Manual Algebraic Geometry and Singularities

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They are homogeneous, meaning that, for every x and y in a component of one of these spaces, there is a homeomorphism carrying x to y. The situation for nonsimply connected manifolds is more complicated.

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In particular, none of our new spaces can have the homotopy type of a torus. In high dimensions, the h- and s-cobordism theorems hold for these topologically exotic manifolds.

Stefan Kebekus The geometry of singularities in the Minimal Model Program and applications to singul

These exotic manifolds complete Siebenmann periodicity in the topological category and remove a discrepancy between the geometric and algebraic classifications of topological manifolds. This work is joint with J. In this survey talk, we discuss four singular toric varieties different equivariant characteristic classes like Chern, Todd, L and Hirzebruch classes, with applications to weighted lattice points counting and Euler-MacLaurin type formulae for lattice polytopes.

Maxim will present a new characterization of perverse sheaves on complex semi-abelian varieties in terms of their cohomology jump loci, generalizing results of Gabber-Loeser and Schnell.

Singularity Theory and Algebraic Geometry - BCAM - Basque Center for Applied Mathematics

He will also discuss propagation properties and codimension lower bounds for the cohomology jump loci of perverse sheaves. As concrete applications, Maxim will mention: a generic vanishing for perverse sheaves on semi-abelian varieties; b homological duality properties of complex algebraic manifolds, via abelian duality; and c new topological characterizations of semi- abelian varieties. This is joint work with Y. Liu and B.

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Geodesic nets are singular objects that are homological equivalents of periodic geodesics. They turn out to be useful for problems about closed geodesics. Geodesic nets can also be used to prove the existence of periodic geodesics on complete, non-compact Riemannian manifolds satisfying an additional, easy-to-state geometric assumption. Various notions of weight play a role in the study of singularities.

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They arise from different fields: characteristic p algebraic geometry, representation theory, metrical decay rates, mixed Hodge theory. We will discuss the relations between these notions and their application to the study of singularities. If there is time, we will discuss ongoing work on a new theory coming from CR geometry. We prove a formula computing the Chern-Schwartz-MacPherson CSM class of an arbitrary sub-scheme of a nonsingular variety in terms of the Segre class of an associated scheme.

This formula generalizes an old result expressing the CSM class of a hypersurface in terms of the Segre class of its singularity sub-scheme. For local complete intersections, the result yields a new expression for the Milnor class. Balmer-Witt groups of constructible derived categories are a natural home for signatures and related invariants of stratified and singular spaces. Woolf will give a survey of the theory and some of its applications. Factorization homology simultaneously generalizes singular homology, Hochschild homology and conformal blocks or observables in conformal field theory.

He will discuss applications of this alpha form of factorization homology in the study of mapping spaces in algebraic topology, bundles on algebraic curves and perturbative quantum field theory. An immediate consequence is a proof of the cobordism hypothesis after Baez-Dolan, Costello, Hopkins-Lurie, and Lurie exactly in the manner of Pontryagin-Thom theory. This is joint work with David Ayala. Intersection cohomology has been introduced for pseudomanifolds and its behavior is mainly derived from sheaf theory.

In this talk, we show how cochain complexes can also be used in an efficient way. We present two cohomologies. A simple calculation on the cone of a manifold shows that they are, in general, different. By using this kind of Lefschetz duality, we specify existence and defects of duality in intersection homology. If part of these results is already known, the originality comes from the use of cup and cap products to express them, as in the case of manifolds.

Jakob Nielsen asked if a finite subgroup of outer automorphisms of the fundamental group of a compact surface can be realized by a group action.

This was proved by Steve Kerckho in It is an open question whether this same statement is true for any compact, negatively curved manifold. Surgery theory, a geometric construction, Cappell UNil groups and the Farrell-Jones conjecture gives positive results in a special case.

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This is the first positive process in Nielsen realization in twenty years. We will also discuss connections with the existence part of the equivariant Borel conjecture. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians working in other areas. In this talk, I will explain for all scientists how singularities are studied in algebraic geometry. In algebraic geometry, we study algebraic varieties, which are figures defined as the zero sets of polynomial equations.

To study an algebraic variety, we often expect that the variety is smooth, that is, the variety locally resembles Euclidian spaces. However, even if we start from smooth varieties, we sometimes encounter non-smooth varieties.


Resolution of Singularities in algebraic geometry

Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians working in other areas. In this talk, I will explain for all scientists how singularities are studied in algebraic geometry. In algebraic geometry, we study algebraic varieties, which are figures defined as the zero sets of polynomial equations.