Page 74 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 74
EMS PRIZE WINNERS
Lefschetz beyond positivity and its implications
Karim Adiprasito, ka@math.ku.dk
University of Copenhagen, Denmark
I will present results of the Hard Lefschetz type that go beyond the classical case of Kaehler
manifolds and projectivity, and survey their applications in combinatorics, geometry and topol-
ogy.
Reciprocity laws for torsion classes
Ana Caraiani, a.caraiani@imperial.ac.uk
Imperial College London, United Kingdom
The Langlands program is a vast network of conjectures that connect many areas of pure math-
ematics, such as number theory, representation theory, and harmonic analysis. At its heart lies
reciprocity, the conjectural relationship between Galois representations and modular, or auto-
morphic forms.
A famous instance of reciprocity is the modularity of elliptic curves over the rational num-
bers: this was the key to Wiles’s proof of Fermat’s last theorem. I will give an overview of some
recent progress in the Langlands program, with a focus on new reciprocity laws over imaginary
quadratic fields.
Smooth compactifications of differential graded categories
Alexander Efimov, efimov@mi-ras.ru
Steklov Mathematical Institute of RAS, Russian Federation
We will give an overview of results on smooth categorical compactifications, the questions of
their existence and their construction. The notion of a categorical smooth compactification is a
straightforward generalization of the corresponding usual notion for algebraic varieties.
First, we will explain the result on the existence of smooth compactifications of derived
categories of coherent sheaves on separated schemes of finite type over a field of characteristic
zero. Namely, such a derived category can be represented as a quotient of the derived category
of a smooth projective variety, by a triangulated subcategory generated by a single object. Then
we will give an example of a homotopically finite DG category which does not have a smooth
compactification: a counterexample to one of the Kontsevich’s conjectures on the generalized
Hodge to de Rham degeneration.
If time permits, we will formulate a K-theoretic criterion for existence of a smooth cate-
gorical compactification, using a DG categorical analogue of Wall’s finiteness obstruction from
topology.
72
Lefschetz beyond positivity and its implications
Karim Adiprasito, ka@math.ku.dk
University of Copenhagen, Denmark
I will present results of the Hard Lefschetz type that go beyond the classical case of Kaehler
manifolds and projectivity, and survey their applications in combinatorics, geometry and topol-
ogy.
Reciprocity laws for torsion classes
Ana Caraiani, a.caraiani@imperial.ac.uk
Imperial College London, United Kingdom
The Langlands program is a vast network of conjectures that connect many areas of pure math-
ematics, such as number theory, representation theory, and harmonic analysis. At its heart lies
reciprocity, the conjectural relationship between Galois representations and modular, or auto-
morphic forms.
A famous instance of reciprocity is the modularity of elliptic curves over the rational num-
bers: this was the key to Wiles’s proof of Fermat’s last theorem. I will give an overview of some
recent progress in the Langlands program, with a focus on new reciprocity laws over imaginary
quadratic fields.
Smooth compactifications of differential graded categories
Alexander Efimov, efimov@mi-ras.ru
Steklov Mathematical Institute of RAS, Russian Federation
We will give an overview of results on smooth categorical compactifications, the questions of
their existence and their construction. The notion of a categorical smooth compactification is a
straightforward generalization of the corresponding usual notion for algebraic varieties.
First, we will explain the result on the existence of smooth compactifications of derived
categories of coherent sheaves on separated schemes of finite type over a field of characteristic
zero. Namely, such a derived category can be represented as a quotient of the derived category
of a smooth projective variety, by a triangulated subcategory generated by a single object. Then
we will give an example of a homotopically finite DG category which does not have a smooth
compactification: a counterexample to one of the Kontsevich’s conjectures on the generalized
Hodge to de Rham degeneration.
If time permits, we will formulate a K-theoretic criterion for existence of a smooth cate-
gorical compactification, using a DG categorical analogue of Wall’s finiteness obstruction from
topology.
72
