Jan 31, 2019
Thursday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


DAG I: the cotangent complex and derived de Rham cohomology
Benjamin Antieau (Northwestern University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this series of lectures, I will give an introduction to derived algebraic geometry aimed at algebraic geometers. The first lecture will introduce simplicial commutative rings and use them to define the cotangent complex and derived de Rham cohomology with several examples. The second lecture will introduce derived stacks and the moduli stack of objects in a derived category. Then, I will give the geometricity theorem of ToënVaquié and describe the cotangent complex to the moduli stack. In the third lecture, we will use the machinery developed in the first two lectures to study three examples: cohomology as maps to a geometric derived stack, the (derived) Picard stack, and the stack of FourierMukai equivalences.
 Supplements

Notes
713 KB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Moduli of canonical models
János Kollár (Princeton University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We give an overview of the current state of the moduli problem for canonical models.
 Supplements

Notes
626 KB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


The uniqueness of Kpolystable Fano degeneration
Chenyang Xu (Massachusetts Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
(Joint with Harold Blum) We want to show that a family of Fano varieties has a unique Kpolystable degeneration. This is one step of the program of constructing a moduli space of Kstable Fano varieties, i.e., proving there is an Artin stack parametrizing Ksemistable Fano varieties, which admits a projective good moduli space parametrizing Kpolystable Fano varieties.
 Supplements

Notes
618 KB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Feb 01, 2019
Friday

09:30 AM  10:30 AM


Birational algebraic geometry in positive characteristic
Karl Schwede (University of Utah)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
This series of introductory talks will discuss some of the issues and pathologies one runs into when exploring birational geometry in positive characteristic. Some time will be spent talking about ideas coming from commutative algebra and their applications, including connections between singularities from the MMP with singularities arising in commutative algebra.
 Supplements

Notes
626 KB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


DAG II: moduli of objects in derived categories
Benjamin Antieau (Northwestern University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this series of lectures, I will give an introduction to derived algebraic geometry aimed at algebraic geometers. The first lecture will introduce simplicial commutative rings and use them to define the cotangent complex and derived de Rham cohomology with several examples. The second lecture will introduce derived stacks and the moduli stack of objects in a derived category. Then, I will give the geometricity theorem of ToënVaquié and describe the cotangent complex to the moduli stack. In the third lecture, we will use the machinery developed in the first two lectures to study three examples: cohomology as maps to a geometric derived stack, the (derived) Picard stack, and the stack of FourierMukai equivalences
 Supplements

Notes
639 KB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Moduli of canonical models
János Kollár (Princeton University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We give an overview of the current state of the moduli problem for canonical models.
 Supplements

Notes
691 KB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Infinity categories and why they are useful, I
Carlos Simpson (Universite de Nice Sophia Antipolis)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this series, we'll introduce infinity categories and explain their relationships with triangulated categories, dg categories, and Quillen model categories. We'll explain how the infinitycategorical language makes it possible to create a moduli framework for objects that have some kind of uptohomotopy aspect: stacks, complexes, as well as higher categories themselves. The main concepts from usual category theory generalize very naturally. Emphasis will be given to how these techniques apply in algebraic geometry. In the last talk we'll discuss current work related to mirror symmetry and symplectic geometry via the notion of stability condition.
 Supplements

Notes
536 KB application/pdf



Feb 04, 2019
Monday

09:00 AM  10:00 AM


Infinity categories and why they are useful, II
Carlos Simpson (Universite de Nice Sophia Antipolis)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this series, we'll introduce infinity categories and explain their relationships with triangulated categories, dgcategories, and Quillen model categories. We'll explain how the infinitycategorical language makes it possible to create a moduli framework for objects that have some kind of uptohomotopy aspect: stacks, complexes, as well as higher categories themselves. The main concepts from usual category theory generalize very naturally. Emphasis will be given to how these techniques apply in algebraic geometry. In the last talk we'll discuss current work related to mirror symmetry and symplectic geometry via the notion of stability condition.
 Supplements

Notes
616 KB application/pdf


10:00 AM  10:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Moduli of canonical models
János Kollár (Princeton University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We give an overview of the current state of the moduli problem for canonical models.
 Supplements

Notes
635 KB application/pdf


11:30 AM  12:30 PM


DAG III: examples
Benjamin Antieau (Northwestern University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this series of lectures, I will give an introduction to derived algebraic geometry aimed at algebraic geometers. The first lecture will introduce simplicial commutative rings and use them to define the cotangent complex and derived de Rham cohomology with several examples. The second lecture will introduce derived stacks and the moduli stack of objects in a derived category. Then, I will give the geometricity theorem of ToënVaquié and describe the cotangent complex to the moduli stack. In the third lecture, we will use the machinery developed in the first two lectures to study three examples: cohomology as maps to a geometric derived stack, the (derived) Picard stack, and the stack of FourierMukai equivalences.
 Supplements

Notes
650 KB application/pdf


12:30 PM  02:30 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:30 PM  03:30 PM


Birational algebraic geometry in positive characteristic
Karl Schwede (University of Utah)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
This series of introductory talks will discuss some of the issues and pathologies one runs into when exploring birational geometry in positive characteristic. Some time will be spent talking about ideas coming from commutative algebra and their applications, including connections between singularities from the MMP with singularities arising in commutative algebra.
 Supplements

Notes
623 KB application/pdf


03:30 PM  04:00 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


The notion of singular support in DAG and its applications I
Sam Raskin (University of Texas, Austin)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
 
 Supplements

Notes
621 KB application/pdf



Feb 05, 2019
Tuesday

09:30 AM  10:30 AM


Infinity categories and why they are useful, III
Carlos Simpson (Universite de Nice Sophia Antipolis)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this series, we'll introduce infinity categories and explain their relationships with triangulated categories, dgcategories, and Quillen model categories. We'll explain how the infinitycategorical language makes it possible to create a moduli framework for objects that have some kind of uptohomotopy aspect: stacks, complexes, as well as higher categories themselves. The main concepts from usual category theory generalize very naturally. Emphasis will be given to how these techniques apply in algebraic geometry. In the last talk we'll discuss current work related to mirror symmetry and symplectic geometry via the notion of stability condition.
 Supplements

Notes
568 KB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Stable birational invariants
Claire Voisin (Collège de France; Institut de Mathématiques de Jussieu)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In these lectures, I will describe the formalism of the "decomposition of the diagonal", Chowtheoretic or cohomological, and the way it controls some more classical stable birational invariants. I will then explain the specialization method and some applications.
 Supplements

Notes
637 KB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


The notion of singular support in DAG and its applications II
Sam Raskin (University of Texas, Austin)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
 
 Supplements

Notes
630 KB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Birational algebraic geometry in positive characteristic
Karl Schwede (University of Utah)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
This series of introductory talks will discuss some of the issues and pathologies one runs into when exploring birational geometry in positive characteristic. Some time will be spent talking about ideas coming from commutative algebra and their applications, including connections between singularities from the MMP with singularities arising in commutative algebra.
 Supplements

Notes
627 KB application/pdf


04:30 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Feb 06, 2019
Wednesday

09:00 AM  10:00 AM


Topological Hochschild homology and topological cyclic homology: from classical to modern  I
Kathryn Hess (École Polytechnique Fédérale de Lausanne (EPFL))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Topological Hochschild homology and topological cyclic homology are important and much studied approximations to algebraic Ktheory. In this series of lectures I will survey various approaches to constructing and computing these remarkable invariants, from the original methods of Bökstedt and of BökstedtHsiangMadsen to the infinitycategory theoretic methods of NikolausScholze.
 Supplements

Notes
926 KB application/pdf


10:00 AM  10:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Stable birational invariants
Claire Voisin (Collège de France; Institut de Mathématiques de Jussieu)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In these lectures, I will describe the formalism of the "decomposition of the diagonal", Chowtheoretic or cohomological, and the way it controls some more classical stable birational invariants. I will then explain the specialization method and some applications.
 Supplements

Notes
638 KB application/pdf


11:30 AM  11:45 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:45 AM  12:45 PM


Rationality problems
Yuri Tschinkel (New York University, Courant Institute)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will discuss recent advances in the study of rationality of higherdimensional varieties (joint work with B. Hassett, A. Kresch, and A. Pirutka).
 Supplements

Notes
1.12 MB application/pdf



Feb 07, 2019
Thursday

09:00 AM  10:00 AM


Topological Hochschild homology and topological cyclic homology: from classical to modern  II
Kathryn Hess (École Polytechnique Fédérale de Lausanne (EPFL))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Topological Hochschild homology and topological cyclic homology are important and much studied approximations to algebraic Ktheory. In this series of lectures I will survey various approaches to constructing and computing these remarkable invariants, from the original methods of Bökstedt and of BökstedtHsiangMadsen to the infinitycategory theoretic methods of NikolausScholze.
 Supplements

Notes
358 KB application/pdf


10:00 AM  10:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Stable birational invariants
Claire Voisin (Collège de France; Institut de Mathématiques de Jussieu)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In these lectures, I will describe the formalism of the "decomposition of the diagonal", Chowtheoretic or cohomological, and the way it controls some more classical stable birational invariants. I will then explain the specialization method and some applications.
 Supplements

Notes
660 KB application/pdf


11:30 AM  11:45 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:45 AM  12:45 PM


Shifted symplectic structures and applications
Tony Pantev (University of Pennsylvania)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will give a brief overview of shifted symplectic and Poisson structures in derived geometry and their quantization. I will survey constructions of these structures on moduli stacks and will discuss several explicit examples. In the rest of the talk I will discuss interesting connections and applications to enumerative geometry, low dimensional topology, and Hodge theory. Time permitting, I will conclude with a sampling of recent results and developments including additivity theorems, connections with Bloch's conductor conjecture, and the Azumaya nature of shifted differential operators in positive characteristic.
 Supplements

Notes
1.76 MB application/pdf


12:45 PM  02:30 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:30 PM  03:30 PM


The notion of singular support in DAG and its applications III
Sam Raskin (University of Texas, Austin)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
 
 Supplements

Notes
612 KB application/pdf


03:30 PM  04:00 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


Extending holomorphic forms from the regular locus of a complex space to a resolution
Christian Schnell (State University of New York, Stony Brook)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Suppose we have a holomorphic differential form, defined on the smooth locus of a complex space. Under what conditions does it extend to a holomorphic differential form on a resolution of singularities? In 2011, Greb, Kebekus, Kovacs, and Peternell proved that such an extension always exists on algebraic varieties with klt singularities. I will explain how to solve this problem in general, with the help of Hodge modules and the Decomposition Theorem. This is joint work with Kebekus.
 Supplements

Notes
647 KB application/pdf



Feb 08, 2019
Friday

09:30 AM  10:30 AM


Topological Hochschild homology and topological cyclic homology: from classical to modern  III
Kathryn Hess (École Polytechnique Fédérale de Lausanne (EPFL))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Topological Hochschild homology and topological cyclic homology are important and much studied approximations to algebraic Ktheory. In this series of lectures I will survey various approaches to constructing and computing these remarkable invariants, from the original methods of Bökstedt and of BökstedtHsiangMadsen to the infinitycategory theoretic methods of NikolausScholze.
 Supplements

Notes
658 KB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Derived categories of cubic fourfolds and noncommutative K3 surfaces
Emanuele Macri (Université ParisSaclay)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The derived category of coherent sheaves on a cubic fourfold has a subcategory which can be thought as the derived category of a noncommutative K3 surface. This subcategory was studied recently in the work of Kuznetsov and AddingtonThomas, among others. In this talk, I will present joint work in progress with Bayer, Lahoz, Nuer, Perry, Stellari, on how to construct Bridgeland stability conditions on this subcategory. This proves a conjecture by Huybrechts, and it allows to start developing the moduli theory of semistable objects in these categories, in an analogue way as for the classical Mukai theory for (commutative) K3 surfaces. I will also discuss a few applications of these results.
 Supplements

Notes
618 KB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


(Derived) moduli of local systems in number theory
Akshay Venkatesh (Institute for Advanced Study)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
If X is a complex variety, we can form a moduli space M of local systems parameterizing representations of pi_1(X).
One would like to do the same in arithmetic situations  if X is a curve over a finite field, or the ring of integers of a number field. Here we can only construct a shadow of M, remembering some of its formal geometry. However, there are many indications that a more satisfactory theory should exist, and I will review three of them in my talk.
 Supplements

Notes
625 KB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



