## Project Details

### Description

Overview:

This is a proposal seeking funding to support the travel and local expenses of early career US mathematicians to attend the conference “Invertible objects and duality in derived algebraic geometry

and homotopy theory” to be held at the University of Regensburg, Germany. The funds will be used exclusively to support junior US based mathematicians whose support from other sources would be too limited to attend the event. Conference speakers and participants from outside of the United States will be supported by the German research foundation (DFG) through the grants SFB 1085: Higher Invariants and SPP 1786: Homotopy theory and algebraic geometry.

Intellectual Merit

In the past fifteen years, we have seen the rapid development of the field of derived algebraic geometry; this conference is about current work and emerging ideas in the field, using the existence and applications of dualities as a theme.

Throughout algebraic topology and algebraic geometry, the existence of a theory of duality reveals deep structure about the objects under study. Basic examples include Poincar´e duality for compact, oriented manifolds or Serre duality for projective varieties. Both of these examples are avatars of much more general dualities: Poincar´e duality can be derived from Spanier-Whitehead duality and bundle theory, and Serre-duality can be obtained from Serre-Grothendeick duality and analysis of quasi-coherent sheaves. A feature of both basic dualities is that they have a natural home in a derived setting and, hence, are natural tools in derived algebraic geometry.

If algebraic geometry is the study of certain spaces with sheaves of rings, then derived algebraic

geometry supplies those same spaces with sheaves of ring objects in some category amenable to

homotopy theory, such as differential graded algebras or E∞-ring spectra. This field has its roots in the 1960s but has grown to maturity in the past fifteen years, initially spurred by the Hopkins-Miller theory of topological modular forms, but now developed into a field in its own right.

Picard groups, as well as the closely related Brauer groups, have long played a central role in commutative algebra, algebraic geometry, and algebraic number theory. Generalizations of these groups to derived or homotopy theoretic settings naturally arise as spaces with additional useful structures. There has been significant recent progress in this area in both theory and computations.

It is the aim of this workshop both to consolidate these advances and to promote new prospective research directions. The workshop will bring together international experts from algebraic topology, derived algebraic geometry, and related areas, such as mathematical physics.

Broader Impacts

We are requesting National Science Foundation funding solely to support the participation of junior research mathematicians from the United States, meaning graduate students, postdoctoral fellows, and junior faculty without other significant support. The direct impact of NSF funding will be the training and career development of a minimum of 15 junior researchers, who will gain the opportunity to participate in a workshop in Europe and, in particular, at the University of Regensburg, which is a major center in derived algebraic geometry. A secondary impact is to further develop collaboration between emerging research groups in this area in the US and Europe, in particular Germany and France.

This is a proposal seeking funding to support the travel and local expenses of early career US mathematicians to attend the conference “Invertible objects and duality in derived algebraic geometry

and homotopy theory” to be held at the University of Regensburg, Germany. The funds will be used exclusively to support junior US based mathematicians whose support from other sources would be too limited to attend the event. Conference speakers and participants from outside of the United States will be supported by the German research foundation (DFG) through the grants SFB 1085: Higher Invariants and SPP 1786: Homotopy theory and algebraic geometry.

Intellectual Merit

In the past fifteen years, we have seen the rapid development of the field of derived algebraic geometry; this conference is about current work and emerging ideas in the field, using the existence and applications of dualities as a theme.

Throughout algebraic topology and algebraic geometry, the existence of a theory of duality reveals deep structure about the objects under study. Basic examples include Poincar´e duality for compact, oriented manifolds or Serre duality for projective varieties. Both of these examples are avatars of much more general dualities: Poincar´e duality can be derived from Spanier-Whitehead duality and bundle theory, and Serre-duality can be obtained from Serre-Grothendeick duality and analysis of quasi-coherent sheaves. A feature of both basic dualities is that they have a natural home in a derived setting and, hence, are natural tools in derived algebraic geometry.

If algebraic geometry is the study of certain spaces with sheaves of rings, then derived algebraic

geometry supplies those same spaces with sheaves of ring objects in some category amenable to

homotopy theory, such as differential graded algebras or E∞-ring spectra. This field has its roots in the 1960s but has grown to maturity in the past fifteen years, initially spurred by the Hopkins-Miller theory of topological modular forms, but now developed into a field in its own right.

Picard groups, as well as the closely related Brauer groups, have long played a central role in commutative algebra, algebraic geometry, and algebraic number theory. Generalizations of these groups to derived or homotopy theoretic settings naturally arise as spaces with additional useful structures. There has been significant recent progress in this area in both theory and computations.

It is the aim of this workshop both to consolidate these advances and to promote new prospective research directions. The workshop will bring together international experts from algebraic topology, derived algebraic geometry, and related areas, such as mathematical physics.

Broader Impacts

We are requesting National Science Foundation funding solely to support the participation of junior research mathematicians from the United States, meaning graduate students, postdoctoral fellows, and junior faculty without other significant support. The direct impact of NSF funding will be the training and career development of a minimum of 15 junior researchers, who will gain the opportunity to participate in a workshop in Europe and, in particular, at the University of Regensburg, which is a major center in derived algebraic geometry. A secondary impact is to further develop collaboration between emerging research groups in this area in the US and Europe, in particular Germany and France.

Status | Finished |
---|---|

Effective start/end date | 3/1/17 → 2/28/18 |

### Funding

- National Science Foundation (DMS-1700795)

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