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 Fleinhardt.

He has also provided insights for possibly solving the P vs. NP problem and published works on H-infinity control of nonlinear systems and computational fluid dynamics, while his current research is in cognitive emergence theory.

Charlie is rather fond of providing excessive explanations (e.g., his discussion of Occam's razor) and reducing complex phenomena to intuitively obvious situations through practical analogies that are quite unlike Larry's metaphysical musings and cosmic metaphors.

Wearing his headset, Dr. Eppes has an intense focusing ability as he voraciously writes equations, often covering several chalkboards with a staccato clacking and the aid of a red chalk holder.

Further, when deeply concentrating on a particular problem, it seems Charlie is unable to provide insights to other topics for the sake that they are simply needed or wanted—he has to write what is in his head.

Charlie also has a 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.

Dr. Fleinhardt has stated that it is a good thing he went into applied mathematics as opposed to engineering, as machines malfunction in his presence, though he was able to take apart and rebuild his father's cell phone to analyze its GPS transceiver.

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).

[2] The episode "Uncertainty Principle" is significant for the backstory it gives on the familial relationships, particularly Charlie's difficulty in dealing with his mother's death from cancer.

While it's unclear how Don reacted, Charlie spent the last three months of his mother's life isolated in the garage, incessantly working on one of the Millennium Prize Problems, specifically P vs NP; it's a point of contention between the brothers.

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 sets 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.

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 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.

Pressures from their colleagues over the inappropriateness of the relationship nearly cost them, but by the middle of the third season their romantic involvement stabilizes and they have grown considerably closer.

With "Pythagorean wit," he dazzles Macmillan and is to serve as a mathematical consultant for the pharmaco-kinetic modelling project, only under the provision 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.

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."

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.

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.

For a time, Don Eppes' team is attempting to make do with Fleinhardt's and Amita's expertise, but Charles' assistance is sorely missed to the point of consulting him secretly.

Even Don supports the idea and stands up to security clearance investigator Carl McGowan (Keith Carradine), stating to go after him, not his brother.

Head of DARPA special projects Jane Karellen (Nancy Travis) knows that he has a limited window to use his genius and tells Charlie that he is one of the top five minds on the planet.

Around this time, Charlie is also presented with a series of letters from previous successive holders of this prestigious office, wherein each celebrated mathematician writes of accomplishments they intend to achieve, passing down the torch to the next in line to do the same.

In the season finale, after the wedding ceremony presided over by Larry, he is happy but worries about disconnecting with Don and offers to have the garage renovated into a guest house for his father.

[16] Early on, Professor Rick Wilson's graduate student, David Grynkiewicz, showed Krumholtz how to write his own equations and even filled in for his hand in several episodes.

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.