Short mathematical proofs for teaching











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I am looking for short proofs in order to illustrate undergraduate notions. Most of the time students struggle with technical exercises without having the time, before going on with the following semester, to realize some great applications or insights about the objects introduced.



I would like to have some such topics to present in detail, in between 10 and 30 minutes on blackboard. To show some examples in my mind:




  • Poisson formula used to prove Minkowski theorem

  • Dimension of spaces of modular forms

  • Equiperimetric inequalities

  • Prime Number Theorem (or weakened versions) by analytic methods


I would like to develop these themes in many other directions: group actions, representation theory, complex analysis, algebraic number theory, local inversion, PDEs, etc.






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    up vote
    4
    down vote

    favorite
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    I am looking for short proofs in order to illustrate undergraduate notions. Most of the time students struggle with technical exercises without having the time, before going on with the following semester, to realize some great applications or insights about the objects introduced.



    I would like to have some such topics to present in detail, in between 10 and 30 minutes on blackboard. To show some examples in my mind:




    • Poisson formula used to prove Minkowski theorem

    • Dimension of spaces of modular forms

    • Equiperimetric inequalities

    • Prime Number Theorem (or weakened versions) by analytic methods


    I would like to develop these themes in many other directions: group actions, representation theory, complex analysis, algebraic number theory, local inversion, PDEs, etc.






    soft-question education big-list







    share|cite|improve this question
























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      I am looking for short proofs in order to illustrate undergraduate notions. Most of the time students struggle with technical exercises without having the time, before going on with the following semester, to realize some great applications or insights about the objects introduced.



      I would like to have some such topics to present in detail, in between 10 and 30 minutes on blackboard. To show some examples in my mind:




      • Poisson formula used to prove Minkowski theorem

      • Dimension of spaces of modular forms

      • Equiperimetric inequalities

      • Prime Number Theorem (or weakened versions) by analytic methods


      I would like to develop these themes in many other directions: group actions, representation theory, complex analysis, algebraic number theory, local inversion, PDEs, etc.






      soft-question education big-list







      share|cite|improve this question













      I am looking for short proofs in order to illustrate undergraduate notions. Most of the time students struggle with technical exercises without having the time, before going on with the following semester, to realize some great applications or insights about the objects introduced.



      I would like to have some such topics to present in detail, in between 10 and 30 minutes on blackboard. To show some examples in my mind:




      • Poisson formula used to prove Minkowski theorem

      • Dimension of spaces of modular forms

      • Equiperimetric inequalities

      • Prime Number Theorem (or weakened versions) by analytic methods


      I would like to develop these themes in many other directions: group actions, representation theory, complex analysis, algebraic number theory, local inversion, PDEs, etc.






      soft-question education big-list




      soft-question education big-list






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      TheStudent

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          For a class on abstract algebra, there is the proof that the complex numbers are an algebraically closed field, following Artin. This is very short and can be presented in $10$ to $30$ minutes on the blackboard. Before going to the next semester, one obtains it as a nice application on Galois theory. For details, see also here:



          Is there a purely algebraic proof of the Fundamental Theorem of Algebra?






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            For a class on abstract algebra, there is the proof that the complex numbers are an algebraically closed field, following Artin. This is very short and can be presented in $10$ to $30$ minutes on the blackboard. Before going to the next semester, one obtains it as a nice application on Galois theory. For details, see also here:



            Is there a purely algebraic proof of the Fundamental Theorem of Algebra?






            share|cite|improve this answer



























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              For a class on abstract algebra, there is the proof that the complex numbers are an algebraically closed field, following Artin. This is very short and can be presented in $10$ to $30$ minutes on the blackboard. Before going to the next semester, one obtains it as a nice application on Galois theory. For details, see also here:



              Is there a purely algebraic proof of the Fundamental Theorem of Algebra?






              share|cite|improve this answer

























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                up vote
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                For a class on abstract algebra, there is the proof that the complex numbers are an algebraically closed field, following Artin. This is very short and can be presented in $10$ to $30$ minutes on the blackboard. Before going to the next semester, one obtains it as a nice application on Galois theory. For details, see also here:



                Is there a purely algebraic proof of the Fundamental Theorem of Algebra?






                share|cite|improve this answer














                For a class on abstract algebra, there is the proof that the complex numbers are an algebraically closed field, following Artin. This is very short and can be presented in $10$ to $30$ minutes on the blackboard. Before going to the next semester, one obtains it as a nice application on Galois theory. For details, see also here:



                Is there a purely algebraic proof of the Fundamental Theorem of Algebra?







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                Dietrich Burde

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