As soon as Zach found a seat, Nylander started off. “Eddie asked me to give an overview of what is known about Ellesmere’s fossils. Existing staff members have already heard most of this, but it’s important that you new people understand not to disturb any fossils you come across. We aren’t experts, so it’s paramount that we leave any prospective fossil sites as pristine as possible to anticipate when formal collecting is permitted and more people with experience are allowed in this region.”
With that admonition, Nylander dimmed the lights and brought up some slides on the screen.
“One important point is that fossil finds on Ellesmere have only happened in the last few decades and, in some cases, just a few years recently. When you consider the limited exploration of Ellesmere, due to the climate and the remote location, this is nothing short of amazing. We can only imagine what fossil treasures are literally lying around, waiting to be discovered.
“Thus far, there are major animal fossil finds in three areas. The first discoveries are also the youngest fossils. In 1961, a John Fyles from the Canadian Geological Survey was mapping some geological formations near Strathcona Fiord northeast of here. On a relatively small exposed slope, he found fossils from what is now believed to have been a beaver pond during the early Pliocene Epoch about four million years ago. Among the fossils that Fyles and others recovered from this exposure are a variety of amphibians, insects, and rodents, including, of course, the resident beavers, a small deer, horses, several small carnivores, badgers and weasels, camels, plus two larger carnivores—a wolf and a bear. Also, fossils of good-size tree trunks, branches, leaves, and cones. Analysis of the fossils indicates similar flora and fauna to northern China at that time. It’s assumed the common species were due to the land connection between Asia and the Americas during part of that era. The climate here at that time was similar to what is found today in the boreal-forest margin environment of the Yukon and Central Alaska—so, still relatively cold but able to support a much more diverse biosphere than Ellesmere today.”
“A second fossil-rich area was first discovered in 1975 by two paleontologists from the Carnegie Museum of Natural History and the Milwaukee Public Museum. It’s near Eureka Sound at the head of the Strathcona Fiord. The finds are from the Eocene Epoch of about fifty million years ago and include a wide range of mammals and reptiles. In addition, fossils of at least seven different species of trees were found. Our single tree stump near the site matches one type from Strathcona: Metasequoia, more popularly called a dawn redwood and related to California’s redwoods. It was thought this type of tree had become extinct tens of millions of years ago. Then, in 1944, an example was discovered growing in a section of China.
“A larger bed of similar tree fossils was near the end of the Stenkul Fiord halfway between Grise Fiord on the southern tip of Ellesmere and Site 23 on the Raanes Peninsula.
“Altogether, about forty fossil sites have been identified near the Strathcona Fiord. They tell us important things about what Ellesmere was like as far back as fifty million years ago and the animals and plants that lived here. However, they are dwarfed in importance by fossils found south of here about halfway to the settlement of Grise Fiord. In 2004, researchers found fossils of a creature fitting what was predicted to be a transitional form between fish and land-dwelling animals, a form up until then not discovered. These creatures were named Tiklaalik, an Inuit name for ‘fresh-water-fish.’ They lived during the Devonian Period about 383 million years ago in shallow waters and came out of the water for at least short periods of time. The traits supporting this transitional form include lungs, the first evidence of a neck, and lobed fins with rudimentary wrists allowing the creature’s movement and supporting its weight on land.”
Jason had been waiting for Nylander to take a breath. His opportunity came when the speaker paused with a beatific expression, assuming the audience shared his amazement.
“Sorry, Bjorn, but are so many different fossils common around the world? If not, why Ellesmere?”
“Well . . . I’m not sure I have an answer for the Pliocene and Eocene fossils, though I suppose everyone’s excitement about those came from finding them so far north, which told us about the climate during those times. However, the Tiklaalik fossils are among the most significant of all because they’re from the pivotal time when animals made the transition from water to land. As you can imagine, the further back in time we go, the harder it is to find fossil evidence that’s survived. Think of it . . . 383 million years.
Zach had questions he couldn’t voice—not in this setting. Was the finding of so many important fossils in Ellesmere somehow related to the Object?
How Long Has the Object Been Here?
Zach raised a question the next day when the Level 3 people met next.
“Nylander gave an interesting overview of Ellesmere fossils last night, but something occurred to me. For those of you who’ve been here for a while, is there any idea how long the Object has been on Earth?”
“That’s just one of the topics the damn thing has been cagey about,” said Charles Adams. “We’ve asked it many times and gotten no answer. We’ve even tried to trick it by asking questions that would let us narrow down a time window . . . but no luck.”
“Was that before we had the VR system working?” asked Zach. “What about recently? I’ve heard you all think Simeon’s answers have changed since then, though not always in useful ways.”
The prior staff spent several seconds looking back and forth among themselves before Mueller spoke.
“I take it no one recently asked that particular question again?”
Shrugs and shakes of heads resulted.
“Mind if I ask it?” asked Zach. “I’m due for a session with Simeon in the VR the day after tomorrow.”
CHAPTER 23
P VERSUS NP
Jason sat at the wall desk in his room, papers covering its surface and the one-person bed. A few papers lay on the floor, either placed there by him or having fluttered there after being lifted by air currents caused by his rapid pacing.
He clutched his hands together to stop the trembling as he stared at the two sheets of paper resting atop those underneath. His breath came in ragged gasps, as he alternately paused before gulping to fill his lungs and followed this with a staccato series of shallow inhalations and exhalations.
When he realized his eyes were dry to the point of pain, he blinked, the first time in many minutes.
“I have to tell someone,” he whispered.
His mind danced through a few options. Sinclair? He wouldn’t follow what I’m talking about. Huxler? He’s smart, but this is out of his field.
Jason shook his head and spoke to the papers. “Even though Mueller, Adams, and Christiansen know more mathematics, this is beyond anything they’re capable of evaluating. Hell, I’m not sure who in the math faculty at MIT could understand what I think I’m seeing. Maybe Foukler, Chiu, and a very few others.”
The dryness of his throat forced its way to his consciousness. Parched, he rose and walked over to a bulge in the papers lying on the bed. He remembered a water bottle in the vicinity. He pushed aside pages to reveal what was hidden. Hands now only lightly shaking unscrewed the top, and he drank the entire pint of water.
He sat back in the chair, closed his eyes, and focused on controlling his breathing. When he opened his eyes, he looked around at the hundreds of paper sheets covered in mathematical notations.
“Sinclair. Even though he won’t understand, it’ll have to be Sinclair, maybe with Huxler and Mueller to help convince the general how important this is—if they understand enough.”
He reached for the phone that was part of the site’s internal communication system but stopped when he noticed the digital clock: 2:47 a.m.
No. He was too tired, his brain fuzzy from straining too long. He couldn’t risk sounding addled. Besides, the others were asleep. Let them be alert when he forced a meeting. He set the alarm for 8:00 a.m., went over to
the bed, swept papers onto the floor, flopped forward, and was asleep as soon as he hit the blanket.
“What’s this about?” asked Mueller as soon as he entered Sinclair’s office to find Huxler and the general talking.
“I was wondering if you knew,” answered Huxler. “Leo and I were trying to come up with the reason Jason accosted me while I was eating breakfast with the meteorologists. He came up to me all agitated about something and saying he had to meet soon as possible—me, Leo, and you. Wouldn’t give more details, just turned and left when I finally suggested 9:00 a.m.”
Sinclair glanced at his watch. “Which is just about now.”
“Oh, good. You’re all here,” said Jason, as he walked in without knocking.
“All right,” said Sinclair. “Let’s all get seated. Jason, what the hell is this all about? Don’t tell me the VR is broke or whatever failure of that kind of technology is called.”
Jason frowned. “The VR system? No, no. That’s finally going well. It’s Freddie and the mathematics he’s been coming up with from Simeon.”
The other three men visibly relaxed, as they learned one of the worst options was not the topic. Sinclair grimaced, and Huxler put a hand on the general’s forearm to forestall his irritation.
“Tell us why you wanted to meet, Jason,” said Huxler in what he intended to be a calming tone.
Jason plopped into the last empty chair, took several deep breaths, and put both hands on his knees. Huxler could see the fingers digging into the cloth.
“I’m sorry,” said Jason. “I’m a bit rattled right now, but it has to do with the mathematics Freddie and Simeon are working with. It involves . . .” He paused with his eyes unfocused, as if gathering his thoughts.
Sinclair raised an eyebrow at Huxler and Mueller. Both men shrugged. Huxler made a calming motion with his right hand, hoping to let Jason proceed at whatever pace he needed.
Moments passed.
“Sorry again,” said Jason. “After Dr. Huxler encouraged me to get closer to Freddie, I started spending as much time as I could with him when I wasn’t helping set up the VR. It didn’t take long for me to understand Dr. Huxler’s claim that Freddie is some kind of mathematics savant like Ramanujan.”
Sinclair’s expression signaled faint recognition.
“Ramanujan was an Indian mathematician. Maybe you saw the movie. You know . . . The Man Who Knew Only Infinity. I also read a book on him. One anecdote I remember is Ramanujan and his Cambridge sponsor, Hardy, riding in a cab . . . maybe in London. Hardy notices the cab’s number is 1729. I guess they used to play number games. Anyway, Hardy comments that it is a boring number, but Ramanujan disagrees strongly. He says it’s the smallest number that can be represented as the sum of two cubes in two different ways.
“And no, don’t ask what those two numbers were. I can’t begin to imagine how someone would look at a number and know that.
“When Ramanujan showed up at Cambridge to work with Hardy, he had notebooks of theorems and solutions. Nothing else! No proofs of any of it. He somehow intuitively knew mathematical relationships without having to carry out formal proofs. Hardy had the devil of a time getting Ramanujan to understand that no one would believe his work unless he showed the steps to reach his conclusions.”
Jason paused as if a new idea suddenly occurred to him.
“You know, that may be related to what I’m going to tell you. Hmmm . . . I’ll think about that. Anyway . . . what made it all the more astounding was that Ramanujan had almost no formal education. Mathematics just seemed to come to him. Freddie’s like that. I love working with him, although I want to tear my hair out at times. He’ll be doing something interesting, and all of a sudden he’ll go off on tangents. Also, like Ramanujan, he doesn’t understand that because he thinks something is right doesn’t mean he doesn’t have to convince people.”
Huxler sensed Sinclair getting more impatient, and he looked for an opportunity to focus Jason on why he called the meeting.
“Jason, does this have to do with the two mathematics papers you told me you might be writing with Freddie?”
“Papers? No, no, I didn’t ask to meet about those. They’re pretty straightforward. Actually . . . quite interesting work once I got Freddie to sit still and go over his notes. I believe we’ll have papers ready to submit to mathematics journals . . . maybe within a couple of months. At first, I was hesitant to put my name on the papers because the core ideas are almost all Freddie’s, but I came to realize that without my contribution and doing much of the derivations, the work would never be published.
“Of course, he didn’t have the necessary details, and he made a couple of minor mistakes I caught, but both papers are short—more refinements of what’s already known. Well . . . I guess I shouldn’t make them sound trivial. Both are slightly different approaches. I can see several more extensive papers developing the use of the new approaches, but that’ll take time . . . maybe six months to a year.”
Huxler shifted in his chair and started to prompt Jason a second time, but Jason moved on quickly.
“No . . . why I called you is that four days ago I was listening to a session Freddie had with Simeon. I admit I was only half paying attention when I caught Simeon saying something about a ‘Millennium Problem.’ That’s when I began listening more carefully and going back to earlier parts of their session. It took me a while to put Simeon’s comment into the perspective of what they were discussing. Then I realized they were dancing around the N versus NP issue.”
Jason stopped talking when he looked at three blank faces.
“You know . . . the Clay Institute’s list of important math problems like the Riemann hypothesis and the Poincaré conjecture.”
Sinclair’s expression changed from annoyed to amused. “Sorry, Jason, but I haven’t the foggiest idea what the hell you’re talking about.”
“Ah,” said Mueller. “I remember. The Millennium Prize Problems are ten or twelve unsolved math problems. Anyone who solves a problem gets acclaim and a big cash prize.”
“A million dollars,” said Jason, “and it’s seven problems. So far, only the Poincaré conjecture has been solved in the twenty-some years since the prizes were announced. Some mathematicians believe most of the rest of the problems can’t be solved. Of course, all that means is no one can see the path to a solution as yet, but it doesn’t mean it couldn’t happen someday. After all, the Poincaré solution was resolved.”
“All right,” said Huxler, “let me see if I understand where you’re going. Did Freddie ask Simeon whether he could solve any of these Millennium Problems?”
Jason shook his head. “I assumed that at first, so I searched all the previous sessions Freddie had with Simeon. The earliest reference to the Millennium Problems was when Simeon mentioned them, not Freddie.”
“Wait a minute,” said Mueller. “Simeon? If Freddie didn’t bring up the topic, who else was it? Adams or Christiansen? I suspect they’re the only ones with enough math background.”
“No, no,” said Jason. “You don’t understand. I searched through ALL the sessions with Simeon. NO ONE mentioned the prizes before Simeon did.”
“So how did Simeon learn about them?” asked Huxler.
“It might be that he learned about them before we built the Faraday cage,” said Mueller. “We’ve never settled exactly how much monitoring of human radio and TV transmissions he had access to before we cut him off from outside contact. The problem with that answer is we think he learned English only through interactions with us. I suppose the Object would have merely recorded transmissions and translated them later.”
“Yes, but the Millennium Problems don’t seem like a topic to come up often on radio or TV,” said Mueller. “Of course, I guess we can’t rule anything out since Simeon so far won’t answer when we ask how long the Object has been on or around Earth.”
“However it happened,” said Sinclair, “what’s it got to do with why you wanted to meet, Jason? So, you think Simeon
is telling Freddie how to solve the other problem you mentioned?”
“The Riemann hypothesis? No . . . one of the others. The N versus NP Problem.”
Sinclair sighed. “I’m right back to where we started. Why should I care if Freddie and Simeon are solving some math problem?”
“Let me try to give you some sense of the problem. In its simplest form, the N versus NP problem asks that if it’s easy to check whether a solution is correct, is it also easy find the solution? So, is it possible that P = NP where P is polynomial time and NP is nondeterministic time?”
Sinclair snorted in disgust, and Huxler took his turn sighing.
“Okay, okay,” said Jason. “It’s hard to explain without going into terminology. Let’s take the classic Traveling Salesman Problem. If a salesman has to visit ‘X’ cities, what’s the shortest route he could he take? The problem is simple to state. All it takes is to determine the distance for all possible routes and pick the shortest one. However, it’s not easy to solve once the number of cities increases. I forget the exact numbers, but with ten cities there are several hundred thousand possible routes and billions for fifteen or so cities. If it was a hundred cities, the number of possible routes is far more than the atoms in the entire universe. So, it’s ‘easy’ to state and ‘hard’ to solve because the only way we know how to do it is to check all possible routes . . . which quickly becomes impossible as ‘X’ increases.
“Now . . . what if a way exists to solve the problem without checking all options? This is what P = NP refers to. Most mathematicians think P can never be equal to NP, but it hasn’t been proven.
“What if we find a proof that P can equal NP, at least under some conditions? This would be an upheaval not only in mathematics but in many areas of technology. It would suggest that problems now considered too complex to solve would be, in fact, solvable. It would be a development at least equal to the invention of the Internet, and a few people even think more important than the discovery of fire . . . though that might be stretching a bit.”
Harbinger (The Janus Harbinger Book 1) Page 29