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John Urschel is used to having to prove himself. In four years at Penn State and three with the Baltimore Ravens, Urschel could never let up. It was a daily challenge to prove to coaches, scouts, and GMs that he was big enough, tough enough, and athletic enough to earn a starting spot on the offensive line.

But when Urschel was accepted into the PhD program for applied mathematics at MIT in 2016, he faced a different challenge — to prove that he wasn’t just a novelty. That he deserved to be learning and teaching with some of the most brilliant mathematical minds on the planet.

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“I guess it’s mission accomplished,” Urschel said by phone on Thursday.

Urschel isn’t going to graduate on time with his doctorate. He’s on track to graduate early — in the spring of 2020, after just 4½ years in the program. The typical math PhD takes five to six years to graduate.

“Obviously it’s been a lifelong dream of mine,” Urschel said of earning his doctorate. “Sometimes you forget. But yeah, I’ve been enjoying my time at MIT.”

It has been two years since Urschel, the smartest man in the NFL, abruptly retired from the league right before 2017 training camp in order to focus on his PhD, and life in general. Urschel has mostly disappeared from the spotlight. He and partner Louisa Thomas have a young daughter together, and Urschel is loving life as a mathematician and father in Boston.

“I’m really happy with where I am in life right now,” Urschel said. “People ask me what am I doing with all my extra time? And it’s like, what extra time?”

But Urschel did find the time for a side project. He has a new memoir from Penguin Press, “Mind and Matter: A Life in Math and Football,” written with Thomas. Urschel, who turns 28 next month, writes about his love of mathematics that blossomed as a youngster, his experiences at Penn State during the Joe Paterno fallout, his three years in the NFL, and his transition back to academia. It’s an entertaining read, though Urschel admits that the book doesn’t fulfill his true vision.

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“I really wanted to write a math book, actually,” he said. “And Penguin sort of told me that if you actually want to reach people, if you actually want to share your love of math, this is not the way to go about it. And you’re going to have to put a lot of yourself into it and you’re going have to share your experiences with math, but also your experiences with football.”

Urschel does share interesting insights about his three years with the Ravens. He also writes about how impressed he was with Bill Belichick and his embrace of mathematics and unconventional wisdom. Urschel was the starting left tackle in the 2015 playoff game when the Patriots unveiled a new eligible-ineligible trick formation that confused the Ravens and ignited a Patriots comeback.

“I’m not claiming Bill Belichick is literally a mathematician, but I would say this is just one example, clearly, of how they think very quantitatively about trying to optimize their winning chances,” Urschel said. “Bill Belichick might not have an expert understanding of, let’s say, game theory, but he’s certainly doing a fantastic job of finding out how to optimize performance in light of suboptimal decisions by other teams and coaches.”

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Interestingly, Urschel writes that a fear of concussions wasn’t the driving force behind his football retirement. Urschel details a concussion he had in the NFL that affected him for several weeks, but he said that the episode actually made him more determined to keep playing football. He said he doesn’t feel any effects of the concussion now, and that his retirement was about a desire to pursue math, raise a family, and to get out of the NFL while he is still in great health.

“Everyone has some stiffness here, an injury there, but by and large, compared to a lot of my friends who are retired, I am extremely healthy,” Urschel said. “Obviously you think about your head, but honestly, I was thinking a lot about my body. When I found out I was going to be a dad, it changed a lot of things for me. I want to be able to walk my daughter down the aisle. I want to have a high quality of life. I had a fun time in the NFL and I wouldn’t change it even if I could. I’m also extremely happy with when I retired.”

Urschel said he has had offers from NFL teams to join their analytics departments, but that he has no interest.

“I’m very much staying away from the sports analytics,” he said. “I have lots of friends who work in the field, and nothing against them, but I think while there are interesting problems, the mathematics is much more interesting in other fields. For example, the mathematics in computer science is much more fascinating to me than the math in football.”

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Instead, he wants to be Professor Urschel. He’ll be applying to universities across the globe this spring after he receives his doctorate.

“It’s not even a given that I will be a professor stateside. I would be open to being a professor at some of the places in the UK or in Switzerland,” he said. “I love solving sort of interesting and tough problems that have to do with our world in some way. And I also love teaching. I love interacting with young people, and teaching people how to be better at math, how to be better at thinking.”

ROLE PLAYING

Patriots coaching jobs are clarified

It’s possible that Bill Belichick will be the Patriots’ de facto defensive coordinator after the departure of Greg Schiano.
It’s possible that Bill Belichick will be the Patriots’ de facto defensive coordinator after the departure of Greg Schiano.Jim Davis/Globe Staff/Globe Staff

■  The Patriots made their coaching staff available to the media on Friday, and clarified most of the new roles. Joe Judge will coach receivers in addition to special teams. Bret Bielema will coach defensive line. Jerod Mayo will handle inside linebackers, DeMarcus Covington will coach outside linebackers, Mike Pellegrino will coach cornerbacks. Cole Popovich will have more responsibility as assistant offensive line coach, and Mike Lombardi, son of former executive Mike Lombardi, will be the assistant QB coach.

One role not mentioned — defensive coordinator. The Patriots still don’t have one after Greg Schiano left abruptly in March after one month on the job.

The buzz I am hearing from a league source is that Bill Belichick will simply call defensive plays himself this fall. This same source also told me Bielema would become defensive line coach, which I was going to write in this space before it was confirmed on Friday.

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Calling plays is not that much more responsibility for Belichick, who has always been involved with the defense, even when Matt Patricia and Brian Flores were the coordinator. Belichick usually coaches the defensive linemen during the game, and often told Patricia and Flores when to call blitzes.

At this point, Belichick as defensive coordinator is the most sensible option. When Schiano left in late March, pretty much anyone else that Belichick would consider for defensive coordinator already had his job for 2019 lined up. Bielema is the only other person on the Patriots’ staff even remotely qualified. Steve Belichick, the coach’s son, is not ready for play-calling yet. Mayo is a first-time coach, and needs at least some experience.

So Belichick probably figures that he’ll get through this year calling plays himself, then figure out a long-term solution next year. Who knows, maybe Patricia will be available to come back?

Related: Patriots are sorting out their coaching staff

■  We’re past free agency, the draft, and half of the offseason program, yet Tom Brady still hasn’t renegotiated his contract. He’s still entering the final year of his deal, set to make $15 million in salary but with a $27 million salary cap number, the fourth-highest in the NFL.

It’s a bit of a head scratcher why the Patriots haven’t given him an extension yet, especially because they could have easily created more salary cap space for him, which they could have used in free agency.

One theory is that Brady usually hammers out his contracts one-on-one with Robert Kraft, and perhaps their paths haven’t crossed much over the past few months. The two know that they can easily get something done in August, when both will be in Foxborough.

But is it possible that Brady finally wants to use a little leverage? His franchise tag number next year will be $32.4 million, or 120 percent of his current cap number. If Brady wanted to, he could play out this year then use that number as a starting point for an extension. Or, even simpler, he could dare the Patriots to franchise tag him next year, and play for $32.4 million.

More likely, Brady gives the team another discount, to ensure some roster security on his end. But with quarterback salaries skyrocketing and the salary cap increasing by $10 million each year, could any of us blame Brady if he finally wanted to use some of his leverage and push the Patriots for a market deal?

■  In an interview with Sports Illustrated, Rams coach Sean McVay said he overanalyzed the Patriots leading up to the Super Bowl.

“You have so much time that you can overprepare and get away from some of the things that helped you get there,” McVay said.

Yet, somehow, McVay and several Rams said shortly after the game that the Rams didn’t expect the Patriots to play so much zone defense, and they didn’t have any adjustments to deal with it.

It appears McVay forgot an important rule when facing the Patriots. Scouting them is all well and good, but scouting your own team is just as important, if not more so. The Patriots are going to find your weaknesses, and attack them relentlessly. McVay should have focused less on his opponent, and focused more on his own team.

ETC.

New interference rule not defined

The controversial missed pass interference call in the NFC Championship Game between the Rams and Saints has made the penalty a reviewable call now.
The controversial missed pass interference call in the NFC Championship Game between the Rams and Saints has made the penalty a reviewable call now.Gerald Herbert/AP/Associated Press

Remember in late March, when the NFL had no intention of making pass interference a reviewable call, then Sean Payton and a bunch of coaches staged a coup, and the owners relented because they wanted to get on their private jets and fly back to their mansions? You’ll be shocked to learn that no one quite understands how this rule is going to be applied this fall.

This rule is a significant one — not only does it open up game-changing pass interference calls to instant replay, it marks the first time that a team can challenge to put a penalty flag on the field when none is originally called. But an item from ESPN on Friday detailed how no one in the league office knows yet how to define “clear and obvious” pass interference, and that this new rule could lead to more penalties, more reviews, and longer games.

“Unless they change it all up, they’re going to use slow motion,” said retired referee John Parry, now with ESPN. “They’re going to use frame-by-frame. They’re going to be technical.”

Parry, who was the referee for the Patriots-Rams Super Bowl, said that instant replay would likely have added three pass interference penalties, including a crucial one that would have been called against Stephon Gilmore on a long pass breakup against Brandin Cooks.

“And that would have been in New England’s red zone. So it would have extended the drive and maybe they put points on the board with four minutes to go,” he said. “That would be a pretty impactful decision.”

Fortunately this rule was only approved for a one-year trial basis in 2019. It sounds like it is going to be a disaster.

XFL has shot to succeed

We wrote a few weeks ago about how developmental football is fundamentally flawed. But the XFL, still a year away from its proposed start date, may have a fighting chance.

The league signed a surprisingly good TV deal, partnering with ABC/ESPN and Fox/FS1. While the Alliance of American Football was mostly tucked away on CBS Sports Network and NFL Network, the XFL will be on channels with a much higher distribution. And the XFL’s deal calls for 24 of the league’s 43 games to be played on broadcast TV, compared with just three games for the AAF.

The partnerships with ESPN and Fox also give the XFL relevance. The two networks, which dominate daytime sports talk, will now include the XFL in their conversations and make the league relevant with free publicity.

Perhaps there’s something to be said from the “second-mover advantage,” with the XFL able to learn from the AAF’s mistakes.

Extra points

The South Carolina senate approved $115 million in tax breaks for the Panthers to move their practice facility over state lines to Rock Hill, S.C. (hometown of Gilmore, Ben Watson, Cordarrelle Patterson, and others). And the Panthers wouldn’t just build a practice facility — borrowing a page from Jerry Jones — Panthers owner David Tepper also wants to build a hotel, conference center, retail, and orthopedic sports medicine center, and host high school football games. “There’s a misunderstanding that we are building a little practice facility,” Mark Hart, the Panthers’ vice president for development, told The State newspaper in April. “But [Tepper] is thinking big.” . . . One of the more bizarre stories of the offseason goes to Jaguars linebacker Telvin Smith, who last week announced on social media that he is taking the 2019 season off to “give this time back to myself, my family, & my health.” But his bosses don’t know what’s going on. “We need to have a conversation with Telvin to understand the situation and the circumstances,” the team said in a statement. Smith, who has skipped voluntary workouts, still has three years and $30 million left on his deal . . . Kyler Murray’s four-year deal with the Cardinals comes with a $23,589,924 signing bonus and is worth $35,158,644, fully guaranteed. Yes, choosing football was the right decision . . . The NFL has paid out nearly $500 million of its $1 billion concussion lawsuit settlement, but believes some ex-players are taking advantage of “doctor shopping” to earn payouts. Per the Associated Press, the NFL said that $46 million has been paid out to players who were recommended by four doctors, each of whom has been dismissed from the program, and requested that players be limited to doctors within 150 miles of their home. “A few were brought to my attention where we had a lawyer from Pennsylvania and a player from Florida going to a doctor in Texas, and that was a red flag,” Judge Anita Brody said in court . . . The Lions slipped from 9-7 to 6-10 last year, and the owner isn’t happy. “I think Mrs. [Martha Firestone] Ford is a very different owner than her husband, and has a lot less patience and a lot less tolerance for mediocrity, which is why we’ve made some of the changes that we’ve made,” Lions president Rod Wood said, via Fox 17. That sound you hear is Matt Patricia and GM Bob Quinn taking a very large gulp.

Ben Volin can be reached at ben.volin@globe.com. Follow him on Twitter @BenVolin. Material from interviews, wire services, other beat writers, and league and team sources was used in this report.