Abstract. Local Langlands correspondence (LLC) is a conjectural parametrization of complex representations of reductive groups over local fields, by Langlands parameters, which are analogies of Galois representations. In these talks I will give the basic definition of these objects, explain the expected properties of LLC, and work explicitly with the case of GL2. If time permits, I will also introduce some recent progress in LLC and the technique used in the proof. Familiarity with basic Lie theory and algebraic number theory will be helpful.
Abstract. The concept of moduli is to analyze families of objects with some fixed invariants/properties. In particular, one of the problems is the construction of the moduli space itself. In this minicourse we will construct the moduli of line bundles on a surface (under the right algebro-geometric adjectives), which is called the Picard scheme. In the course of doing so, we will also construct the moduli of curves (divisors) on a surface. Using the same arguments one can in general show the existence of the moduli of sheaves of ideals/ moduli of subvarieties known as the Hilbert schemes, and the same construction of the Picard scheme works in any dimension. Also, we will make some remarks on the smoothness of those schemes. A first course in algebraic geometry is helpful (basics of schemes and cohomology). We will follow Mumford's book "Lectures on Curves on an Algebraic Surface".
Abstract. I will give an introduction to Serre duality and Grothendieck duality of coherent sheaves from a category theoretic point of view, following some recent advances. Prerequisites are basic algebraic geometry and category theory, though I will also recall the relevant categorical inputs to the theory.
Abstract. Let X be a smooth projective variety over F_q (for q a prime power), so X is globally cut out by a finite set of homogeneous polynomials with coefficients in F_q. One important arithmetic question is to determine how many simultaneous solutions exist in F_{q^n} for each n. One can put this data together into a zeta function Z_X(t) \in Q[[t]] whose logarithmic derivative is precisely the 'naive' generating function for N_n. Information about the zeros and poles of Z_X(t) can yield information about the growth rate of its logarithmic derivative in a precise way, and vastly generalizing this, Weil conjectured very special structural properties for Z_X(t) informed by certain computations he worked out for curves.
Moreover, Weil noted that if you could define a cohomology theory mimicking certain natural topological properties, then his conjectures (minus the Riemann hypothesis) would more or less follow formally. Such a cohomology theory was realized by Grothendieck and Artin in the 1960s in the form of \ell-adic cohomology, which is loosely speaking a 'repackaging' of etale cohomology. We will introduce and develop the theory of etale cohomology and apply it to the study of \ell-adic cohomology, focusing on certain key properties of etale cohomology most relevant to proving the Weil conjectures. We will mostly follow the exposition in Milne's "Lectures on Etale Cohomology," among other references.
[For a more detailed description of the course, click the "Notes" link to the right.]
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