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DTSTART;TZID=America/New_York:20180102T090000
DTEND;TZID=America/New_York:20180518T170000
DTSTAMP:20260728T203131
CREATED:20230904T080137Z
LAST-MODIFIED:20250304T172359Z
UID:10000048-1514883600-1526662800@cmsa.fas.harvard.edu
SUMMARY:Simons Collaboration on Homological Mirror Symmetry
DESCRIPTION:The Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality\, and fully exploring its applications. This program is funded by the Simons Foundation. \nMirror symmetry\, which emerged in the late 1980s as an unexpected physical duality between quantum field theories\, has been a major source of progress in mathematics. At the 1994 ICM\, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or Landau-Ginzburg model). \nWe envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints: \n\ncategorical algebraic geometry and non-commutative (nc) spaces: in this language\, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.\nthe Strominger-Yau-Zaslow (SYZ) approach\, which provides a global geometric prescription for the construction of mirror pairs.\nLagrangian Floer theory and family Floer cohomology\, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.\n\nThe Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester: \n\nJacob Bourjaily (Neils Bohr Institute)  4/1/2018 – 4/14/2018\nColin Diemer (IHES)  2/25/2018 – 3/10/2018\nCharles Doran (University of Alberta) 5/13/2018 – 5/25/2018\nBaohua Fu (Chinese Academy of Sciences)  4/15/2018 – 4/28/2018\nAndrew Harder (University of Miami)  4/15/2018 – 4/28/2018\nShinobu Hosono (Gakushuin University) 2/25/2018 – 3/10/2018\nAdam Jacob (UC Davis) 3/5/2018 – 3/16/2018\nTsung-Ju Lee (National Taiwan University) 4/18/2018 – 5/13/2018\nIvan Loseu (Northeastern University) 1/21/2018 – 2/3/2018\nCheuk-Yu Mak (Cambridge University) 4/1/2018 – 4/15/2018\nDaniel Pomerleano (Imperial College) 3/19/2018 – 3/23/2018\nMauricio Romo (Tsinghua University) 4/1/2018 – 4/18/ 2018\nEmanuel Scheidegger (Albert Ludwigs University of Freiburg) 2/22/2018 – 3/22/2018\nDmytro Shklyarov (Technische Universität Chemnitz) 3/4/2018 – 3/17/2018\nAlan Thompson (University of Cambridge) 4/15/2018 – 4/21/2018\nWeiwei Wu (University of Georgia) 4/27/2018 – 5/6/2018\nMatt Young (Chinese University of Hong Kong) 1/15/2018 – 2/9/2018\nJeng-Daw Yu (National Taiwan University) 4/2/2018 – 4/6/2018\nMinxian Zhu (Yau Mathematical Sciences Center\, Tsinghua University) 1/ 22/2018 – 2/25/2018\n\nAs part of their CMSA visitation\, HMS focused visitors will be giving lectures on various topics related to Homological Mirror Symmetry throughout the Spring 2018 Semester.  Click here for information. \n\n\nThe Collaboration will include two workshops hosted by The Center. The workshops will take place January 10-13\, 2018  and April 5-7\, 2018 at CMSA. Click here for more information.
URL:https://cmsa.fas.harvard.edu/event/simons-collaboration-on-homological-mirror-symmetry-2/
CATEGORIES:Programs
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DTSTART;TZID=America/New_York:20180123T170000
DTEND;TZID=America/New_York:20180123T170000
DTSTAMP:20260728T203131
CREATED:20240213T103131Z
LAST-MODIFIED:20240213T103131Z
UID:10002436-1516726800-1516726800@cmsa.fas.harvard.edu
SUMMARY:2018 HMS Focused Lecture Series
DESCRIPTION:As part of their CMSA visitation\, HMS focused visitors will be giving lectures on various topics related to Homological Mirror Symmetry throughout the Spring 2018 Semester. The lectures will take place  on Tuesdays and Thursdays in the CMSA Building\, 20 Garden Street\, Room G10. \nThe schedule will be updated below. \n\n\n\nDate\nSpeaker\nTitle/Abstract\n\n\nJanuary 23\, 25\, 30 and February 1  \n3-5pm \n*Room G10*\nIvan Losev  \n(Northeastern)\nTitle: BGG category O: towards symplectic duality  \nAbstract: We will discuss a very classical topic in the representation theory of semisimple Lie algebras: the Bernstein-Gelfand-Gelfand (BGG) category O. Our aim will be to motivate and state a celebrated result of Beilinson\, Ginzburg and Soergel on the Koszul duality for such categories\, explaining how to compute characters of simple modules (the Kazhdan-Lusztig theory) along the way. The Koszul duality admits a conjectural generalization (Symplectic duality) that is a Mathematical manifestation of 3D Mirror symmetry. We will discuss that time permitting. \nApproximate (optimistic) plan of the lectures: \n1) Preliminaries and BGG category O. \n2) Kazhdan-Lusztig bases. Beilinson-Bernstein localization theorem. \n3) Localization theorem continued. Soergel modules. \n4) Koszul algebras and Koszul duality for categories O. \nTime permitting: other instances of Symplectic duality. \nPrerequisites: \nSemi-simple Lie algebras and their finite dimensional representation theory. \nSome  Algebraic geometry. No prior knowledge of category O/ Geometric \nRepresentation theory is assumed. \nScanned from a Xerox Multifunction Device\n\n\nFebruary 27\,  \nand March 1 \n3-5pm\nColin Diemer  \n(IHES)\nTitle: Moduli spaces of Landau-Ginzburg models and (mostly Fano) HMS.  \nAbstract: Mirror symmetry as a general phenomenon is understood to take place near the large complex structure limit resp. large radius limit\, and so implicitly involves degenerations of the spaces under consideration. Underlying most mirror theorems is thus a mirror map which gives a local identification of respective A-model and B-model moduli spaces. When dealing with mirror symmetry for Calabi-Yau’s the role of the mirror map is well-appreciated. In these talks I’ll discuss the role of moduli in mirror symmetry of Fano varieties (where the mirror is a Landau-Ginzburg (LG) model). Some topics I expect to cover are a general structure theory of moduli of LG models (follows Katzarkov\, Kontsevich\, Pantev)\, the interplay of the topology  of LG models with autoequivalence relations in the Calabi-Yau setting\, and the relationship between Mori theory in the B-model and degenerations of the LG A-model. For the latter topic we’ll focus on the case of del Pezzo surfaces (due to unpublished work of Pantev) and the toric case (due to the speaker with Katzarkov and G. Kerr). Time permitting\, we may make some speculations on the role of LG moduli in the work of Gross-Hacking-Keel (in progress work of the speaker with T. Foster).\n\n\nMarch 6 and 8  \n4-5pm\nAdam Jacob  \n(UC Davis)\nTitle: The deformed Hermitian-Yang-Mills equation  \nAbstract: In this series I will discuss the deformed Hermitian-Yang-Mills equation\, which is a complex analogue of the special Lagrangian graph equation of Harvey-Lawson. I will describe its derivation in relation to the semi-flat setup of SYZ mirror symmetry\, followed by some basic properties of solutions. Later I will discuss methods for constructing solutions\, and relate the solvability to certain geometric obstructions. Both talks will be widely accessible\, and cover joint work with T.C. Collins and S.-T. Yau.\n\n\nMarch 6\, 8\, 13\, 15  \n3-4pm\nDmytro Shklyarov  \n(TU Chemnitz)\nTitle: On categories of matrix factorizations and their homological invariants  \nAbstract: The talks will cover the following topics: \n1. Matrix factorizations as D-branes. According to physicists\, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg (LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed topological field theories underlying the LG B-models and then talk about their refinements. \n2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds\, and a natural first step towards proving the assertion would be to try to attach similar Hodge-like data to abstract derived categories. I will talk about some recent results in this direction and illustrate the approach in the context of the LG B-models. \n3. Hochschild cohomology of LG orbifolds. The scope of applications of the LG mod- els in mirror symmetry is significantly expanded once we include one extra piece of data\, namely\, finite symmetry groups of singularities. The resulting models are called orbifold LG models or LG orbifolds. LG orbifolds with abelian symmetry groups appear in mir- ror symmetry as mirror partners of varieties of general type\, open varieties\, or other LG orbifolds. Associated with singularities with symmetries there are equivariant versions of the matrix factorization categories which\, just as their non-equivariant cousins\, describe D-branes in the corresponding orbifold LG B-models. The Hochschild cohomology of these categories should then be isomorphic to the closed string algebra of the models. I will talk about an explicit description of the Hochschild cohomology of abelian LG orbifolds.\n\n\nApril 10 & 12  \n3-4pm\nMauricio Romo  \n(IAS)\nTitle: Gauged Linear Sigma Models\, Supersymmetric Localization and Applications  \nAbstract: In this series of lectures I will review various results on connections between gauged linear sigma models (GLSM) and mathematics. I will start with a brief introduction on the basic concepts about GLSMs\, and their connections to quantum geometry of Calabi-Yaus (CY). In the first lecture I will focus on nonperturbative results on GLSMs on closed 2-manifolds\, which provide a way to extract enumerative invariants and the elliptic genus of some classes of CYs. In the second lecture I will move to nonperturbative results in the case where the worldsheet is a disk\, in this case nonperturbative results provide interesting connections with derived categories and stability conditions. We will review those and provide applications to derived functors and local systems associated with  CYs. If time allows we will also review some applications to non-CY cases (in physics terms\, anomalous GLSMs). \nLecture notes\n\n\nApril 17\, 19\, 26  \n3-5pm\nAndrew  Harder  \n(University of Miami)\nTitle: Perverse sheaves of categories on surfaces  \nAbstract: Perverse sheaves of categories on a Riemann surface S are systems of categories and functors which are encoded by a graphs on S\, and which satisfy conditions that resemble the classical characterization of perverse sheaves on a disc. \nI’ll review the basic ideas behind Kapranov and Schechtman’s notion of a perverse schober and generalize this to perverse sheaves of categories on a punctured Riemann surface. Then I will give several examples of perverse sheaves of categories in both algebraic geometry\, symplectic geometry\, and category theory. Finally\, I will describe how one should be able to use related ideas to prove homological mirror symmetry for certain noncommutative deformations of projective 3-space. \n \n \n \n\n\nMay 15\, 17  \n1-3pm\nCharles Doran  \n(University of Alberta)\n\n\n\n\n\n\nLecture One:\nTitle: Picard-Fuchs uniformization and Calabi-Yau geometry\nAbstract:\n\n\n\n\n\n\nPart 1:  We introduce the notion of the Picard-Fuchs equations annihilating periods in families of varieties\, with emphasis on Calabi-Yau manifolds.  Specializing to the case of K3 surfaces\, we explore general results on “Picard-Fuchs uniformization” of the moduli spaces of lattice-polarized K3 surfaces and the interplay with various algebro-geometric normal forms for these surfaces.  As an application\, we obtain a universal differential-algebraic characterization of Picard rank jump loci in these moduli spaces.\n\nPart 2:  We next consider families with one natural complex structure modulus\, (e.g.\, elliptic curves\, rank 19 K3 surfaces\, b_1=4 Calabi-Yau threefolds\, …)\, where the Picard-Fuchs equations are ODEs.  What do the Picard-Fuchs ODEs for such families tell us about the geometry of their total spaces?  Using Hodge theory and parabolic cohomology\, we relate the monodromy of the Picard-Fuchs ODE to the Hodge numbers of the total space.  In particular\, we produce criteria for when the total space of a family of rank 19 polarized K3 surfaces can be Calabi-Yau.\n\n\n  \n\nLecture Two:\nTitle: Calabi-Yau fibrations: construction and classification\nAbstract: \nPart 1:  Codimension one Calabi-Yau submanifolds induce fibrations\, with the periods of the total space relating to those of the fibers and the structure of the fibration.  We describe a method of iteratively constructing Calabi-Yau manifolds in tandem with their Picard-Fuchs equations. Applications include the tower of mirrors to degree n+1 hypersurfaces in P^n and a tower of Calabi-Yau hypersurfaces encoding the n-sunset Feynman integrals. \nPart 2:  We develop the necessary theory to both construct and classify threefolds fibered by lattice polarized K3 surfaces.  The resulting theory is a complete generalization to threefolds of that of Kodaira for elliptic surfaces.  When the total space of the fibration is a Calabi-Yau threefold\, we conjecture a unification of CY/CY mirror symmetry and LG/Fano mirror symmetry by mirroring fibrations as Tyurin degenerations.  The detailed classification of Calabi-Yau threefolds with certain rank 19 polarized fibrations provides strong evidence for this conjecture by matching geometric characteristics of the fibrations with features of smooth Fano threefolds of Picard rank 1.
URL:https://cmsa.fas.harvard.edu/event/2018-hms-focused-lecture-series/
CATEGORIES:Seminars
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