Mathematicians have disproved a major conjecture about the relationship between curvature and shape.

n an old Indian parable, six blind men each touch a different part of an elephant. They disagree about what the elephant must look like: Is it smooth or rough? Is it like a snake (so thinks the man touching the trunk) or a fan (as the man touching the ear proposes)? If the blind men had combined their insights, they might have been able to give a correct account of the nature of the elephant. Instead, they end up fighting.

For decades, topologists have hoped to avoid falling into a similar trap. They thought they could characterize mathematical shapes by synthesizing numerous local measurements. But newly discovered, paradoxically curved spaces show that this isn’t always possible. “Things can be much more wild than what we thought,” said Elia Bruè of Bocconi University in Italy, who worked with two other mathematicians to demonstrate this.

Topologists stretch and compress the shapes they study. An infinitely thin rubber band, from a topological perspective, is equivalent to a circle, because you can easily deform it into a circular shape. Topologists tend to characterize shapes according to their global properties: Do they have holes, like a doughnut? Do they go on forever, like an infinite plane, or are they “compact” like the surface of a sphere? Do their “straight” lines go on indefinitely — making them what mathematicians call “complete” — or are there dead ends?

But as with the elephant in the parable, it can be hard to directly perceive the global nature of topological shapes. And so mathematicians want to understand their relationship to local geometric properties, like curvature. What can you say about a shape’s global topology, given information about how it curves at every point?

In 1968, John Milnor, a renowned mathematician then at Princeton University, conjectured that an average sense of a complete shape’s curvature was enough to tell us that it couldn’t have infinitely many holes. For the next 50 years, many results supported his claim. “You were tempted to believe it was true, because it was true in so many realistic cases,” said Jeff Cheeger of the Courant Institute of Mathematical Sciences at New York University. “And how in God’s name could you construct a counterexample to it?”

In this area of mathematics, said Vitali Kapovitch of the University of Toronto, “the Milnor conjecture was probably the biggest open problem.”

And so in 2020, Bruè and two colleagues set out to prove it. They ended up finding a counterexample instead — and built an entirely new kind of topological shape in the process. “It’s fantastic work,” Cheeger said. “A landmark.”

Full Article

posted by f.sheikh

(Israeli Prime Minister Benjamin Netanyahu, left, and National Security Minister, Itamar Ben-Gvir in Israel’s parliament last year. Credit…Abir Sultan/EPA, via Shutterstock)

Radical forces in Israeli society have moved from the fringes to the mainstream and put Israel’s democracy in peril. Here are the takeaways from our investigation.

For decades, most Israelis have considered Palestinian terrorism the country’s biggest security concern. But there is another threat that may be even more destabilizing for Israel’s future as a democracy: Jewish terrorism and violence, and the failure to enforce the law against it.

Our yearslong investigation reveals how violent factions within the Israeli settler movement, protected and sometimes abetted by the government, have come to pose a grave threat to Palestinians in the occupied territories and to the State of Israel itself. Piecing together new documents, videos and over 100 interviews, we found a government shaken by an internal war — burying reports it commissioned, neutering investigations it assigned and silencing whistle-blowers, some of them senior officials.

It is a blunt account, told in some cases for the first time by Israeli officials, of how the occupation came to threaten the integrity of the country’s democracy.

Settlers Pursuing a Theocratic State Have Become Lawmakers

Officials told us that once fringe, sometimes criminal groups of settlers bent on pursuing a theocratic state have been allowed for decades to operate with few restraints. Since Prime Minister Benjamin Netanyahu’s coalition government came to power in 2022, elements of that faction have taken power — driving the country’s policies, including in the war in Gaza.

The lawbreakers have become the law.

Bezalel Smotrich, the finance minister and the official in Netanyahu’s government with oversight over the West Bank, was arrested in 2005 by the Shin Bet domestic security service for plotting road blockages to halt the Israeli withdrawal from Gaza. He was released with no charges. Itamar Ben-Gvir, Israel’s national security minister, had been convicted multiple times for supporting terrorist organizations and, in front of television cameras in 1995, vaguely threatened the life of Prime Minister Yitzhak Rabin, who was murdered weeks later by an Israeli student.

Full Article of Take away

Original Full Article

posted by f.sheikh

To the question “Is she still relevant?” her Celebration Tour, which concluded this month, is proof that she is. Madonna performed before the largest audience ever gathered to watch a female artist and staged the single biggest free stand-alone concert in history: 1.6 million people turned Rio de Janeiro’s Copacabana Beach into a dance floor on May 4. According to Billboard, her six-month, 80-show tour grossed $225.4 million, making her the only woman in history to gross more than $100 million during each of six concert tours. (The only solo male in that category is Bruce Springsteen.)

But there’s so much more to her triumph than numbers. That a 65-year-old female pop star pulled off this tour and, despite our increasingly intolerant times, the performance was her most relentlessly and delightfully queer since 1990’s groundbreaking Blond Ambition Tour would be unimaginable, except that it was Madonna. The Celebration Tour proved that Madonna wasn’t afraid of drawing attention to her long career; she owned it proudly.

Full Article

posted by f.sheikh

From dawn till dusk, and dusk till dawn, she toils within the confines of an unjust patriarchal framework as a grandmother, mother, sister, lover, girlfriend, wife, partner, professional, worker and relationships in any other name. When misfortune strikes or accountability evades a man’s actions, the burden of blame often unjustly falls upon her shoulders. This societal phenomenon perpetuates the cycle of inequality, relegating women to the role of scapegoats for the failings of a system that thrives on their oppression. Whether it’s a trivial mishap or a grave error, the default response is to point fingers at the woman rather than addressing the root cause of the issue. This ingrained behaviour reflects not only a systemic imbalance of power but also a deep-seated reluctance to confront the inherent flaws within the patriarchal social, economic, political, religious, cultural, and family structure itself.

It becomes evident that the scapegoating of women worldwide serves to uphold the status quo, enabling men to avoid responsibility and perpetuate their dominance over women both in public and private sphere. This pattern of domestication of women is deeply entrenched in societal norms and expectations, making it challenging to dismantle without concerted effort and awareness. Furthermore, the consequences of this blame-shifting extend beyond individual interactions, shaping broader cultural attitudes and reinforcing gender stereotypes.

Blaming a woman’s cooking skills for a man’s stomach upset, faulting her attire and lifestyle for instances of rape, or holding her responsible for a man’s diminished libido based on her dress—all these instances underscore a disturbing pattern of shifting blame onto women for the shortcomings and misdeeds of men in society. This reflexive inclination to attribute fault to women, regardless of the circumstances, reflects a deep-seated bias ingrained within patriarchal structures.

When familial bonds falter due to the irresponsible and unaccountable behaviours of lazy, irresponsible men, it’s often she who bears the brunt of the blame, despite her potential role as a victim of such behaviours. This societal norm not only perpetuates injustice but also absolves men of their responsibility to introspect and address their actions. While she works tirelessly to sustain the household and support her family, the ingrained societal norm is to hold her accountable for any perceived failures or inadequacies, regardless of her actual culpability. This systemic bias not only undermines her efforts but also perpetuates a cycle of exploitation, inequality, and injustice.

Full Article

posted by f.sheikh