Explore advanced topics in commutative algebra and algebraic geometry through this comprehensive mathematical school focusing on prime characteristics and their connections to complex algebraic geometry. Delve into four major lecture courses covering vector bundles and tight closure, singularities in prime characteristics, F-signature and Hilbert-Kunz multiplicity, and the study of singularities via Big Cohen-Macaulay (BCM)-algebras, perfectoid spaces and Frobenius splitting. Master fundamental concepts including F-singularities and their applications of characteristic p methods to singularity theory, polynomial invariant rings in modular invariant theory, and local cohomology of invariant rings. Investigate Groebner deformations and F-singularities, singularity invariants in positive characteristic, and the Frobenius test exponents in prime characteristic. Examine F-threshold of filtration of ideals, F-rationality of blow-ups, and perturbations of ideals in local rings. Study epsilon multiplicity in graded dimension two and density functions, the weak Lefschetz property of monomial ideals, and mixed multiplicities of ideals. Analyze Hilbert functions of Artinian local complete intersections, Betti cones over fiber products, and regularity of symbolic and ordinary powers of weighted oriented graphs. Learn about symbolic powers of edge ideals, invariants of binomial edge ideals via linear programming, and tight Hilbert polynomial and F-rational local rings. Explore affine semigroups of maximal projective dimension, bounds for reduction numbers of primary ideals in dimension three, and the v-number of monomial ideals. Investigate Betti sequences of projective closures of affine monomial curves, images of linear derivations and Mathieu-Zhao subspaces, (A,d)-modules, Hochschild homology and higher derivations, and upper bounds on Hilbert coefficients.

Syllabus

Vector bundles and tight closure
F-singularities — applications of characteristic p methods to singularity theory
Polynomial invariant rings in modular invariant theory
Local cohomology of invariant rings
Groebner deformations and F-singularities
Singularity invariants in positive characteristic
Singularities via BCM Algebras
The Frobenius test exponents in prime characteristic
F-threshold of filtration of ideals
The F-rationality of Blow-up
Vector bundles and tight closure
F-singularities — applications of characteristic p methods to singularity theory
Perturbations of ideals in local rings
Epsilon multiplicity in graded dimension two and density functions
A density function for a Noetherian filtration of homogeneous ideals
Singularity invariants in positive characteristic
Singularities via BCM Algebras
The Weak Lefschetz Property of monomial ideals and Mixed Multiplicities of ideals
The Hilbert functions of Artinian local complete intersections
Vector bundles and tight closure
F-singularities — applications of characteristic p methods to singularity theory
Betti cones over fiber products
On regularity of symbolic and ordinary powers of weighted oriented graphs
Singularity invariants in positive characteristic
Singularities via BCM Algebras
Symbolic powers of edge ideals of weighted oriented graphs
Invariants of binomial edge ideals via linear programs
Tight Hilbert polynomial and F-rational local rings
Vector bundles and tight closure
F-singularities — applications of characteristic p methods to singularity theory
Affine semigroups of maximal projective dimension
Bounds for the reduction number of primary ideals in dimension three
Affine semigroups of maximal projective dimension
The v-Number of Monomial Ideals
Betti sequence of the projective closure of affine monomial curve
Singularity invariants in positive characteristic
Singularities via BCM Algebras
Image of linear derivations and Mathieu-Zhao subspaces
(A,d)-modules, Hochschild homology and higher derivations
Upper bounds on two Hilbert coefficients