- Presidents Obama and Putin have been at odds all year
- Russia angered Obama when it gave temporary asylum to Edward Showden
- Putin signed an anti-gay propaganda law that offended Obama
- Obama asserts Syria uses chemical weapons, but Putin wants proof
(CNN) -- Presidents Barack Obama and Vladimir Putin of Russia meet face-to-face this week after a long summer of antagonism.
Will the friction between them ease? Anyone fascinated by how power relationships evolve will be watching the two world leaders during this week's Group of 20 nations summit in Russia. Although the pair have no formal meetings scheduled, they likely will have opportunities to chat.
To say that the two men are at odds is putting it mildly.
Edward Snowden: fugitive or refugee?
The Obama administration made former National Security Agency contractor Edward Snowden perhaps the most wanted man in the world. Snowden faces espionage charges after he admitted leaking top-secret details about U.S. surveillance programs.
Are Obama and Putin 'frenemies'? 
Why Russia is sticking by Syria The U.S.-Russia relationship became strained when Snowden was revealed to be hiding in a Moscow airport, where the Russian government allowed him to hole up in a transit zone for weeks.
Putin declined to hand Snowden over to the United States and declared Snowden a "free man" biding time in the airport.
After Snowden inquired about asylum in Latin American countries, Russia stunned Obama: it decided to grant temporary asylum to Snowden.
Angered, Obama threatened to skip this week's G-20 summit in St. Petersburg. He obviously changed his mind: Obama and Putin shook hands and exchanged words briefly during the summit's opening Thursday.
It was polite -- perhaps a tad forced -- moment, and it went no further than that.
Gay rights
Obama has often been referred to as America's first "gay president" because of his aggressive advocacy of gay rights and same-sex marriage.
Putin, however, signed a propaganda law passed by parliament that bans the public discussion of gay rights and relationships where children might hear it. Violators can be fined and, if they are foreigners, deported.
Critics say the law is so vague that someone can be prosecuted for wearing a rainbow T-shirt or holding hands in public with someone of the same sex.
Obama declared "nobody's more offended than me" by the Russian law.
Syria: is chemical warfare for real?
The backdrop to the economic G-20 summit is Obama's international campaign to begin limited bombing of Syria for the regime's alleged use of chemical weapons against its own people in a civil war.
Putin hasn't been quick to embrace U.S. assertions that Syrian President Bashar al-Assad's government is using chemical weapons. Putin wants high proof that al-Assad -- an ally of Russia -- has indeed engaged in such warfare.
Putin called the chemical weapon allegations "absurd" because he said the al-Assad regime holds the upper hand over rebels.
The skepticism is powerful because Russia is a permanent member of the United Nations Security Council and, if the United States and allies seek U.N. approval to bomb Syria, Russia can veto such a resolution, a power that Russia has invoked repeatedly the past two years.
On Wednesday, the eve of the G-20 summit, Putin said he "doesn't exclude" supporting a U.N. resolution for military force against Syria -- but only if there's irrefutable proof of chemical weapons by the Syrian government.
What's one to make of all this?
Look to history, one analyst said.
"I think Putin especially sees himself as a Cold War guy," said Julia Ioffe, senior editor of the New Republic, who was a Moscow correspondent for Foreign Policy. "He's not shed that mentality. He sees himself as a foil and counterweight to America and the world."
