We usually meet on Wednesday in 508 Math at 7:00 PM and the talk takes place at 7:30 PM in 507 Math.
May 29 *Wednesday* Sicong Zhang (Columbia Undergrad): Complex manifolds.
Abstract: In this talk, we introduce the leading actor of this summer: complex differentiable manifolds. Basic facts about them are covered in details.
June 5 *Wednesday* Yifei Zhao (Columbia Undergrad): Comparison I: derivatives on complex vs. real manifolds; introduction to Kähler manifolds.
Abstract: we continue with basic constructions on a complex manifold, including real and holomorphic tangent bundles. Metric aspects of them are
covered in detail. We define the Chern connection on a holomoprhic vector bundle, and show that for the Chern connection to agree with the Levi-Civita
connection, the Kähler condition has to be imposed. Next, we study general aspects of Kähler manifolds. This include, in particular, Hodge theory on
Kähler manifolds and its miraculous implications on determining various cohomology groups associated to a Kähler manifold.
June 12 *Wednesday* John Yu (Columbia Undergrad): Complex algebraic varieties.
Abstract: in this talk, we start introducing the leading actress of this summer: complex algebraic varieties. We prove the weak and strong Nullstellensatz
using Noether normalization, for an arbitrary algebraically closed field. We also discuss the Zariski topology, and define affine and quasi-affine varieties.
June 19 *Wednesday* John Yu: Sheaves and sheaf cohomology.
Abstract: we finish the discussion on affine and quasi-affine varieties, their coordinate ring, function field, and regular functions. We then turn to
sheaves, and prove their local nature (a sheaf morphism is bijective if and only if it is so on stalks), and the failure of right-exactness of the global section
functor. The machinary to remedy this failure, sheaf cohomology, will appear in subsequent talks.
June 26 *Wednesday* Cancelled (due to window replacement).
July 3 *Wednesday* Yifei Zhao: Comparison II: function fields; Siegel's theorem on algebraic independence of meromorphic functions.
Abstact: we first finish up the discussion on sheaves, and use them to construct the four ringed spaces we are interested in, namely real manifolds,
complex manifolds, algebraic varieties, and complex analytic spaces, in a somewhat unified manner. We illustrate the use of sheaves (& sheaves of
modules) in a concrete example of canonical line bundles over the projective spaces. Then we discuss meromorphic functions, and show that any (n+1)
meromorphic functions on a n-dimensional complex manifold are algebraically dependent (Siegel's theorem) in detail.
July 10 *Wednesday* John Yu: Comparison III: projective embedding; Chow's theorem.
Abstact: a proof of Chow's theorem based on a theorem in several complex variables (of Remmert and Stein) is presented. We prove Chow's theorem
for complex analytic spaces rather than just complex manifolds. Several applications are mentioned but not proved.
July 17 *Wednesday* Yifei Zhao: Comparison III: projective embedding; Kodaira's embedding theorem.
Abstact: we finally motivate and define sheaf cohomology, and discuss how it is related to topological cohomology (Cech-to-de Rham). As
applications, we discuss the sheaf exponential sequence and the first Chern class of a holomorphic line bundle. We then define positivity of line bundles,
and sketch a proof of Kodaira's embedding theorem: a line bundle on a complex manifold is positive if and only if it is ample. The proof techniques
involve blowing-up at a point, and Kodaira's vanishing theorem, both of which we introduce only briefly.
July 24 *Wednesday* (Bonus talk) Justin Ripley (Columbia undergrad): Quantum mechanics, quantum information and complex projective spaces.
Abstract: We will briefly review the basic ideas of quantum mechanics (ie the Hilbert space formalism, the Born rule, the Schrodigner equation, and the
density matrix). After the summary, we will see how to look at the basic formalism of quantum mechanics as state vectors living in a complex projective
space. This geometric viewpoint will allow a convenient visualization of the simplest nontrivial physical state in quantum mechanics: the qubit, which is a
two-dimensional quantum system. The qubit will be compared to the bit, which is the basic unit of information in classical information theory. We will then
see how a qubit can hold much more information than a bit, but that extra information can only be accessed in a quantum ensemble. This will be
demonstrated via a simple proof of Bell's inequality.
July 31 *Wednesday* Yifei Zhao: Comparison IV: coherent cohomology; the GAGA theorems.
Abstract: we define coherent sheaves over an arbitrary ringed space, and specialize to the case of complex algebraic varieties and analytic spaces (in
the sense of Serre) and define the analytification of an algebraic variety. We then state a few technical lemmas about this construction, including the
canonical map from the analytification to the variety itself, and that it induces a functor from coherent sheaves over the variety to those over its
analytification. We then state GAGA's main theorems, but only prove the equivalence of cohomology groups (GAGA's first theorem).
August 7 *Wednesday* (Bonus talk) Xin Chen: Elliptic functions and modular forms.
Abstract: we will start with elliptic functions, which are functions of a complex variable that have two independent periods (unlike trigonometric
functions which only have one period) and give some well known examples. Next, we will discuss how elliptic functions are connected to elliptic curves
and use elliptic functions to prove the remarkable relationship between the sum of the 7th powers of the divisors of a number and the sum of the 3rd
powers of the divisors of the same number, namely $$\sigma_{7}(n) = \sigma_{3}(n) + 120\sum_{k=1}^{n-1}\sigma_{3}(n - k)\sigma_{3}(k).$$ We will also
talk briefly about modular forms.