Even if she knew that the Young Scholars Program at the University of Chicago was a highly competitive program, she couldn't sit around and wait. Immediately after the press conference of the coach, Gen rushes to his office to have a frank discussion relative to this summer camp.
"Since the time you talk about attending the University of Chicago, here's my chance to see if attending it will actually be for me. It's decided, I will apply to the YSP program there" Gen announces her coach.
"The only reason why I wouldn't see you attend Harvard, Princeton, Yale, or Stanford is because, even though you can undoubtedly excel there, these universities are unhealthily careerist. Chicago appeared less careerist to me. That said, most people attending the YSP camp, from what my stepsister in Chicago told me, live in the region. They will certainly want a student or two from other regions, after all, having a variety of talents will benefit everyone in there" Trent explains to her.
"Suppose that Chicago is my first choice. What backup plans for me?" a confused Gen asks him.
"It's not a secret that Tulane is very favorable to Louisiana students, compared to its peers. But with you, it seems that it's either reach for the stars, or go to college on a full-ride scholarship. You have a very imbalanced profile, which is a double-edged sword for college admissions. For example, Georgetown, which is known first and foremost for law, will receive hordes of applicants that will dream of political science or history. But I will put G-Town as a match in your situation: practically no one outside of the Greater Washington at your level of mathematical talent wants to go there for mathematics!"
"If I go to YSP, will your stepsister lodge me for the entire duration of the camp? I might be going there all expenses paid... I know no one in Chicago!"
"My stepfamily in Chicago will probably not believe me before meeting with you as for how a rural Louisiana school could produce a student that can solve for PDEs in open air in eleventh grade! I must talk about it to her, but in the meantime, I will write you a recommendation letter for it"
Everyone imagines the Deep South as a land where people can barely count... but we know that the most common strategy for those aiming for the Ivies or equivalent is simply to blanket the most extracurriculars and to take as many advanced placement courses as they possibly can, without regard to coherence in extracurricular activities or courses between them. With the consequences this carries: sleep disorders, chronic fatigue, anxiety disorders, depression...
In the meantime, Krista resurfaces and gives a rendezvous around the school's stadium. Thus, she texts Geneviève, whose share of the popular attention has clearly diminished now that we are in the vacation period, that is, the mathletic off-season. « My parents are OK for me to transfer to your school; would you like to meet again at dinnertime? Around the stadium is OK for you? » Krista texts Geneviève, with this fresh as a rose memory of this PDE solved in open air.
« It would be fantastic; I would love it if Marcia came to see you as well » she responds. If Krista is to be part of the mathletics team of the school, then I believe it's a good idea to introduce her to a potential future teammate, and I feel that, for the next season, we would like to fixate a roster for the Square Root of the Answer before any other contest. We saw it this year, the SRA is a tournament that requires an entire year of preparation. However, it's the most demanding contest on the calendar in terms of required background; one must, at a minimum, take calculus BC concurrently. But will she be part of this only for the SRA or will she participate in other tournaments? She thinks, while waiting on Marcia's answer, who takes her time to think about it.
At my old school, I was practically a one-eyed person in a kingdom of blind people. I was the captain of a very bad team. Unfortunately, the neighboring parishes are not much better, Krista thinks, while in both towns, calculus is only a chimera to the eyes of the public, hence the aura that this material could have. And solving for PDEs in open air? It was a circus-worthy novelty for them.
« Marcia agrees. Around the stadium tonight then » Geneviève adds once Marcia answers her. When she arrives at the stadium at night...
"I am Krista. Surely you remember me from this pep rally; I was the only one that talked about partial differential equations that night. Now that my parents let me transfer..."
"Krista, you have certainly noticed that this town is very different from other towns for miles around. When the football team can barely win a game in a season... speaking of football I wonder what effect the departure of Dylan and the long snapper will have. Frankly the long snapper that took AP Chemistry with me was very bad" Marcia comments.
"I would have loved to discuss differential equations with other people that night, you included, but people had enough questions about calculus as it was, single and multi-variable. Here's our chance to talk about it in greater detail, especially for Marcia here that I believe would benefit" Gen then tells the other two girls.
"Remember the form of the solutions to the PDE that you solved in open air?" Krista asks them.
"Yes, I know, if we have a linear combination of these solutions, then the linear combination is also a solution. From this idea comes what we call the Fourier series" Gen answers Krista's question.
"I would never have had this discussion at my old school; I would have been called crazy from the moment I talk about partial differential equations. Even talking about partial derivatives there gave me goosebumps. With you two I can do so without complex"
"I understand also that you can express any periodic function in a Fourier series" Gen mentions another implication of Fourier series.
"Speaking of series as solutions to differential equations, ordinary or partial, it allows to get solutions we can't obtain otherwise" Krista follows up on that.
"I don't remember seeing this in problems that our teacher gave us for extra credit! Our teacher stuck to what was on the official content of the Square Root of the Answer!" Marcia complains.
I already start seeing the problem with Marcia on the team! Even though, virtually everywhere else in Louisiana, except for the magnet schools of the big two cities, she would have been in a very similar situation to my old school, her mathematical immaturity would cause us issues especially if we participate in proof contests! Krista thinks, while Marcia thought the local public seemed to prefer more « computational » contests, to which the SRA belongs. « Proof contests » seemed reserved to a very advanced level, the best known being the USAMO (to qualify one must do well on the computational AMC12 and AIME), the IMO (of which the USAMO is a qualification step) and the ARML. For which everyone in the parish knows that it would take fifteen students like Geneviève or Krista to even compete, and that combing the entire state is necessary to do so, and still...
"You are discussing differential equations while none of the competitions we are going to enter will contain that! You were perhaps my friend for the past three years, but I feel this season will go sour!" Marcia keeps complaining.
"Really not hard to solve for homogeneous equations with constant coefficients; the only real complication is to find all the roots of the characteristic polynomial, which comes from the form of the solution, which is e to the ax. You want to solve for a" Gen keeps discussing differential equations.
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"But what happens when you have a root whose multiplicity is greater than one?" Marcia asks, who seems capable of following.
"Then you will have a solution of the form C-one times e to the ax plus C-two times x e to the ax. The sum from n equals one to the multiplicity of the root de C-n times x to the n minus one, times e to the ax, for the root whose multiplicity is greater than one. We know that, by the fundamental theorem of algebra, that the solution to the characteristic polynomial exists but that no general formula exists for a polynomial of degree five or higher" Krista answers Marcia.
"Krista, you evoked the fact that we could express the solution to a differential equation as a series! I assume that it is like finding the recurrence relationships in a sequence... As an aside I can say that, if you fear having to hide your intelligence here, it will be unnecessary. But if you talk about power series, you need to consider convergence and convergence radius"
"I do not doubt it at all, especially not when this town gives an adulation to its mathletics team which, everywhere else, would be kept for the football team. After all, no other town will even give the opportunity to solve PDEs in open air in front of such a large audience; if it's not an overt display of the intellectual capabilities of the person that solves this problem to the eyes of the public, I don't know what that is" Krista makes a long-winded comment.
"Other than a monumental trick, but what would be the goal to perform a trick that size?" Marcia asks, knowing however that the qualities required to excel even in computational contests are still qualities envied by many.
"Krista, you forced my hand that day. As much as I could have tried to hide my intelligence, after this pep rally, it would be useless for me to do so. I think you can no more hide it than Marcia could. We might be wondering who else will be on the team this season. You're the first to have transferred here for that, there might be another parishioner or two, but to the eyes of the public the pressure is there to find another multi-variable calculus enthusiast for the SRA, which is the chief competition here"
"For the last season, we took the entire state by surprise, everyone expected a quick and direct elimination in the first round of the SRA or of the AMC12 on our part. It will no longer be just our town that will watch us. From now on, the entire Louisianan mathletic community will have us in their sights" Marcia puts Krista back in context about the evolution of expectations.
To the eyes of the public, as it relates to mathematics, the more advanced the material, the smarter the person capable of doing it is, Marcia ruminates while the other two girls discuss how reversing a gradient amounts to solving an exact PDE, and how solving an ODE with constant coefficients happens with an inhomogeneous differential equation. And finally, how to calculate the Fourier coefficients. But once the mathematics are over, the two girls discuss their plan for the rest of the summer.
"Let's cross our fingers as for the me and the YSP camp at the University of Chicago. Do your teachers, or your parents, or the booster club even, push you to try math camps?" Gen asks Krista about math camps.
"No. My parents know that I have mathematical talent but believe these camps are useless for college. But you're a welcome change from my old teammates. At my old school, with my old teammates, the Square Root of the Answer would be a one-person show and forget about the state title!" Krista starts crying.
"If we ignore the unhealthy environment of your old school, how would you describe your old teammates?"
"The other girl on the team, Annette, was there just because her boyfriend, Trevor, was there; Trevor would make Marcia look like a genius. The other boys weren't any better than him. We just kept to the AMC-twelve, and I was the only one in the last ten years to qualify for the AIME there. How much do you think I got on the AIME?"
"Cat got my tongue"
"Five; I hold the school's record on it"
"Tomorrow the summer courses start, and I will be a math tutor for the month"
"I will instead work for the school's daycare for the summer"
"Nice to have met you"
During that time, the days come and go for her while awaiting the response from the teacher and the University of Chicago for the camp, with Trent that starts thinking about how wonderful an experience it would be for his stepsister, but also how a good performance at camp will help her, if Geneviève attends, with undergraduate admissions at the University of Chicago. At the same time, Marcia is occupied, in the coming days, to tutor students for the ACT rather than for a summer school subject, especially considering there are two summer dates for the make-up ACT, and that parents are ready to pay for their children to take semi-private courses with her. After all, one or two more points can mean going up one tier on the TOPS ladder and therefore more scholarships.
The following week's edition of the parish newspaper contains a preseason ranking of the various mathletics teams of the state as a joke and hoax ranking, with the local school that sits at the top, but that still places the Bâton-Rouge Magnet School second, and New Orleans as well as Caddo Parish magnet schools that share the remaining top-5 spots. But it's starting from positions 6-10 that the « joke and hoax » nature of the ranking starts to be felt. Joke and hoax that catches Geneviève in the middle of a tutorial session.
"I just read the parish newspaper and, even though we are at the top of the state in mathematics, frankly, the neighboring parish is eleventh? Are you kidding me? We played there last year and, even though we lost on the football field, from what I've seen from their fans, they were nothing special academically" Ted questions, a rising junior fullback.
"My God! If you have questions, please ask them! That's what I'm here for" Gen exclaims in front of him.
"We want to solve a system of two equations with two unknowns"
"Does your system contain a parabola or two lines? Because what follows will differ somewhat. Should you have a parabola and a line, you subtract the line from the parabola and then you solve the quadratic equation that arises from that. If b-squared minus four a-c is smaller than zero, it's not that there is no solution, but the solution is a complex number"
"Complex number?" a confused Ted asks.
"A complex number contains a real part and an imaginary part, with i defined as the square root of negative one"
"Look, the coach strongly advised me to take the second algebra course during the summer so that I'm not bothered by it during the football season. It's the only reason why I'm even here" Ted comments.
Meanwhile, in the neighboring parish which Ted, the fullback, alludes to, this brand new mathletics ranking of the state's schools seems to cause headaches to its coach, who wonders what to do to move up the ranking for next season. Ouch: our direct rivals in the parish next to us just reinforced their position with the transfer of the captain of another school's team! They no longer have rivals on their own turf, we cannot rely on the Square Root of the Answer to score points, this contest is their preserve. We will need to participate in other contests than just the Purple Comet or the AMC12! Our eleventh position is, I must admit, a surprise, but we must get real, we must not expect to rise to the top-5. We will need several weekly sessions, as well as extra credit, to succeed, Brian, the coach of the neighboring parish team, thinks while preparing his plan for the next season.
But, at the end of the week, Trent asks to see Geneviève at his office, with news concerning the logistics of the camp as well as the mathletics team ranking of Louisiana high schools.
"Unsurprisingly we are the best in the state, but are you aware of the weight that you bear as the captain of the team? Are you aware that the whole town, even the parish, will have their eyes on us? The public might forgive us a poor performance on a contest they never heard about, but if we lose the Square Root of the Answer..." Trent warns Gen.
"Yes, I know that I'm the face of this team! Since our victory at the state stage of the SRA. But couldn't it be possible that, should we lose the SRA, that's it's the rest of the state that got better?" Gen harbors some doubts.
"I suppose. However, if we put this contest aside, my stepsister agrees to lodge you at the condition that you attach all the grocery and transportation bills to the expense report. I hope that you're comfortable with geometry because you will spend four weeks hearing about geometry if you attend YSP"
If I don't attend YSP, how will that affect my applications to submit in a few months, at Chicago or elsewhere? Will that affect my performance during the next season? I have been able to get by without it, but I know that those who dream of capitalizing on their high school years to attend the major universities will have multiple extracurriculars in hand. Yet attending and doing well in it will tell Chicago that I have what it takes to succeed. I could then definitely say that my two anchor points will effectively be so, she thinks while her mother combs the mail, and she finds a letter from the University of Chicago. « Honey, I hope you're comfortable with the wind...» her mother answers by text message, even though, in reality, Chicago isn't exceptionally windy. « You will spend the entire month of July in Chicago »
"That was fast; my mother just notified me I was accepted to the camp!"
"Even though you know it's rare that anyone attends a summer camp for an academic extracurricular, at least here, you know that the attitude is different for sports, and certainly football. So have a good camp"