Hot on the heels of his Pyongyang summit with North Korean leader Kim Jong Un, South Korean President Moon Jae-in briefed Donald Trump on the outcome of the high-profile, three-day meeting on Monday.

Both presidents, in New York for the UN General Assembly, sounded positive on the prospects of improved ties with a nation they both hope – and expect – will denuclearize.

In his meeting with Moon, Trump said “we’re making tremendous progress,” according to a press statement released to reporters by the White House. “Chairman Kim has been really very open and terrific, frankly. And I think he wants to see something happen.”

Moon, who is working to play a brokering role between Kim and Trump, said at the meeting: “There was also a message from Chairman Kim that he wanted me to convey to you,” though he did not reveal what that was. Adding that he would brief Trump on the summit, he added: “I hope it will contribute to the efforts to achieve complete denuclearization and peace on the Korean Peninsula, as well as to the US-North Korea dialogue at your second summit with Chairman Kim.”

Trump, who is facing major domestic political pressure ahead of mid-term elections in November, has consistently talked up his approach toward North Korea as a foreign policy success and has compared his approach favorably to that of his predecessors.

He continued that theme Monday, saying: “I see tremendous enthusiasm on behalf of Chairman Kim for making a deal, and I think that that’s something that’s very good. I think we’ve made more progress than anybody has made in – ever, frankly, with regard to North Korea.”

A second Kim-Trump summit – following their groundbreaking meeting in Singapore in June – is now the subject of widespread speculation. Trump addressed the issue, confirming that it would, indeed, go ahead – though he offered no details on either a time or place.

“We’ll be having a second summit with Chairman Kim in the not too distant future,” he said. “And I think within a fairly – pretty short period of time – that will be announced and it will be location to be determined.”

Is there any common ground?

Prior to that summit, US Secretary of State Mike Pompeo – whose last trip to North Korea was cancelled by Trump on the grounds that Pyongyang was backpedaling on denuclearization – told US journalists on Monday that he expected to travel to North Korea for talks soon.

He is also expected to speak to North Korean officials at the UN over the next few days and confirmed there is ongoing discussions between North Korean and US officials, both public and closed-door.

There is much to discuss. Despite upbeat signaling by Moon and Trump, and despite unilateral steps by both sides, there still appears to be little – if any – bilateral convergence between Pyongyang and Washington on what the denuclearization process actually entails.

Moon, in his meeting with Trump on Monday, stated: “North Korea’s decision to relinquish its nuclear program has been officialized to a degree that not even those within North Korea can reverse.”

However, that decision not yet been backed by the kind of actions which the United States considers critical to the process.

Moon’s meeting with Kim generated multiple positive outcomes: upbeat vibes, relationship building and multiple photo opportunities between the leaders and their officials, hostility reduction and confidence-building measures along the DMZ and the related Northern Limit Line in the Yellow Sea, agreements to upgrade economic and infrastructure ties at a time when such ties become feasible and an apparent softening in Washington’s stance toward Pyongyang, which had been strained in recent weeks.

Yet despite the glowing optics, the overall denuclearization process – at least, as the United States understands it – remains vague.

Kim agreed to permit international inspectors to visit a missile-engine testing and launch side which he is in the process of unilaterally dismantling. But he did not agree to provide a list of his nuclear arms and facilities, nor on any kind of verification protocol, two steps which the US insists are the key starting points of the denuclearization process.

While Kim also agreed to shut down his central nuclear facility at Yongbyon, he made that conditional upon un-defined “corresponding measures” from the United States.

Moreover, while there had been hopes in some South Korea quarters that the two Koreas would sign a bilateral “end of war” declaration, no such outcome, beyond a range of tactical agreements on military measures, was reached.

This may be in recognition of the fact that, absent a US commitment to sign a peace treaty to replace the 1953 armistice that ended the 1950-53 Korean War, such a declaration holds little water.