Fall 2024 topics wishlist:
Quantum Character Varieties
Schbert Calculus
(p-adic) Hodge Theory
Why CY manifolds/K3 surfaces are important?
Motives for babies
Mirror Symmetry and Lefschetz Principle
Applications of Model Theory in Number Theory
Etale fundamental groups and the section conjecture
Intersection (Co)homology
Hilbert's 10th Problem
Classification of surfaces & MMP
Galois/Automorphic Representation
Geometric Langlands Program
Bundles on Curves: Birkoff-Grothendieck Theorem, Atiyah's Classification of vector bundles on elliptic curves, Harder-Narasimhan Filtration, Seshadri's Theorem, Higgs bundles, Nonabelian Hodge Theory, Stack Bun_G(C)
Analytic Langlands
Witten-Kapustin's paper on Langlands and S-duality
Cohomological Field Theory
Brill-Noether Theory: Tevelev's Scattering amplitudes of stable curves
Spectral curves in matrix model
Derived Algebraic Geometry
Fourier-Mukai Transformation and Bondal-Orlov Theorem
Iwasawa Theory
Eichler-Shimura Theory
Serre's modularity conjecture
Local-Global Principles for Torsors
Intersection theory
Brauer-Manin obstructions for curves and surfaces
Deformation Theory
GAGA Principle and Rieman existence theorem
Log Geometry
Kollar's Bend and Break
Rank of elliptic curves
Deligne-Illusie on degeneration of Hodge spectral seqeunce and Kodaira vanishing theorem
Rational varieties
Tropical Geometry, correspondence theorem
Monstrous Moonshine
Fall 2023 topics wishlist:
toric varieties
(p-adic) Hodge Theory
Tate's thesis (Fourier analysis on number fields)
K3 surfaces
moduli spaces+problems
rigid/formal geometry
Hecke operators/correspondences (translating NT and geo)
vertex algebras
combinatorial Hodge theory
arakelov theory
derived AG
log geometry
tropical geometry
classification of surfaces (over C)
what is... (geometric) Langlands? a stack? a motive? a perfectoid space?
Mumford's GIT
Intro to Schubert calculus
(algebraic) K-theory
Hilbert Irreducibility and generalizations
McKay correspondence
F-theory
Spring 2023 topics wishlist:
What is Etale cohomology?
Weil conjectures
What are motives? (Explain like I am 5)
Geometry and arithmetic of K3 surfaces
Brauer-Manin obstruction for K3 surfaces
Toric varieties for dummies
p-adic Hodge theory for beginners
Arithmetic schemes/surfaces (Arakelov theory)
Who were the Fields medalists? (pick a Fields medalist and introduce their work)
Perfectoid spaces
Mazur’s theorem on torsion in elliptic curves
Duality theorems in topology, AG and arithmetic
Monodromy representations
Moduli of curves - introduction and the irreducibility theorem
Gromov-Witten theory
Kodaira-Parshin construction (related to moduli of curves + Mordell conjecture)
What are Chow rings?
Fano varieties and Fano problems
Top 10 things you need to know about rational points on varieties
What are Neron models?
Good reduction of abelian varieties (Serre - Tate)
Jouanolou device
Bridgeland stability
Hensel’s lemma, Henselian rings
Tate’s thesis
Modular forms
Fermat’s theorem / Galois representation
Knots and primes
Iwasawa theory
Inverse Galois theory (Shafrevich’s theorem on solvable groups)
Bruhat-Tits
What is a Calabi-Yau manifold?
Inflexion of linear series on curves
Fukaya category
Representation Theory of Reductive p-adic groups (or harmonic analysis on p-adic groups)