Undergraduate Student Seminar (2020 summer)
Undergraduate Student Seminar (2020 summer)
This seminar is an opportunity for undergraduate students to learn several basic topics in number theory, distribution theory, and some topics beyond calculus and linear algebra.
The main goals are
Algebraic method in number theory.
Distribution theory and Fourier analysis.
Stokes' theorem on manifolds.
陳奕勳 Metric Space and Topological Space. note
王羿璁 Group, Cyclic group, and symmetry group. note
陳奕勳 Initial Topology, Final Topology, and Topology on Function Space. note
姚皓勻 Isomorphism Theorem of Group and Lagrange Theorem. note
莊秉勳 Paracompact and Partition of Unity. note
姚皓勻 Polynomial Ring and Finite Field. note
顏逸儒 Lebesgue Measure in Euclidean Space. note
姚皓勻 Ring and Modules. note
顏逸儒 Lebesgue Integral and Convergence Theorems. note
莊秉勳 Category Language. note
The main reference is A Course in Arithmetic written by Serre, J-P.
Speaker: 王羿璁
Chevalley-Warning Theorem and Quadratic Reciprocity. note
p-adic Number. note
Hilbert Symbol. note
Quadratic Form and Hasse–Minkowski Theorem. note
Integral Quadratic Forms with Discriminant ±1. note
Speaker: 莊秉勳 Combinatorial Nullstellensatz and Applications. note handnote
The main references are Analysis of linear partial differential operators i: distribution theory and fourier analysis written by L. Hormander, Functional analysis written by Walter Rudin and Functional analysis written by Peter Lax.
Speaker: 陳奕勳
Partition of Unity in Euclidean Space. note
Introduction to Distributions. note
Convolution on Distributions. note
Uniform Bounded Principle and Completeness of Distribution Space. note
Schwartz Kernel Theorem and Introduction to Fourier Analysis. note
Tempered Distributions and Its Fourier Transform . note
Fourier Transform of Gaussian Functions. note
The Method of Stationary Phase and Stirling's Formula Pro.
夏維澤 Two Great Theorems on Permutations. note exercise
Reference: John D. Dixon. “The probability of generating the symmetric group”. In: Mathematische Zeitschrift 110.3 (1969), pp. 199–205.
廖松毅 Three Squares and Fermat Polygonal Number Theorem. note
Reference: Jean-Pierre Serre. “Part I, Chapter VI, Appendix”. In: A course in arithmetic. Springer, 1985.
John D. Dixon. “The probability of generating the symmetric group”. In: Mathematische Zeitschrift 110.3 (1969), pp. 199–205.
Jean-Pierre Serre. “Part I”. In: A course in arithmetic. Springer, 1985.
Raoul Bott and Loring W. Tu. Differential forms in algebraic topology. Springer, 1986.
Glen E. Bredon. “Chapter I, II”. In: Topology and geometry. Springer, 1995.
Peter D. Lax. Functional analysis. Wiley, 2002.
David Steven. Dummit and Richard M. Foote. Abstract algebra. John Wiley & Sons, 2004.
Elias M. Stein and Rami Shakarchi. “Chapter 1, 2”. In: Real analysis: measure theory, integration,and Hilbert spaces. Princeton University Press, 2005.
Joseph J. Rotman. An introduction to homological algebra. Springer Science and Business Media, 2009.
L. Hormander. Analysis of linear partial differential operators i: distribution theory and fourier analysis. Springer, 2012.
Walter Rudin. “Chapter 1,6”. In: Functional analysis. McGraw-Hill, 2014.