The goal of my talk will be to introduce the local and the global
the Class field theory. Though definitions of all basic objects
(inverse limits, p-adic numbers, adeles, ideles, etc.) will be briefly recalled,
it is recommended that the participants review these notions
before the lecture.
The talk will be independent of the first lecture
The main goal of my talk will be to introduce adeles and ideles and to formulate the global class
field theory both for number fields and function fields.
Key words: adeles, ideles, class field theory, algebraic curves.
Abstract: A meet-tree is a partial order such that the set of vertices below any vertex is linearly ordered, and for every pair of vertices there is a greatest element smaller than or equal to each of them. I'll talk on a work in progress with Itay Kaplan and Tomasz Rzepecki mainly showing that the universal homogeneous countable meet-tree admits generic automorphisms.
Abstract: Modular forms are historically the first example of automorphic
forms, and are still studied today as they have many applications. In
this talk I want to introduce modular forms, give some examples, and, if
time permits, explain the connection to elliptic curves, objects we
already met in the first lecture.
Title: An improved bound for Hadwiger’s covering problem via thin shell inequalities for the convolution square. Abstract: A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number N(n) such that every convex body in {\mathbb R}^n can be covered by a union of the interiors of at most N(n) of its translates. Despite continuous efforts, the best-known general upper bound for this number remain as it was more than half a decade ago, and is of the order of \binom{2n}{n}n\ln n.
Speaker: Ehud de Shalit Title:Periods and Hodge theory, complex and p-adic P-adic theory p-adic Hodge theory started with Tate in the 1960's, and witnessed two periods of great expansion: one by Fontaine and his school, the other one more recently, by Scholze and his collaborators. It has found spectacular applications in arithmetic algebraic geometry. Scholze got the Fields medal last August for this work. By analogy with the first lecture we shall try to explain the main ideas, without going into details
Speaker: Ehud de Shalit Title:Periods and Hodge theory, complex and p-adic Complex theory This will be a survey of complex Hodge theory, with applications to period maps and moduli and with an emphasis on the example of abelian varieties. It will cover classical results of Riemann, Siegel, Hodge and Griffiths
This is a new seminar, whose official name is "Topics in number theory and algebraic geometry". At least in the beginning the goal of the seminar will be to give a (relatively) gentle introduction to various topics, which should be accessible to beginning but motivated graduate students. The seminar has a number in the shnaton (80942), so graduate students can get a credit for it. First talk: The goal of this organizational/introductory talk will be to describe areas of mathematics, connected to Langlands program.
Abstract:
The Chabauty method is a remarkable tool which employs p-adic analitic methods (in particular Colman integration.) To study rational points on curves. However the method can be applied only when the genus of the curve in question is larger than its Mordell-Weil rank. Kim developed a sophisticated "nonableian" generalisation.
We shall present the classical methid, and give an approachable introduction to Kim's method.
I'm basically going to follow http://math.mit.edu/nt/old/stage_s18.html
The Whitehead conjecture asks whether a subcomplex of an aspherical 2-complex is always
aspherical. This question is open since 1941. Howie has shown that the existence of a finite counterexample implies (up to the Andrews-Curtis conjecture) the existence of a counterexample within the class of labelled oriented trees. Labelled oriented trees are algebraic generalisations of Wirtinger presentations of knot groups.
Title: Szego theorem for measures on the real line: optimal results and
applications.
Abstract: Measures on the unit circle for which the logarithmic integral
converges can be characterized in many different ways: e.g., through
their Schur parameters or through the predictability of the future from
the past in Gaussian stationary stochastic process. In this talk, we
consider measures on the real line for which logarithmic integral exists
and give their complete characterization in terms of the Hamiltonian in
De Branges canonical system. This provides a generalization of the
Speaker: Bruno Benedetti, U. Miami
Title: Local constructions of manifolds
Abstract: Starting with a tree of tetrahedra, say you are allowed to
recursively glue together some two boundary triangles that have nonempty
intersection. You may perform this type of move as many times you want.
Let us call "Mogami manifolds" the triangulated 3-manifolds (with or
without boundary) that can be obtained this way. Mogami, a quantum
physicist, conjectured in 1995 that all triangulated 3-balls are Mogami.
This conjecture implies an important one in discrete quantum gravity
