Topics Wishlists

Fall 2025 topics wishlist:

• Abelian varieties

• Classification of surfaces

• Fourier-Mukai Transforms

• Etale cohomology

• Deformation Theory

• Crystals

• Mirror symmetry (any kind)

• Moduli of curves

• Joyce VOA

• Condensed Math

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)