Dr. Charles Edward Eppes is portrayed as a young mathematical genius and professor of applied mathematics at the fictional California Institute of Science, CalSci (primarily based on Caltech, where some filming and mathematics consulting is done). As a world-class mathematician, Charlie helps his brother Don Eppes solve many of his perplexing FBI cases, sometimes with the help of his best friend and colleague Larry Fleinhardt and his on-again off-again girlfriend and former student Amita Ramanujan, who further refines Charlie's approach and helps him stay focused. Charlie has consulted for the National Security Agency (NSA), in part as a cryptanalyst, for nearly five years, having attained a (now inactive) TS/SCI security clearance.
It was his paper on the Eppes convergence, which concerned asymptotics of Hermitian random matrices, that made him a star in his field. Following a seminar that heavily criticized this seminal piece many years after its initial publication, Charlie realized that his work with the FBI has prevented him from doing research significant to other mathematicians and now hopes to spend decades on cognitive emergence theory ("the mathematics of the brain") to rectify this certain inequity, which has delighted Larry. Although he was a child prodigy, Charlie now moreover fears the fact that his best years in his research will never come ahead of schedule again.
In the "Decoy Effect", it is revealed Charlie is a multiple Ph.D. Following his five-year studies on random matrices, Charlie worked on sequences with orthogonal symmetry. He has also provided insights for possibly solving the P=NP problem and published works on H-infinity Control of Non-Linear Systems and computational fluid dynamics, while his current research is in cognitive emergence theory. Charlie is a recipient of the Milton Prize. He has presented seminars on harmonic analysis and the zeros of random orthogonal polynomials and given lectures on group theory and Kac–Moody algebras. Charlie has taught courses on calculus, chaos theory, fluid dynamics, game theory and probability at CalSci in addition to giving guest lectures on applied probability. The lecture in which he converted the classroom into a miniature casino for analyzing probabilities is considered an "Eppes Classic". Also, Charlie has taken over Larry's computational physics class when he was asked to do so, and has given a joint lecture on circular motion and the Coriolis effect with Larry.
He is extremely talented in chess, as it requires both his father and brother to play against him and a distraction to defeat him. Charlie also has vast understanding of theoretical physics, often assisting Larry with his multi-dimensional supergravity theory and papers on gravity waves, and biology, extending to knowledge of ciliate protozoa and the spread of infectious diseases. While brilliant in some areas, he is lacking in others. He is apparently a bad speller (e.g., he misspells "anomaly" and "conceited") and does not know the meaning of "defenestration" (for which Larry chastises him stating that the idealization is to be a Renaissance man and that even math and physics majors had to have a course on English). Thus, his father likes playing Scrabble with him. He drives a silver Toyota Prius
In "Prime Suspect," Charlie purchases the beautiful Craftsman family home from his father, who continues to live with him. Now 30 years old, Charlie wants to be responsible and take care of his father but still believes that much of the pressures involving their dad has been put on his shoulders as Don doesn't seem to have enough time. Realizing this, Alan set his sights on moving out to accomplish things on his own, with Charlie seemingly supporting the idea, but has since chosen to stay as he favors his son's company. Recently, Charlie is bothered by his father trying to impose his will on him with maintaining the house, as he is a full tenured professor at one of the most prestigious institutions in the country working on "life-altering" mathematics, i.e., in solving crimes. Alan just wants him to be responsible and not end up like Larry, though Mr. Eppes respects Fleinhardt. Even he has thought of this possibility and decides to do more around the home. After Charlie compares his own situation to that of Einstein his father looks into information on the physicist and understands his son's predicament. Charlie also feels guilty about the amount of time his parents, particularly his mother, who always was attuned to his way of thinking, spent with him as a child; he even asks his mother (JoBeth Williams) in a dream if she regrets the time away from Don and Alan because of the special attention he needed growing up.
Charlie's research often interferes with his relationships: as with Amita on their first date, for all they could talk about is mathematics; Fleinhardt says that it is a common interest and they should not struggle to avoid the subject. Charlie and Amita had several false starts. Charlie has also spent some time with his ex-girlfriend, Susan Berry (Sonya Walger), an attractive neuroscientist from London. He had lived with Susan for two years, and Larry described this as his very own Berry's phase. However, she later reveals that she is currently involved with someone else and has to return to England. Charlie attempts to start a relationship with Amita once more, though her job offer at Harvard University strains this possibility and makes him distraught for some time. Eventually, she decides to take the alternative offer at CalSci with the hope to begin a romantic relationship with him, though his fear of possible failure causes him to question whether he wants this second chance; Don cautions him about such an attitude. Amita notices his ambivalence and isn't certain if she wants to back out of the relationship, but he then pushes it forward. Pressures from their colleagues over the inappropriateness of the relationship nearly cost them, but by the end of the third season their romantic involvement stabilizes and they have grown considerably closer. He feels rejected when Amita does not want him to meet her father, due to his expectations about the men she dates. Initially, he wonders if it is based on him being Jewish, but the problem is that he is not Indian-born. Alan explains that Mr. Ramanujan would like him after meeting him. Charlie and Amita state that they love each other, and have even decided to move in together, though, the actual living arrangement has not been determined. When Amita's parents finally meet him, they are rude, but warm to him later. Alan says that Charlie will have to marry her.
Unforeseen complications with his work emerge as the new Chair of the CalSci Physics, Mathematics, and Astronomy Division, Dr. Mildred Finch, in the episode "Waste Not", makes Charlie head of the Ph.D. admissions committee against his wishes, bogging him down with more work, and gives him pressures about using the school's supercomputer for FBI work or missing classes for such. This leads to Charlie being confrontational, but she calms him when she says she just wants him to be "the Sean Connery of the mathematics department." In the episode "Take Out", Charlie and Millie attend a black tie reception for CalSci fundraising and bond. Soon after, Charlie is asked by Millie to meet with Macmillan Pharmaceuticals, which according to Amita has a reputation for exploiting third-world countries. Amita serves as his conscience in this matter, and he goes to see Dr. Finch about it. With "Pythagorean wit ," he dazzles Macmillan and is to serve as a mathematical consultant for the pharmaco-kinetic modelling project, only under the proviso that CalSci will administer the trials and monitor the drug at every step, all computational analyses are conducted by Charlie and his team, and 5% of the gross will go to third-world AIDS organizations chosen by Professors Eppes and Ramanujan . In "Democracy", as part of his duties, he recruits the young fantasy baseball and sabermetrics fanatic Oswald Kittner (Jay Baruchel), who shows great promise, to attend CalSci.
When Larry announces his leave of absence to board the International Space Station (in the episode "Brutus"), Charlie is shocked and upset. He is in denial about Larry's ambitions and he thinks reason of the risks involved will assert itself firmly in Larry's mind. He reasons that apparently Larry wouldn't do such a thing. Larry remands his few prized possessions to him, and he is grateful for the gesture. When he tells him his concerns, Larry is angered. Amita says to Larry, "[Charlie] has never dreamt of something he couldn't reach, so he has no idea what it is like to want something he is not able to get. So how could he understand how much [Larry] you would give up when this chance comes along?" Larry understands and decides to ignore Charlie's protestations. Though neither apologize, they are on good terms. Charlie says to Amita that he does not know what he would do without him, as he peruses through Larry's precious items that were bestowed upon him. In "Killer Chat", he says that he was glad that Larry's dream could come true, but feels conflicted about being relieved when he discovers that Larry might have been scrapped from the mission when NASA learned of his eccentric indulgences such as sleeping in the campus steam tunnels. However, Charlie agrees with Megan in wanting to help Larry and personally vouches for him. Eppes' words and affiliation with the NSA gets him back on the mission, thus, repaying a debt as Larry helped launch him into the academic firmament. Earlier, he had given him back his lucky t-shirt.
In "The Art of Reckoning", Charlie is initially overjoyed to see his friend Larry return, but Charlie is dismayed with Larry's lack of enthusiasm in life, so Charlie doesn't object to him readjusting at a monastery. His concern is apparent, and he says he just wants his friend back.
Since then, in "Trust Metric", Charlie is pleased to see his friend doing well, feeling that he needed a friend, and is glad to have focused on teaching while Don did not include him with FBI work; though, on previous occasions, he expressed the desire to be involved. Granger's escape from the prison bus causes him to become active in helping his brother once more, allowing him a chance to use set covering deployment.
Dr. Finch tells him to publish, and so he has renewed interest in old research, having decided to publish one of his eleventh-grade papers he started at nine years old, "The Mathematics of Friendship," with an addendum. A publishing company has turned his work into a classic book for those not mathematically inclined, while opting for a title with more pizazz, "Friendship, As Easy as Pi." Charlie takes joy in the belief that this book will allow his thoughts to reach a much wider audience than before. By the episode "In Security," the published book appears with the title "The Attraction Equation" and a dapper photo on the back cover of him holding a sculpture of a stellated icosidodecahedron with bevelled edges. A decision theoretic approach to relationships is covered in the book. His proud father hands copies to friends and Larry sells signed copies on eBay. He apparently has some fans and gives into a televised interview.
In "Checkmate", Charlie is training in weapons and tactics in LA's FBI school, where he is shown to have a high skill in marksmanship at the range, impressing many of his FBI training peers as well as his instructor. He is given a certificate of marksmanship by the FBI. It is unclear whether Charlie's certification will allow him to take part in dangerous situations with the rest of the team, or that he will now be allowed to carry a service pistol. In the episode "Pay to Play", Charlie convinces Don to let him join the team when they go to arrest a suspect, mentioning that he passed the FBI course.
In the season four finale "When Worlds Collide", Charlie helps an innocent colleague accused of terrorism by sending genetic research to scientists at Pakistani universities which is prohibited, with full cognizance of the consequences. As a result, Charlie is arrested, loses his security clearance and ultimately loses his ability to help Don on FBI cases. Once again, Charlie holds strong to his ideals. In the previous season's "Burn Rate", Charlie has strong opinions concerning genetic engineering, believing bomb suspect, fellow prodigy, physicist and Feynman student Emmett Glaser's ideas about genetic predeterminism are rational, not incendiary, and helps to clear his name.
The charges against Dr. Eppes are dropped. At present, Don Eppes' team is attempting to make due with Fleinhardt's and Amita's expertise, but Charles' assistance is sorely missed to the point of consulting him secretly. For his part, Charles has been convinced by his lawyer and father to attempt to have his clearance restored. Even Don now supports the idea and stands up to security clearance investigator Carl McGowan (Keith Carradine), stating to go after him, not his brother.
CharlieVision (as labeled by the show's creators) is the mode in which Charlie's insights are displayed on-screen. It consists of fast-paced visions or cutscenes often characterized by false-color images that integrate his analogies and mathematical models, usually followed by him rushing off to tell Don about his new insights. 'CharlieVision' is not to be confused with "audience visions," in which Charlie's voice uses an analogy to simplify a mathematical concept while corresponding images are flashed on screen.
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