting federal and state taxes and also essential expenses like housing, utilities, groceries, auto and health insurance, fuel, and childcare in each state from the paychecks for a family of four with two adult breadwinners who work full time and earn the state's median hourly income.
Visual Capitalist's Dorothy Neufield then revisualized the results to focus on how much that of the modeled families' income remained. Here's her version of the Common Sense Institute's map:
Here's her analysis of the most and least affordable states:
In top-ranked states like Iowa, households keep nearly 35% of their income, about $2,900 per month. In Hawaii, that figure drops to just 9%. That’s a difference of more than $2,000 per month in disposable income.
[...]
Midwestern states dominate the rankings, largely due to lower housing and childcare costs.
Iowa ranks first, with households keeping 34.7% of their income, followed by South Dakota (34.6%) and North Dakota (33.5%).
[...]
In the least affordable states, families spend up to 91% of their income on essentials and taxes, leaving little room for savings or unexpected expenses.
Hawaii families are most strained, with 9% of income left, followed by California at 10.9%. Between 2019 and 2025, California households saw one of the largest declines in affordability across states.
Massachusetts, despite high incomes, ranks near the bottom. Childcare alone consumes 24% of household income, showing how a single cost category can erode income advantages.
The Common Sense Instutute also looks at how each state's affordability has changed from 2019 to 2025. They find that Kansas, New Mexico, and Utah have seen their cost of living fall the most, while Rhode Island, Massachusetts, and California have seen the biggest escalation in living expenses over these years.
References
Dorothy Neufield. Mapped: Where Americans Keep Most of Their Paycheck. Visual Capitalist. [Online article]. 27 April 2026.
Labels: data visualization, personal finance
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