conformal maps - 2023 ACF Nationals

Question

Joseph Liouville (“zhoh-ZEFF l’yoo-VEEL”) proved a theorem that classifies all of these functions in a Euclidean (“yoo-CLID-ee-in”) space of degree 3 or higher. For 10 points each:

[10h] Name these functions that exist between any simply connected open subset of the complex plane and the unit disk. These functions have the property of constancy of dilation at every point in their domain.

ANSWER: conformal maps [or conformal mappings or conformal transformations; prompt on biholomorphic maps or holomorphic maps or bijective holomorphic maps by asking “a holomorphic function that is a bijection is what other type of function?”] (The sentence about the complex plane and the unit disk refers to the Riemann mapping theorem.)

[10m] Liouville’s theorem states that conformal maps are all higher dimensional analogues of transformations named for this mathematician. A number theoretic function named for this mathematician outputs one on square-free positive integers with an even number of prime factors.

ANSWER: August Ferdinand Möbius [accept Möbius function]

[10e] Conformal maps preserve these values between curves. Polar coordinates specify points using a distance and one of these values.

ANSWER: angles

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Summary


2023 ACF Nationals	04/22/2023	Y	21	20.00	100%	67%	33%

Data


Chicago C	Florida B	0	10	10	20
Ohio State A	McGill A	0	10	10	20
Rutgers A	Houston A	0	10	10	20
Claremont A	NYU A	0	0	10	10
Harvard A	Chicago A	0	0	10	10
Minnesota A	Maryland A	10	10	10	30
North Carolina A	Imperial A	10	10	10	30
Penn A	Vanderbilt A	0	0	10	10
Stanford A	Brown A	10	10	10	30
Texas A	Yale A	0	10	10	20
South Carolina A	Yale B	0	0	10	10
Columbia B	Florida A	10	10	10	30
Cornell A	Johns Hopkins A	10	10	10	30
Michigan A	Penn State A	0	0	10	10
UC Berkeley B	Rutgers B	10	10	10	30
Columbia A	WUSTL B	0	10	10	20
Northwestern A	Minnesota B	0	0	10	10
Indiana A	Purdue A	0	10	10	20
WUSTL A	Toronto A	0	0	10	10
MIT A	Duke A	0	10	10	20
Chicago B	UC Berkeley A	10	10	10	30