Basic Notions

  • 2021 Jan 14

    Basic Notions: Kobi Peterzil (U. of Haifa) "The Pila-Zannier method: applications of model theory to Diophantine geometry".

    4:00pm to 5:15pm

    Location: 

    Zoom

     A family of problems in Diophantine geometry has the following
    form: We fix a collection of "special" algebraic varieties among which the
    0-dimensional are called "special points". Mostly, if V is a special variety
    then the special points are Zariski dense in V, and the problem is to prove
    the converse: If V is an irreducible algebraic variety and the special
    points are Zariski dense in V then V itself is special.

    Particular cases of the above are the Manin-Mumford conjecture

  • 2021 Jan 07

    Basic Notions: Kobi Peterzil (U. of Haifa) "The Pila-Zannier method: applications of model theory to Diophantine geometry".

    4:00pm to 5:15pm

    Location: 

    Zoom

     A family of problems in Diophantine geometry has the following
    form: We fix a collection of "special" algebraic varieties among which the
    0-dimensional are called "special points". Mostly, if V is a special variety
    then the special points are Zariski dense in V, and the problem is to prove
    the converse: If V is an irreducible algebraic variety and the special
    points are Zariski dense in V then V itself is special.

    Particular cases of the above are the Manin-Mumford conjecture

  • 2020 Dec 31

    Basic Notions: Ari Shnidman "Randomness in arithmetic: class groups."

    4:00pm to 5:15pm

    Location: 

    Zoom

    For everynumber field K, there is a finite abelian group C called the
    class group, which serves as an obstruction to unique factorization.
    Since Gauss, number theorists have tried to understand questions such
    as how often is C trivial, or how often C contains an element of fixed
    order (as K varies). In the 1970's, Cohen and Lenstra observed
    empirically that when the degree and signature of K is fixed, the
    isomorphism class of C adheres to a natural probability distribution.
    I'll discuss these Cohen-Lenstra heuristics and survey what is known,

  • 2020 Dec 24

    Basic Notions: Ari Shnidman "Randomness in arithmetic: class groups."

    4:00pm to 5:15pm

    Location: 

    Zoom
    For every number fieldK, there is a finite abelian group C called the
    class group, which serves as an obstruction to unique factorization.
    Since Gauss, number theorists have tried to understand questions such
    as how often is C trivial, or how often C contains an element of fixed
    order (as K varies). In the 1970's, Cohen and Lenstra observed
    empirically that when the degree and signature of K is fixed, the
    isomorphism class of C adheres to a natural probability distribution.
    I'll discuss these Cohen-Lenstra heuristics and survey what is known,
  • 2020 Dec 03

    Cancelled - Basic Notions: Zeev Rudnik "The Robin eigenvalue problem: statistics and arithmetic"

    4:00pm to 5:15pm

    Location: 

    Zoom
    Abstract:  

    Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions.  I have recently started to explore the statistics of these Robin eigenvalues for planar domains, and the fluctuations of the gaps between the Robin and Neumann spectrum, in part driven by numerical experimentation.

  • 2020 Nov 26

    Basic Notions: Zeev Rudnik "The Robin eigenvalue problem: statistics and arithmetic"

    4:00pm to 5:15pm

    Location: 

    Zoom
    Abstract:  Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions.  I have recently started to explore the statistics of these Robin eigenvalues for planar domains, and the fluctuations of the gaps between the Robin and Neumann spectrum, in part driven by numerical experimentation.
  • 2020 Nov 19

    Basic Notions: Jake Solomon "Geometric stability"

    4:00pm to 5:15pm

    Abstract: Thestudy of geometric stability begins with Mumford's geometric invariant theory.The Kempf-Ness theorem establishes a connection between geometric invarianttheory and symplectic quotients. An infinite dimensional analog of theKempf-Ness theorem leads to a deep connection between algebraic geometricstability and special metric geometries. Examples of this connection includethe work of Donaldson and Uhlenbeck-Yau on the Kobayashi-Hitchin correspondenceand work of Yau, Tian, Donaldson and many others on extremal Kahler metrics.

  • 2020 Nov 12

    Basic Notions: Jake Solomon "Geometric stability"

    4:00pm to 5:15pm

    Abstract: The study of geometric stability begins with Mumford's geometric invariant theory. The Kempf-Ness theorem establishes a connection between geometric invariant theory and symplectic quotients. An infinite-dimensional analog of the Kempf-Ness theorem leads to a deep connection between algebraic-geometric stability and special metric geometries. Examples of this connection include the work of Donaldson and Uhlenbeck-Yau on the Kobayashi-Hitchin correspondence and work of Yau, Tian, Donaldson, and many others on extremal Kahler metrics.

  • 2020 May 28

    Basic Notions: Michael Tenkin "Resolution: classical, relative and weighted"

    4:00pm to 5:15pm

    Location: 

    Join Zoom Meeting https://huji.zoom.us/j/98768675115?pwd=WnZOZUpuVmpoNGkrYWQxanNVWkQzUT09,

    Until recently there was known an essentially unique way to resolve singularities of varieties of 
    characteristic zero in a canonical way (though there were different descriptions and proofs of correctness). 
    Recently a few advances happened -- 
    1) The algorithm was extended to varieties with log structures and even to morphisms -- This requires to consider 
    more general logarithmic blow ups and results in a stronger logarithmic canonicity of the algorithms even for resolution 

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