It is well known that a generic function on a compact smooth manifold has only Morse critical points (i.e. locally equivalent to a non-degenerate quadratic form). Counting gradient lines between critical points (saddle connections) one obtains a differential on so called Morse complex, giving a way to calculate cohomology of the underlying manifold with integer coefficients. About 40 years ago S.P.Novikov proposed a generalization of Morse theory when the function is replaced by a closed 1-form.
I will talk about new developments in Morse-Novikov theory when the 1-form is real part of a holomorphic 1-form rescaled by a non-zero complex parameter t . A remarkable feature of the holomorphic situation is that for a generic value of the argument of t there is no saddle connections at all, and one obtains a canonical basis in Morse-Novikov cohomology, represented by "infinite chains" which are typically everywhere dense.
For countably many special values of the argument of t one obtains a canonical change of the basis, giving a new elementary example of so-called Wall-Crossing structure (originally discovered in much more complicated theory of Donaldson-Thomas invariants). In concrete terms, one obtains a relation between some explicit rational matrix-valued functions in several variables, and some counting problems in dynamical systems. I will also review the connection of new theory to resurgent series, like e.g. Stirling formula for the asymptotic of Gamma function at infinity.
Date & Time : September 1(Wednesday), 2(Thursday), 10 - 11 am
Speaker : Prof. Henri Darmon (McGill University)
Title : The theory of complex multiplication, and beyond
The theory of complex multiplication gives an explicit construction of abelian extensions of quadratic imaginary fields (or more general CM fields) from the values of modular functions on modular curves (or more general higher dimensional Shimura varieties) at CM points. We will describe an approach to extending this theory beyond the setting of CM fields, in which modular functions are replaced by ``rigid meromorphic cocycles”.
Date & Time : July 30(Friday), 3 - 4 pm
Speaker : Prof. Caucher Birkar (Tsinghua, Cambridge)
The classification of algebraic varieties is at the heart of algebraic geometry. With roots in the ancient world the theory saw great advances in dimensions one and two in the 19th century and the first half of 20th century. It was only in the 1970-80’s that a general framework was formulated, and by the early 1990’s a satisfactory theory was developed in dimension 3. The last 30 years has seen great progress in all dimensions.
In this talk I will try to give a historical perspective and discuss the theory in general terms. I will explain how the theory is based on birational transformations and moduli considerations.
Boundedness and moduli spaces play important roles in algebraic geometry. The subject has evolved in different directions depending on the type of objects being parameterized. In this talk I will focus on boundedness and moduli of algebraic varieties. In particular, I will discuss Fano and Calabi-Yau varieties as well as minimal models of arbitrary Kodaira dimension.
The shift locus S_d is the space of conjugacy classes of degree d polynomials f(z) in one complex variable for which all the critical points tend to infinity under repeated application of f. When d=2 this is the complement of the Mandelbrot set. Although S_d is a very complicated space geometrically, it turns out one can get a surprisingly concrete description of its topology; for example, S_2 is homeomorphic to an open annulus (this is equivalent to the famous theorem of Douady-Hubbard that the Mandelbrot set is connected). I would like to discuss two very explicit ways to capture the topology of S_d, one via the combinatorics of laminations (Butcher paper) and one via algebraic geometry (Sausages). As a corollary of this explicit description one can show that S_d has the homotopy type of a complex of half its real dimension, and is a K(pi,1) if d is 3 or 4.