Ludmil Katzarkov - US grants

Katzarkov 9700605 This project supports work on three problems on non-abelian Hodge theory and its applications. The first problem is concerned with work on the Shafarevich conjecture. The principal investigator will work on non-abelian analogs of the theorems of the (p,p) cycles, of the fixed pard theorem, and of the nonabelian Clemens Schmidt sequence. The second problem concerns a study of algebraic cycles on algebraic surfaces. The principal investigator will use the action of a certain infinite dimensional Heizenberg algebra on the kernel of the Albanese map to try to prove the Bloch conjecture. The third problem concerns work on families of K3 surfaces. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

9801647 Katzarkov This award provides support for a conference on motives, polylogarithms, and Hodge theory to be held at the University of California Irvine. Twenty years ago, S. Bloch gave his famous lecture series on the theory of dilogarithms in Irvine. These insightful lectures laid the foundation of a theory that combined Grothendieck's theory of motives with algebraic K-theory and number theory. Remarkably, each topic and example from Bloch's original treatment has become an active area leading to new discoveries or growing into a subject in its own right. Two major programs have recently emerged in the boundary zone between algebraic geometry, algebraic K-theory and Hodge theory. The first is stable homotopy theory of algebraic varieties, and the second is non-abelian Hodge theory. Twenty years later is a good time to look back and see where the fields stand. By gathering researchers in all of these fields, the organizers hope to stimulate a creative atmosphere that will lead to many new exciting discoveries. This conference is in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

Katzarkov
9875383

This award funds programs for research and education with special emphasis in algebraic geometry and nonabelian Hodge theory. The investigator plans to study the actions of homotopy groups and homology groups on homotopy types of varieties using developments in nonabelian Hodge theory as introduced by C. Simpson. The investigator hopes to shed light on the interplay between the geometric structure of algebraic varieties and the representation theory of their fundamental groups. In addition to his research, the investigator proposes a number of broadly based activities for the promotion of education at all levels. This includes the mentoring of high school students through the Mathematical Circles initiative, which brings these students to campus for tutoring and help in competitive exams. Moreover, the investigator intends to enhance opportunities for graduate students and postdoctural fellows at Irvine by organization of research workshops in algebraic geometry.

The National Science Foundation strongly encourages the early development of academic faculty as both educators and researchers. The Faculty Early Career Development (CAREER) Program is a Foundation-wide program that provides for the support of junior faculty within the context of their overall career development. It combines in a single program the support of quality research and education in the broadest sense and the full participation of those traditionally underrepresented in sciences and engineering. This program enhances and emphasizes the importance the Foundation places on the development of full, balanced academic careers.

Algebraic Geometry Summer Institute (Proposal DMS-0456683.)

A large scale summer institute in algebraic geometry, three weeks in
duration, will be held during July-August 2005 at the University of
Washington, Seattle. Organizational support will be provided by the
American Mathematical Society and the Clay Mathematics Institute. Each
week will have a general topic as a focus and at the same time will
include a component of core "classical" algebraic geometry. The foci
will be: Mathematical Physics in Algebraic Geometry; Combinatorics,
Commutative Algebra and Classical Algebraic Geometry; and Arithmetic Algebraic
Geometry. Three plenary morning lecture series will be held in each
week, lecturers having been selected as leading researchers who are at the
same time exemplary expositors. In addition, leading specialists will be
invited to serve as seminar leaders, and will select high impact
lectures given by the best lecturers for afternoon seminars. Additional
activities will include the CMI distinguished lectures and an evening graduate
program.

Algebraic geometry, the study of geometric shapes defined by polynomial
equations, is a vast generalization of the high-school topic "analytic
geometry". It has enjoyed much cross-fertilization in past decades with
areas in pure mathematics, applied mathematics, computing and
theoretical physics. One way American algebraic geometers have maintained the
vibrancy of the field is by holding a large scale, NSF-funded summer institute
about every decade, an institute which aims to bring participants up to
the forefront of research in the salient direction the field has taken
in the previous years, and to project directions which the field is
expected to take in the following years. These institutes have had a crucial
formative role on the subject, with enormous impact especially on the
emerging careers of young researchers. This summer institute project
aims to follow in their footsteps.

Abstract DMS-0506503Katzarkov


The conference Algebraic Geometry, Symplectic Geometry and Physics, to be held on December 15 - 20, 2004 at University of Miami will be focused on the following topics:

1) Understanding Kontsevich's approach for computing Gromov Witten
invariants from Fukaya category.
2) Understanding Homological Mirror Symmetry for manifolds of general type.
3) Trying to extend Hori's and Vafa's approach of constructing Landau Ginzburg model to the case of generalized symplectic manifolds.

The conference will stimulate new lines of interaction between algebraic geometry, symplectic geometry and string theory. It will stimulate further collaboration between mathematics community and physics community in general as well as the collaboration between University of Miami and FIU. The activities in the conference will help to prepare graduate students and postdocs in their research in these fields.

The Department of Mathematics at the University of Miami will host a
Winter Conference and School in December 2005 entitled "Arithmetic,
Algebraic and Symplectic Geometry". The conference will focus on
promotion and incorporation of recent advances in higher dimensional
algebraic geometry into arithmetic and symplectic research directions
and the creation of new bridges between projective geometry and
foliation theory, between singularity theory and reconstructive Galois
problems and between the geometry of coherent sheaves and symplectic
topology. The award will support junior researchers and graduate
students from U.S. institutions and will enable the dissemination of
geometric techniques among the new generation of young scientists.

The conference will address difficult classical questions about the hidden symmetries of the physical world, the large scale structure of space-time and the properties of matter at
high energies. This activity aims to advance our understanding of the
mathematical methodology necessary to handle such questions and to
produce results applicable in stringy physics, soft condensed matter theory
and mathematical biology.

Homological Mirror Symmetry (HMS) was initially introduced by Kontsevich
as a duality between Morse theory and de Rham theory.
After many wonderful physics papers at the beginning (Vafa, Hori,
Kapustin) many algebraic and symplectic Geometers have turned to the
subject. Rigorous mathematical proofs of HMS were given in several cases by D.
Auroux, L. Katzarkov, P.Seidel, M. Kontsevich, D. Orlov,
Y. Soibelman, K. Ueda. Bondal, Ruan, Abuzaid.


This research will concentrate on some striking applications of HMS to birational geometry, Gromov -Witten theory, Symplectic Geometry as well as some rather promising connections with Hodge theory. The expected results will help streamline
classical ideas of algebraic geometry in the new light of
collaboration between Mathematicians and String Theorists.

The main aim of this collaborative project is an in-depth study of
the homological mirror symmetry conjecture. Auroux, Katzarkov,
Kontsevich, Orlov and Seidel will lead a concerted effort to formulate
and understand homological mirror symmetry in systematic manner, and
extend it to varieties of general type and to noncommutative varieties.
This will require some foundational work in homological algebra,
noncommutative geometry, and symplectic geometry. Another goal is to
investigate applications of mirror symmetry to classical problems in
algebraic geometry (for example studying the rationality of certain
algebraic varieties) and symplectic topology (in particular, Lagrangian
submanifolds).


From a wider perspective, the project aims to provide a mathematical
counterpart to some recent advances in theoretical physics. The
contribution that mathematics can make is to verify the soundness and
consistency of physical intuition, and to prepare the general ground
on which further development can occur. This is particularly important
in those situations where developments in physics suggest the presence
of deep and complex structures, which are difficult to detect by direct
experiment. At the same time, this effort will make it possible to answer
some purely mathematical (geometric) questions, some of which have been
open for a long time.
The collaborative effort will be carried out through regular meetings
and workshops, and by fostering interaction between leading experts in
the field and younger mathematicians (graduate students and postdocs);
dissemination of knowledge in this rapidly evolving area of mathematics
will be facilitated by regularly held winter and summer schools and
conferences.

Topological Quantum Field Theory studies maps from Riemann surfaces to targets - algebraic stacks. Homological Mirror Symmetry allows us to look deeper into geometry of these stacks and algebraic varieties. In particular this project studies two main questions:

1. Rationality and unirationality of algebraic varieties. Program maximum in this direction is understanding non-rationality of generic four dimensional cubic and non-unirationality of generic three dimensional quartic.

2. Theory of algebraic cycles - computations of Chow and Griffiths groups, integral and Hodge cycles. Program minimum in this direction is understanding Bloch's conjecture for Chow groups of surfaces of general type and Griffiths groups of so called Fano - Calabi Yau manifolds.


By now Topological Quantum Field Theory and Homological Mirror Symmetry
have been established as closed and beautiful mathematical parts of
String theory. Regardless of the success of the experiments in the high energy collider in CERN and confirmation of String Theory as a theory of everything, one can still use Topological Quantum Field Theory and Homological Mirror Symmetry in order to get nice mathematical consequences. In particular one can use them to get answers of some long standing questions in Algebraic geometry.

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

This FRG builds on various recent successes in mirror symmetry and tropical geometry. On one hand, the Strominger-Yau-Zaslow conjecture has led to work by Kontsevich, Soibelman, Gross, Siebert, Zharkov and others to view mirror symmetry in terms of integral affine manifolds and tropical data on them. On the other hand, Mikhalkin's pioneering work on holomorphic curve counting using tropical geometry demonstrated that Gromov-Witten invariants were accessible by tropical methods, and increasingly, tropical methods are being seen as a tool for studying algebraic varieties. The aim of this FRG is to make further connections between tropical geometry and mirror symmetry with the aim of creating a new synthesis of these two fields. Some of the research directions of this project include the study of mirror symmetry for Fanos and manifolds of general type; Lagrangians and SYZ fibrations in this new setting; non-archimedean integrable systems in hyper-Kaehler manifolds; tropical enumerative geometry of real and complex curves; the development of non-commutative Hodge theory, tropical homology and its relation to classical homology and the Hodge conjecture; Welschinger invariants, open Gromov-Witten theory and their applications to mirror symmetry. These should greatly extend the realm of mirror symmetry and applications of tropical geometry to the classical algebro-geometric world. Members of this FRG include: Ricardo Castano-Bernard (Kansas State), Mark Gross (San Diego), Ilia Itenberg (Strasbourg), Ludmil Katzarkov (Miami), Viatcheslav Kharlamov (Strasbourg); Maxim Kontsevich (I.H.E.S & Miami), Diego Matessi (Alessandria); Grigory Mikhalkin (Geneva), Yan Soibelman (Kansas State), Jake Solomon (Hebrew U), Ilia Zharkov (Kansas State).

During past 25 years there has been intensive interaction between string theory and geometry which has led to a creation of entirely new mathematical areas. String theory also suggested that "conventional"
geometry emerges from the quantum theory at certain limits. Then various "string dualities" give equivalent but mathematically very different descriptions of the same physical quantities. A beautiful and deep example illustrating all these ideas is mirror symmetry. An important component of this project is to expand the existing collaborative links of the FRG members exploring tropical methods of mirror symmetry into a solid collaborative network of postdocs, graduate students and experts in an integrated research-training environment. This includes the organization of workshops, summer schools and local seminar series at the local FRG nodes. International exchange of young and senior researchers is one of the key aspects of this project bringing together well-established groups in Europe and in the U.S. The participation of mathematicians and young researchers from underrepresented groups will be promoted.

This conference will take place January 23-27, 2012 in Miami. The conference webpage is at http://www.math.berkeley.edu/~auroux/miami2012.html


Kontsevich's formulation of Homological Mirror Symmetry (HMS) and its subsequent extension to more general settings involve algebraic and geometric tools such as Lagrangian intersection theory, some new derived categories, or noncommutative Hodge structures. Recently, Homological Mirror Symmetry has transformed from a phantasy to a leading field in modern geometry. This was achieved in part due to constant exchange of information among the leaders of the field of Homological Mirror Symmetry (Kontsevich, Soibelman, Orlov, Auroux , Abouzaid, Fukaya, Seidel, Zaslow, Pantev, Gross, Siebert, Mikhalkin, Zharkov, Thomas, Smith, Katzarkov, Bridgeland, Frenkel, Okounkov, Bezrukavnikov, Bridgeland, Pandharipande)with physicists (Vafa, Hori, Gukov, Kapustin, Neitzke, Douglas, Diaconescu). This exchange was mainly done during month-long activities taking place every winter in Miami. In addition, many young people were attracted to this field: Sheridan, Preigel, Izik, Pandit, Favero, Kerr, Ballard, Diemer, Dykerhoff. This conference will enable these fruitful exchanges to continue.

While mirror symmetry initially arose from phenomena in string theory, this very active subject at the interface between mathematics and physics has acquired increasing mathematical stature. Numerous works have shown the relevance to mirror symmetry of new and subtle mathematical structures. This conference will extend the success of previous activities by branching out in a new field of dissemination - creating a new University of Miami, Cinvestav, Campinas institute featuring two conferences per year and exchange of postdocs and students. These conferences will be happening in winter and will be partially supported by University of Miami, Cinvestav, and Campinas.

This Pan-American Advanced Studies Institutes (PASI) award, jointly supported by the NSF and the Department of Energy (DOE), will take place in February 1-10, 2013 at the Institute of Physics of the Federal University of Rio Grande do Norte in Natal, Brazil. Organized by Dr. Ludmil V. Katzarkov, the 10-day institute will bring together leading researchers in the world in the area of homological mirror symmetry, a deep and still mysterious part of the evolving interaction between algebraic geometry and string theory. Experts from a variety of disciplines including symplectic geometry, algebraic geometry, topology, physics, mathematical physics, representation theory, and other areas of geometry are expected to participate. The opportunity for interaction, joint research, and reporting of recent progress at this singular event will help generate further breakthroughs in one of the most challenging, fastest moving, and very compelling areas in algebraic geometry and its interaction with mathematical physics.

This PASI is expected to enhance the infrastructure for research and education on the topics mentioned above through collaborations developed before, during and beyond the activity, and by sharing resources. The PI, Dr. Kartzarkov has a track record of organizing meetings that bring together outstanding individuals in nearby fields, leading in turn to important developments. The co-PI, Dr. Maxim Kontsevich, a Fields medalist, is one the initiator of many of the ideas that will be discussed at the PASI and his influence on the field is far reaching. The PASI will attract a large number of junior researchers, and give them access to the pace-setters in the field. Their influence on graduate students and young researchers is likely to help steer the field's development in the future.

The PIs will continue their foundational work on Homological Mirror Symmetry (HMS), to develop the structures and theories involved in HMS, and to build on applications of these theories. One notable direction is the theory of higher symplectic structures, which brings out the duality between the ``stacky'' directions and the ``derived'' directions of most moduli problems in algebraic geometry. Exploiting the full depth of these structures will require a careful study of the moduli of various kinds of categorical and higher categorical entities. This is one of the main areas of expertise of all the PIs. Kontsevich introduced one of the main tools, derived schemes. Katzarkov came up with the idea that moduli of LG models and its monodromy can be interpreted as stability conditions and spectra.

These activities fit into a more general and global philosophy designed to accompany Geometry in the 21st Century. The study of Geometry in the 20th Century was devoted, in large part and with astounding success, to the classification and parametrization of geometrical objects. However,these objects, of various kinds, were uniformly viewed somehow as ``sets of points''. Along the way, the relationship with categorical structures grew steadily, leading to the many inputs into our program as discussed above. The PIs themselves played a pivotal role in much of the progress that was made at the turn of the century. With PI Kontsevich's introduction of HMS, a subtle change was introduced, in that ``Geometry'' began to be seen within a categorical structure. And the concurrent development of the theory of higher stacks meant that geometric structures were no longer viewed just as ``sets of points'' but rather as objects enclosing a higher structure. This project is highly connected with theoretical physics. As we head into the second decade of the 21st Century, elementary particle physics is on the crux of a profound revolution to be brought about by the new experimental results coming out of the LHC at CERN. These will serve to identify which of the multitude of theoretical possibilities which are currently open, best address quantum field theory at the high energy scale. And for those theories, to tell which are the right parameters. So there will soon be a lot of work to do on the theoretical side, and this will surely require new tools and a new approach. With the relationship between HMS and supersymmetric theories, with the relationship between higher categories and TQFT, with the relationship between partition functions and nonabelian cohomology, the kinds of geometrical objects which we are going to investigate in this project are becoming crucial for understanding these new panoramas in theoretical physics. The project has an educational component - conferences and educating postdocs, This component has been hugely successful in the past and with more funding we plan to bring it to the next level.

This project will investigate wall-crossing formulas for a wide class of invariants which appear in a priori different situations in mathematics and physics. Mathematically, those invariants are typically described as virtual Euler characteristics of some moduli spaces. The wall-crossing phenomenon is related to the presence of real codimension one "walls" in the space of parameters, where the invariants jump. In the case of Donaldson-Thomas invariants, the walls live in the moduli space of Bridgeland stability conditions on the ppropriate Calabi-Yau categories. Similar walls also occur in the theory of representations of quivers and cluster algebras. In mirror symmetry, walls correspond to jumps in the number of pseudo-holomorphic discs bounded by the torus fibers of an SYZ Lagrangian fibration. In supersymmetric gauge theories in physics, the number of BPS states jumps across "walls of marginal stability". The Kontsevich-Soibelman wall-crossing formulas for Donaldson-Thomas invariants thus occur in the physics literature on topics such as moduli spaces of vector ultiplets of 4-dimensional supersymmetric theories and supersymmetric black holes. Since these various wall-crossing formulas look so similar, one can ask for a common formalism. The aim of the FRG is to study the underlying "wall-crossing structures" and demonstrate hat the above-mentioned similarities are not coincidental, but rather reflect a deep underlying theory.

It is a frequently encountered situation in mathematics and physics that numerical quantities which in principle depend on various parameters actually are constant for general parameter values (they are "invariants"), but jump along certain "walls" in the parameter space. Wall-crossing formulas describe these "jumps" quantitatively. The subject of wall-crossing has recently become a very active one due to its relevance to a number of different areas of mathematics and physics. The aim of this project is to develop the concept of "wall-crossing structure" rigorously and apply it to problems both old and new in which wall-crossing formulas appear. The results arising from this project will be in demand by both the mathematics and physics communities. The FRG will also build a research community around this coordinated effort, involving a mix of junior and senior researchers, training opportunities for graduate students, and the rganization of several workshops.

This project supports a conference on Homological Mirror Symmetry to be held at the University of Miami, Florida, from January 28 - February 1, 2013. More information can be found on the conference website: http://math.berkeley.edu/~auroux/miami2013.html

While mirror symmetry initially arose from phenomena in string theory, this active area at the interface between mathematics and physics has been shown to be relevant to numerous mathematical structures. In particular, methods from Lagrangian intersection theory and Floer homology theory, integrable systems and wall-crossing, derived and higher categories, and non-commutative Hodge structures are all intimately connected with homological mirror symmetry. The wide range of topics in homological mirror symmetry research necessitates venues such as this, particularly for early-career researchers to be introduced to and stay abreast of current topics. The main topics of this conference will be wall crossing, stability Hodge structures and topological quantum field theories. There will be 3 lecture courses, by M. Kontsevich, by M. Abouzaid and by S. Keel. There will be two discussion sessions.

The conference will result in dissemination of results and getting a young wave of researchers to join these projects. All considerations on giving opportunities to underrepresented groups were taken. The conference will also serve to facilitate interactions between US-based researchers and international participants who will be attended the conference. This will be particularly beneficial for US-based graduate students and should lead to a fruitful exchange of ideas among representatives of many mathematical communities. In addition, the conference will be a means of "giving back to physics." Many of the ideas at the beginning of the subject come from String theory - Mirror Symmetry. It is time for the mathematical aspect to give back to physics. The singularities on the discriminant loci we study correspond to a new physics theory of 6d type. In this way the circle comes to a close.

The main goals of the Conference on Homological Mirror Symmetry, to be held on January 27-February 1, 2014 at the University of Miami, will be dissemination of results and encouragement a wave of young, early career researchers to join these projects. The wide range of topics appearing in HMS research necessitates venues for the open exchanges of ideas in order for graduate students and early career researchers to stay afloat on current topics. The subject has developed extremely rapidly in recent years, and this conference will really help engage young researchers.

Over the last several years, a variety of new categorical structures have been discovered by physicists. Furthermore, it has become transparently evident that the higher categorical language is beautifully suited to describing cornerstone concepts in modern theoretical physics. As we head into the second decade of the 21st century, modern geometry and theoretical physics are more intertwined than ever before. The convergence of ideas from mathematics and physics is accelerating at the same time as elementary particle physics is on the cusp of a profound revolution to be brought about by the new experimental results coming out of the Large Hadron Collider (LHC). These will serve to identify among the multitude of theoretical possibilities currently open, which ones best address quantum field theory at the high energy scale. At the same time, a lot of mathematical work remains to be done to provide a suitable framework for the new physical theories that are being proposed. The geometric objects which we investigate today are the foundations for such a framework: homological mirror symmetry is the mathematical realization of dualities among super-symmetric theories, and higher categories are the mathematical foundation for quantum field theories. These new flavors of geometry, in which categorical structures play a primordial role, will certainly continue to play a fundamental role in 21st century theoretical physics.

The conference website: http://math.berkeley.edu/~auroux/miami2014.html

Category theory is an algebraic approach that formalizes the study of mathematical structures. This project lays the foundations of classical geometric methods in category theory. Today, in the second decade of the 21st century, modern geometry and theoretical physics are more intertwined than ever before. The convergence of ideas from mathematics and physics is accelerating at the same time as elementary particle physics is on the cusp of a profound revolution to be brought about by the new experimental results coming out of the Large Hadron Collider (LHC). At the same time, a lot of mathematical work remains to be done to provide a suitable framework for the new physical theories that are being proposed. The geometric objects investigated in this project are the foundations for such a framework: homological mirror symmetry is the mathematical realization of dualities and higher categories, the analogues of classical manifolds, are the mathematical foundation for quantum field theories. These new flavors of geometry on which this project is based will continue to play a fundamental role in the future development of theoretical physics.

The proposed approach is based on the pioneering works by Seidel, Ein, Lazarsfeld, Mustata, Nakamaye, Popa, and Budur. The PIs will go further and conjecture that the categorical multiplier ideal sheaf is related to the Orlov spectrum of the category. Developing K-calculus and making it rigorous the PIs will break new ground in studying classical questions in Algebraic Geometry, including questions of rationality of projective varieties. In particular the PIs plan to consider some more than hundred years old questions about nonrationality of conic bundles and four dimensional cubics. The applications go beyond the scope of Algebraic Geometry. Classical questions in Sympletcic Geometry will be studied as well - using invariants of Fukaya category one can try to distinguish symplectic manifolds with the same Seiberg - Witten invariants. The approach connects K-calculus with so called tasting configurations used in the study of the existence of Kahler-Einstein metrics. An intriguing question is that of finding a connection between Orlov spectra and the existence of Kahler-Einstein metrics. Another direction of the project is the investigation of the connection of K-calculus with physics, for which a starting point is the interpretation of monodromy data of the K - calculus as limited stability conditions.

This award supports participation in a conference on Homological Mirror Symmetry held at the University of Miami from January 26 through January 31, 2015. Mirror symmetry originated in physics as a duality between superconformal quantum field theories. Maxim Kontsevich interpreted this duality in a consistent, powerful mathematical framework called Homological Mirror Symmetry (HMS). The ideas put forth by Kontsevich have led to dramatic developments in how the mathematical community approaches ideas from theoretical physics, and indeed our conception of space itself. Several cutting edge developments have happened in the last year.

The conference on Homological Mirror Symmetry at the University of Miami will serve as a perfect opportunity for a dissemination of these developments and encouragement of a wave of young, early career researchers in this subject area. The wide range of topics appearing in HMS research necessitates venues for the open exchanges of ideas in order for graduate students and early-career researchers to stay informed on current topics. The subject has developed extremely rapidly in recent years, and this conference will have an immense broader impact by bringing young people into it. More details on the conference can be found at https://math.berkeley.edu/~auroux/miami2015.html.

This award supports participation in a conference on Homological Mirror Symmetry to be held at the University of Miami from January 27 through February 31, 2020. Mirror symmetry originated in physics as a duality between superconformal quantum field theories. This duality has been interpreted in a consistent, powerful mathematical framework called Homological Mirror Symmetry (HMS) that has led to dramatic developments in how the mathematical community approaches ideas from theoretical physics, and indeed our conception of space itself.

Today HMS is established as one of the most powerful frameworks for building bridges between seemingly unrelated parts of modern geometry. Traditionally HMS has been incredibly successful in discovering novel phenomena and proving unexpected results in symplectic and contact topology informed by algebraic geometry. The goal of this conference is to highlight the flow of ideas in the opposite direction: from symplectic to complex geometry. The conference will summarize recently developed connections between Hodge Theory and HMS which have striking algebraic applications and give new insights into rationality problems. In particular, the conference will focus on new cutting-edge directions of investigation: (1) birational invariants and gluing of D-modules; (2) extending the scope of Homological Mirror Symmetry; (3) categorical Kaehler geometry; and (4) topological recursion. The conference website is https://schms.math.berkeley.edu/events/miami2020/.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Birational geometry is a classical mathematical discipline whose roots go back to ancient Greece. Nevertheless, it still offers many difficult unsolved questions. The core part of this project is to tackle these questions with cutting-edge modern methods coming from the homological mirror symmetry program. Homological mirror symmetry is a deep geometric duality that originates in quantum field theory and has been used in studying novel phenomena and proving unexpected results in symplectic geometry suggested by algebraic geometry. This project aims to use homological mirror symmetry to introduce new applications of symplectic topology to algebraic geometry and to answer classical open questions in birational geometry. A postdoctoral fellow will be involved in various aspects of this research project.

This project will use an approach based on categorical Kähler geometry. The most notable application of this approach is toward proving the non-rationality of generic, four-dimensional cubics, which is arguably the central problem in algebraic geometry. More specifically, a detailed study of the singularities of quantum D-modules produces a completely new type of birational invariant. This new invariant is a canonical decomposition of the cohomology of a four-dimensional cubic based on simultaneous use of both (algebraic and symplectic) sides of homological mirror symmetry. The example of four-dimensional cubics is only the tip of the iceberg. There are many other potential applications of this approach, for example to uniformization problems.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.