Periodic Reporting for period 2 - MSAG (Mirror symmetry in Algebraic Geometry)
Período documentado: 2023-04-01 hasta 2024-09-30
Mirror symmetry is a mathematical and physical phenomenon discovered by
string theorists in 1989. In string theory, space-time is 10 dimensional
rather than 4 dimensional, with six of these dimensions being extremely
small. To picture this concretely, one might instead imagine the
universe in fact is shaped like a water hose. A small ant crawling on the
surface of the hose might perceive the universe as two-dimensional,
but a larger being might only see the length of the hose and not
the small circumference of the hose. Nevertheless, the geometry of
the circular cross-section of the hose impacts the physics of the
observed universe. In string theory, instead of the one-dimensional
circle, one has a very small six-dimensional geometric object,
known as a Calabi-Yau manifold. Mirror symmetry can then be expressed
as the observation that such Calabi-Yau manifolds come in pairs,
called mirror pairs, which give rise the same observed four-dimensional
physics.
While this phenomenon was originally very much part of theoretical
physics, mathematicians have become fascinated by it because it has
revealed profound insights into the geometry of Calabi-Yau manifolds
and beyond. Mirror symmetry identifies radically different calculations
carried out on the two members of a mirror pair, producing powerful tools
for carrying out difficult calculations. The subject has now become
central to modern mathematics, lying at the intersection of multiple
fields and driving innovation in all of these fields. These innovations
have lead to solutions to many diverse long-standing problems in mathematics.
One of the central approaches to understanding mirror symmetry from a
mathematical perspective has been developed by the PI and Bernd Siebert,
creating what is now known colloquially as the Gross-Siebert program.
The main goal of the grant is to both further develop the program
and to find applications of it to other areas of mathematics.
string theorists in 1989. In string theory, space-time is 10 dimensional
rather than 4 dimensional, with six of these dimensions being extremely
small. To picture this concretely, one might instead imagine the
universe in fact is shaped like a water hose. A small ant crawling on the
surface of the hose might perceive the universe as two-dimensional,
but a larger being might only see the length of the hose and not
the small circumference of the hose. Nevertheless, the geometry of
the circular cross-section of the hose impacts the physics of the
observed universe. In string theory, instead of the one-dimensional
circle, one has a very small six-dimensional geometric object,
known as a Calabi-Yau manifold. Mirror symmetry can then be expressed
as the observation that such Calabi-Yau manifolds come in pairs,
called mirror pairs, which give rise the same observed four-dimensional
physics.
While this phenomenon was originally very much part of theoretical
physics, mathematicians have become fascinated by it because it has
revealed profound insights into the geometry of Calabi-Yau manifolds
and beyond. Mirror symmetry identifies radically different calculations
carried out on the two members of a mirror pair, producing powerful tools
for carrying out difficult calculations. The subject has now become
central to modern mathematics, lying at the intersection of multiple
fields and driving innovation in all of these fields. These innovations
have lead to solutions to many diverse long-standing problems in mathematics.
One of the central approaches to understanding mirror symmetry from a
mathematical perspective has been developed by the PI and Bernd Siebert,
creating what is now known colloquially as the Gross-Siebert program.
The main goal of the grant is to both further develop the program
and to find applications of it to other areas of mathematics.
A significant amount of effort was devoted to one of the key questions
in the Gross-Siebert program. This is joint work with Daniel Pomerleano
and Bernd Siebert. This question will get at the heart of
why the difficult calculations above on a mirror pair are related.
On the way, we have encountered significant technical difficulties which
still need to be overcome, but these problems are now identified and
we have a program for going forward.
Roughly half of the stated problems in the proposal now have either been
solved or seen significant progress. In many cases, this constitutes
work of research students employed on the grant. This includes recently
graduated students Yu Wang, Samuel Johnston, and Evgeny Goncharov, who
have made significant progress advancing the goals of the grant. New research
students Peter Zaika and Xuanchun Lu are now making headway on problems
in the proposal which had not yet been touched. In addition, the grant
has employed post-doctoral research assistants Tim Graefnitz, Renata
Piciotto and Fatemeh Rezaee, and joint projects involving applications
of the program to various other mathematical questions are ongoing.
in the Gross-Siebert program. This is joint work with Daniel Pomerleano
and Bernd Siebert. This question will get at the heart of
why the difficult calculations above on a mirror pair are related.
On the way, we have encountered significant technical difficulties which
still need to be overcome, but these problems are now identified and
we have a program for going forward.
Roughly half of the stated problems in the proposal now have either been
solved or seen significant progress. In many cases, this constitutes
work of research students employed on the grant. This includes recently
graduated students Yu Wang, Samuel Johnston, and Evgeny Goncharov, who
have made significant progress advancing the goals of the grant. New research
students Peter Zaika and Xuanchun Lu are now making headway on problems
in the proposal which had not yet been touched. In addition, the grant
has employed post-doctoral research assistants Tim Graefnitz, Renata
Piciotto and Fatemeh Rezaee, and joint projects involving applications
of the program to various other mathematical questions are ongoing.
While the general Gross-Siebert construction of mirror pairs had been
developed before the beginning of the grant, the last two and a half
years have seen the development of a much more profound understanding
of this construction. In particular, we are developing practical tools
for making use of the construction, whcih was initially very theoretical.
The central goal of understanding the equivalence of the two sets of
calculations on two mirror pairs remain, and we hope to complete this
during the remaining timespan of the grant.
developed before the beginning of the grant, the last two and a half
years have seen the development of a much more profound understanding
of this construction. In particular, we are developing practical tools
for making use of the construction, whcih was initially very theoretical.
The central goal of understanding the equivalence of the two sets of
calculations on two mirror pairs remain, and we hope to complete this
during the remaining timespan of the grant.