• 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 10

# Basic Notions: David Kazhdan "On the Langlands conjecture for curves over local fields."

4:00pm to 5:15pm

## Location:

Zoom

In this talk  I  present conjectures and results of joint work with Pavel Etingof and Edward Frenkel on an analytic version of the Langlands correspondence for curves over local fields.

• 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