벡터공간, 기저, 행렬, 행렬식, 내적, 기저의 직교화, 선형변환, 고유값, 고유벡터, 대각화, 정규형식, 조단형식 등: ‘선형대수학개론’과 ‘선형대수학’에서 배우는 내용
vector spaces, basis, matrices, determinant, inner product, linear transformation, eigenvalues, eigenvectors, diagonalization, canonical form, Jordan form, etc: contents of ‘Introduction to Linear Algebra’ and ‘Linear Algebra’

compactness, 연결성(connectedness), 분리공리(separation axioms), 수열(sequence)과 급수(series): 수렴, 절대수렴(absolute convergence), 연속함수의 성질(properties of continuous functions), 평균값정리(Mean value theorem), 미적분학의 기본정리(Fundamental theorem of calculus), 함수로 이루어진 수열과 급수(Sequence and series of functions), 고른수렴(uniform convergence), Stone-Weierstrass 정리(Stone-Weierstrass theorem), 미분(differentiation)과 리만 적분(Riemann integration), 벡터값함수(vector valued function)와 벡터장(vector field), 중적분(multiple integral)과 변수변환(change of variable), 음함수 정리(implicit function theorem), 선적분(line integral)과 면적분(surface integral), Green’s theorem, divergence theorem, Stokes’ theorem

집합과 논리, 위상공간, 거리공간, 연속함수, 곱공간, 몫공간, 연결성, 컴팩트성, 분리가정, 유리손 보조정리, 거리공간화, 함수공간
sets and logic, topological spaces, metric spaces, continuous functions, product spaces, quotient spaces, connectedness, compactness, axioms of separations, Urysohn Lemma, metrization, function spaces

곡선(curves): 정칙(regular), 단위속도 매개변수(arc length parameter), 곡률(curvature), 비꼬임(torsion), Frenet formula,
곡 면(surfaces): 정칙(regular), 접평면(tangent plane), Gauss map, 기본형식(fundamental forms), 모양연산자(shape operator), 주곡률(principal curvature), Gaussian curvature, 평균곡률(mean curvature), 방향성(orientation), 넓이(area), 선직면(ruled surface), 극소곡면(minimal surfaces), 등장사상(isometry), Gauss 곡률에 관한 정리, 평행이동(parallel transport), 측지선(geodesic), 회전각도에 관한 정리, 측지선 곡률(geodesic curvature), Gauss-Bonnet theorem (local or global), 벡터장(vector field), 벡터장의 지수(Index of a vector field)

변수독립성(variable independence), 변수변환(variable transformation), 다변수정규분포(multivariate normal distribution), 분포적 수렴과 확률적 수렴(convergences in distribution and in probability), 중심극한정리(Central limit theorem), 통계적 검정(statistical testing), 최대우도추정(maximum likelihood estimation), 추정량 품질(quality of estimator), 라오-크레머 하한(Rao-Cramer lower bound), 우도비 검정(likelihood ratio test), 지수족 분포(Exponential family of distributions), 통계량 충분성과 완전성(sufficiency and completeness of statistics), 회귀모형(regression model)

Cauchy-Riemann Condition, Cauchy Theorem, Cauchy Integral Formula, Liouville Theorem, Maximum Modulus Theorem, Open Mapping Theorem, Isolated singularity, Pole, Laurent expansion, Residue Theorem, Linear fractional transformation, Conformal mapping

군(group), 순열군(permutation group), 부분군(subgroup), 순환군(cyclic group), 정규부분군(normal subgroup), factor group(잉여군), homomorphism(동형사상), 유한가환군의 기본정리(fundamental theorem of finitely generated abelian group), Sylow theorems, 자유군(free group), 군 작용(group action), free Abelian group, 환(ring), 부분환(subring), ideal, 잉여환(factor ring), 다항식환(polynomial ring), 행렬환(matrix ring), 소아이디얼(prime ideal), 최대아이디얼(maximal ideal), Chinese remainder theorem

유클리드 정역(Euclidean domain), 중심아이디얼체(principal ideal ring), 유일인수분해정역(unique factorization domain), irreducibility and factorization in domains, 체의 확대(field extension), 분해체(splitting field), 정규 확대(normal extension), 분해 가능 확대(separable extension), 유한체 (finite field), 갈루아 확대(Galois extension), 갈로아 군(Galois group), 갈루아 이론의 기본 정리(fundamental theorem of Galois theory), 갈로아 이론(Galois theory)의 응용

monotone class theorem, measure space, Lebesgue-Stieltjes measure, Lebesgue integral, limit theorems, types of convergence, product measure, signed measure, Radon-Nikodym theorem, differentiation, Lp space, Fourier transform, Riesz representation theorem