I’m in the provincial headquarters building in downtown Kirkuk — the oil-rich district of northern Iraq that is the most disputed corner of this country. The provincial leaders — Sunnis, Kurds, Turkmen and Christians — have come to meet America’s top military officer, Adm. Mike Mullen, the chairman of the Joint Chiefs of Staff, with whom I am tagging along. All 11 Iraqi leaders are seated on one side of a conference table and local U.S. officials have provided me a color-coded guide, identifying each Iraqi politician, their political tendencies and religious affiliation. Each Iraqi leader tells the admiral, through an Arabic translator, why his or her community deserves to have this or that slice of Kirkuk, until it comes to a Kurdish representative, who announces in English: “I want to tell a joke.”

It’s my lucky day.

“After Saddam was ousted in 2003,” said Deputy Provincial Council Chairman Rebwar Talabani, “there was an elderly citizen who wanted to write a letter to the new government to explain all his sufferings from the Saddam era to get compensation. But he was illiterate. As you may know, outside our government offices we have professional letter-writers for illiterate people. So the man told the letter-writer all of his problems. ‘In the ’50s, they destroyed my house,’ he said. ‘In the ’60s, they killed two of my sons. In the ’70s, they confiscated my properties,’ and so on, right up to today. The letter-writer wrote it all down. When he was done, the man asked the letter-writer to read it back to him before he handed it to the governor. So the letter-writer read it aloud. When he got done, the man hit himself on the head and said, ‘That is so beautifully done. I had no idea all this happened to me.’ ”

Talabani’s joke seemed to have been directed as much to his fellow Iraqis as to Admiral Mullen. My translation: “Everyone here has a history, and it’s mostly painful. We Iraqis love to tell our histories. And the more we do, the better they get. But with you Americans leaving, we need to decide: Do we keep telling our stories, or do we figure out how to settle our differences?”

And that is my take-away from this visit: Iraqis know who they were, and they don’t always like it, but they still have not figured out whom they want to be as a country. They are exhausted from years of civil strife and really don’t want to go there again. Yet on the big unresolved issues — how will power be shared in Kirkuk, how will the Sunnis who joined the “awakening” be absorbed into the government, how will oil wealth and power be shared between provinces and the central government — the different ethnic communities still don’t want to compromise much either.

I am amazed in talking to U.S. Army officers here as to how much they’ve learned from and about Iraqis. It has taken way too long, but our soldiers understand this place. But what about Iraqis? There are now many Iraqis embedded with U.S. forces in Kirkuk. In the dining hall on the main base, I like to watch the Iraqi officers watching the melting pot of U.S. soldiers around them — men, women, blacks, whites, Asians, Hispanics — and wonder: What have they learned from us? We left some shameful legacies here of torture and Abu Ghraib, but we also left a million acts of kindness and a profound example of how much people of different backgrounds can accomplish when they work together.

We are going to find out just what Iraqis have learned soon. As Admiral Mullen told the Iraqi leaders around that table: “The U.S. is not going to solve” Iraq’s problems. That is the job “of a sovereign nation.” So Iraqis better get to work, because “on the current withdrawal plan, coalition forces will not be here in 18 months.”

That’s an important message — otherwise Iraqis will delay forever resolving their big, nation-shaping disputes. We can’t do it for them — but our diplomats could do more to help them forge those compromises. We have special envoys for Iran, Afghanistan, Pakistan and Arab-Israeli affairs, but for Iraq — a country key to the Middle East in which we have lost so many lives and are spending a trillion dollars — there is no special envoy, or secretary of state, totally focused on securing a decent outcome here. Vice President Joe Biden is overseeing Iraq policy, but he has too many other things to do. Iraq needs a big, tough, full-time mediator.

Senior Iraqi officials are too proud to ask for our help and would probably publicly resist it, but privately Iraqis will tell you that they want it and need it. We are the only trusted player here — even by those who hate us. They need a U.S. mediator so they can each go back to their respective communities and say: “I never would have made these concessions, but those terrible Americans made me do it.”

After we invaded and stabilized Bosnia, we didn’t just toss their competing factions the keys. President Bill Clinton organized the Dayton peace talks and Richard Holbrooke brokered a deal that has lasted to this day. Why are we not doing in Iraq what we did in Bosnia — when the outcome here is 100 times more important?