Tropical and log corals with a view towards symplectic cohomology
We will discuss an algebraic-geometric approach to some open invariants arising naturally on the A-model side of mirror symmetry. We will first overview of the use of logarithmic geometry in the Gross-Siebert program. We then will discuss various illustrations of the use in open invariants, including a description of the symplectic Fukaya category via certain stable logarithmic curves. For this, our main object of study will be the degeneration of elliptic curves, namely the Tate curve. However, the results are expected to generalise to higher dimensional Calabi-Yau manifolds.
Kernels in noncommutative algebraic geometry
I will discuss joint work with Blake Farman on derived Morita theory for noncommutative projective schemes. A consequence is Morita statement along the lines of Orlov in the commutative case.
Quantum master equation on cyclic cochains and categorical higher genus Gromov-Witten invariants
The construction of cohomology classes in the compactified moduli spaces of curves based on the quantum master equation on cyclic cochains will be reviewed. For the simplest category consisting of one object with only the identity morphism it produces the generating function for products of the psi-classes. The talk is based on the speaker's works “Modular operads and Batalin-Vilkovisky geometry” (MPIM Bonn preprint 2006-48 (04/2006)) and “Noncommutative Batalin–Vilkovisky geometry and matrix integrals” (preprint Hal-00102085 (09/2006)).
Calabi-Yau moduli space metric for hypersurfaces in weighted projective spaces
It is known that the Lagrangian of massless fields in superstring theories compactified on a Calabi-Yau 3-fold can be computed in terms the Special geometry on the CY moduli space. It is the reason why we need to know the Weil-Peterson metric on the CY moduli space. For the case where the CY is given by a hypersurface in a weighted projective space there are only few explicit computations of this metric. I will talk about a new method to compute the moduli space metric in the case of hypersurfaces in weighted projective spaces using a connection with the invariant Frobenius ring structure of the corresponding Landau-Ginzburg model. In the talk I will explain this method and illustrate its efficiency for the case 101 moduli space of the quintic threefold.
Motivic realizations of dg-categories, matrix factorizations and vanishing cycles
Given a cohomology theory for schemes, it is often hard to generalize this notion to dg-categories (aka noncommutative spaces). One method consists in approximating a noncommutative space by the geometric stack of objects inside it, which gives rise to a motive. In this talk we will explain how to use Morel—Voevodsky’s homotopy theory of schemes and realization functors in order to define some cohomology theories for noncommutative spaces (Betti, l-adic). Given a LG model over a discrete valuation ring with perfect residue field, with potential induced by a uniformizer, we will see how the l-adic cohomology of the associated category of matrix factorizations is given by the inertia invariant part of vanishing cohomology. (Joint work with M. Robalo, B. Toën, G. Vezzosi.)
Categorifying non-commutative deformation theory
I will discuss a categorification of the non-commutative deformation theory of n objects in an abelian category. A suitable abelian category plays the role of the non-commutative base for a deformation in this approach. A motivation coming from a categorical description of flops will be outlined. I will give sufficient conditions for the pro-representability of the deformation functor. I will also construct a pro-representing hull for the deformation functor in general situation and discuss how to recover the functor from the hull.
Schobers for Grothendieck resolutions
I will discuss Grothendieck resolutions related to semisimple Lie algebras and relevant systems of flops and other birational transformations. Then I describe a Schober of a Grothendieck resolution, i.e. a categorified perverse sheaf on the complex vector space stratified by a system of hyperplanes of Dynkin type. This is a joint work with M. Kapranov and V. Schechtman.
Higher Kac-Moody algebras
absolutely indecomposable representations of a set quiver with varying dimension vectors, record the root multiplicities of the associated Kac-Moody Lie algebra. This begs the question: is there some Lie-theoretic interpretation of the other coefficients of the Kac polynomials? The answer to this question comes via Donaldson-Thomas theory. It was shown by Mozgovoy that the Kac polynomials themselves can be considered as refined DT invariants of special quivers (endowed with potential). The Jacobi algebras associated to such data can be thought of as “nc-Landau-Ginzburg models”. Recent work with Sven Meinhardt on the categorification of DT theory shows how to upgrade this statement to a Lie-theoretic interpretation for the entire Kac polynomials (not just the constant coefficients). If there is time I will explain how recent work of McGerty and Nevins, along with purity results on the Borel-Moore homology of preprojective stacks, suggests a conjectural approach to “Borcherdsifying” the resulting extended Kac-Moody Lie algebra.
Some new categorical invariants
In a joint work with L. Katzarkov, viewing \(D^b(K(l))\) as a non-commutative curve, where \(K(l)\) is the \(l\)-Kronecker quiver, we observe that it is reasonable to count non-commutative curves in any category which lies in a small neighborhood (with respect to our topology) of a given non-commutative curve. Examples show that this idea (non-commutative curve-counting) opens directions to new categorical structures and connections to number theory and classical geometry. We give a general definition, which specializes to the non-commutative curve-counting invariants.
Holomorphic Chern-Simons Theory and Noncommutative Geometry
In this talk I will explain how quiver gauge theories in even dimensions (up to ten) have holomorphic Chern-Simons theory as their common origin. The quiver gauge theories realize non-commutative resolutions of CY n-folds. Finally, I will explain the relationship between supersymmetric indices of the field theories and negative cyclic homology of the CY n-folds. For four dimensional quiver guage theories in the large-N limit, the relationship between the supersymmetric index of the quiver theory and negative cyclic homology gives a mathematical proof of the matching of local operators between the field theory and its holographic dual under the AdS/CFT correspondence.
Algebraicity and algebraizability
We will recall the notion of algebraizability of perfect complexes on a formal neighborhood of a closed subscheme: these are perfect complexes which are generated (via cones, shifts and direct sumands) by the restrictions of perfect complexes on the ambient scheme. We will state several results stating that in some situations algebraizability is equivalent to algebraicity of certain power series. A special case is covered in the recent preprint arXiv:1711.00756.
On the construction of LG models on coadjoint orbits
In this talk we describe the LG models associated to coadjoint orbits of complex simple Lie groups. We also discuss its Fukaya-Seidel category in low dimensional examples as well geometric informations about the mirror manifold.
Semi-stability and iterated logarithms
A very important tool in the study of algebraic vector bundles is the notion of (semi-) stability. It turns out that any semistable vector bundle admits a canonical filtration whose subquotients are direct sums of stable bundles. This weight-type filtration depends only on the lattice of vector bundles of the same slope contained in a given bundle and is defined in the general context of modular lattices. It has an analytic interpretation in terms of certain gradient flows and this is how we originally discovered it. Conjecturally, the filtration describes the asymptotics of Donaldson's heat flow on the space of metrics on a semistable bundle. All this is joint work with Katzarkov-Kontsevich-Pandit (arXiv:1706.01073).
Toric Schobers and D-modules
Many classical mirror symmetry results can be recast using the more recent language of perverse sheaves of categories and schobers. In this context, I will explain a Riemann-Hilbert type conjectural connection with the D-modules naturally appearing in mirror symmetry.
Mirror Symmetry of Branes and the Quest of Hyperbolic 3-manifolds
We discuss the computation of certain normal functions on the mirror quintic Calabi-Yau threefold in a semi-stable degeneration limit. In this limit the normal functions are described as elements of higher Chow groups. Physically this amounts to computing the domain wall tension between certain B-branes on the mirror quintic in the large complex structure limit. By mirror symmetry we expect that these normal functions/domain wall tensions have a geometric meaning on the quintic Calabi-Yau threefold for suitable A-branes. As we discuss, the number theoretic structure of the analyzed normal functions in this limits suggests a relation to hyperbolic 3-manifolds. This talk is based on work in progress with Dave Morrison and Johannes Walcher.
\(\mathbb{P}^n\)-functors and cyclic covers
I will give and explain the definition of a (non-split) \(\mathbb{P}^n\)-functor, prove that its P-twist is unique, and give our first ever example of such functor using the geometry of a cyclic \((n+1)\)-to-\(1\) cover ramified in a divisor. This is a joint work with Rina Anno (Kansas).
Irrationality of the motivic zeta function
Kapranov defined a motivic zeta function \(\zeta(X)\) of a variety \(X\) as a power series with coefficients in the Grothendieck ring of varieties. It is a motivic lift of the classical Hasse-Weil zeta function of \(X\). Kapranov proved that \(\zeta(X)\) is rational if \(\dim X\) is at most one. He then also conjectured that rationality holds for all varieties \(X\). Jointly with Michael Larsen we disproved this conjecture in the paper math/0110255. However our method does not prove irrationality if one considers the localization of the Grothendieck ring of varieties by inverting \(\mathbb{L}\) – the class of the affine line. Actually Denef and Loeser conjectured that \(\zeta(X)\) is rational in this localized ring. I will report on our recent theorem with Larsen which claims irrationality even after inverting \(\mathbb{L}\).
Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces I, II, III
Let \((E,\overline{\partial}_E,\theta)\) be a stable Higgs bundle of degree 0 on a compact connected Riemann surface. Once we fix a flat metric \(h_{\det(E)}\) on the determinant of \(E\), we have the harmonic metrics \(h_t\) \((t>0)\) for the stable Higgs bundles \((E,\overline{\partial}_E,t\theta)\) such that \(\det(h_t)=h_{\det(E)}\).
In this series of talks, we will discuss two results on the behaviour of \(h_t\) when \(t\) goes to \(\infty\). First, we show that the Hitchin equation is asymptotically decoupled under some assumption for the Higgs field. We apply it to the study of the so called Hitchin WKB-problem. Second, we discuss the convergence of the sequence \((E,\overline{\partial}_E,\theta,h_t)\) in the case where the rank of \(E\) is 2. We explain a rule to describe the parabolic weights of a “limiting configuration”, and we show the convergence of the sequence to the limiting configuration in an appropriate sense.
In the talk I, we will give an overview. In the talks II and III, we will give more details without assuming that the audience have listened to the talk I.
Some phase-transitions and their q-analogs on powers of the quantum dilogarithm
The quantum dilogarithm has been important object to study in mathematics and physics. We consider phase-transitions on the powers of the quantum dilogarithm, assuming statistical-mechanical interpretations on all of these powers. We discuss the phase-transitions by its multiplicative q-analog. We then state some conjectural positivity of q-polynomials and additive q-analogs of the phase-transitions. This is a work-in-progress.
Center manifold theory for the Yang-Mills-Higgs flow
Quantum periods of orbifold del Pezzo surfaces
Recent work of Coates, Corti, Kasprzyk et al. investigated Mirror Symmetry for del Pezzo surfaces with isolated cyclic quotient singularities: on one side there are such surfaces with their quantum cohomology, whereas on the other side there are polygons and Laurent polynomials. In this talk, I will review a couple of conjectures in this field and I will provide some evidence for the case where the surfaces has only a particular type of singularity, based on joint work with Alessandro Oneto.
Degenerations from Floer cohomology
I will explain how, under suitable hypotheses, one can construct a flat degeneration from the symplectic cohomology of log Calabi-Yau varieties to the Stanley-Reisner ring on the dual intersection complex of a compactifying divisor. I will explain how this result connects to classical mirror constructions of Batyrev and Hori-Iqbal-Vafa as well as ongoing work of Gross and Siebert.
HKR theorems
In this talk I will survey recent progresses in derived geometry. Building on the techniques of derived analytic geometry, I will explain how we can generalize the Hochschild-Kostant-Rosenberg theorem in the setting of (not necessarily derived) analytic spaces. Here “analytic” means both complex and non-archimedean. This is joint work with F. Petit and J. Antonio.
Cluster algebras, \(SL(N, \mathbb{C})\) flat connections and 3-manifolds
\(SL(N, \mathbb{C})\) Chern-Simons theory have been getting some increasing interest in math and physics in the last years. I'll describe some recently developed techniques inspired by string theoretical constructions, that makes use of the connection between higher Teichmuller theory and cluster algebras to produce topological invariants of hyperbolic 3-manifolds. As time allows I'll talk about connections with other subjects in this context. They include volume conjecture, defects in quantum field theory and spectral networks.
The hemisphere partition function and LG orbifolds
We consider LG orbifolds and their corresponding gauged linear sigma models. We discuss the relation between the hemisphere partition function in the LG phase of the gauged linear sigma model and Chern characters of equivariant matrix factorizations of the LG potential and generating functions of FJRW invariants.
Hodge-Tate conditions for Landau-Ginzburg models
In the context of mirror symmetry for Fano manifolds, Katzarkov-Kontsevich-Pantev proposed some conjectures for Landau-Ginzburg models. In this talk, I will give a sufficient condition for Landau-Ginzburg models to satisfy some versions of their conjectures. This condition is called Hodge-Tate condition and can be checked for some examples of Landau-Ginzburg models. I will also explain that this condition is natural in the context of mirror symmetry.
Specialization of stable rationality
In this talk based on joint work with J. Nicaise I will explain how stable rationality behaves in families. The main result is that in characteristic zero stable rationality is preserved under specialization in smooth and nodal families. The proof is based on the Grothendieck ring of varieities and various forms of resolution of singularities. The method has been recently used to by Kontsevich and Tschinkel to prove similar results for specialization of rationality.
Log schemes, root stacks and parabolic bundles
Log schemes are an enlargement of the category of schemes that was introduced by Deligne, Faltings, Illusie and Kato, and has applications to moduli theory and deformation problems. Log schemes play a central role in the Gross-Siebert program in mirror symmetry. In this talk I will introduce log schemes and then explain recent work joint with D. Carchedi, S. Scherotzke, and M. Talpo on various aspects of their geometry. I will discuss a comparison result between two different ways of associating to a log scheme its etale homotopy type, respectively via root stacks and the Kato-Nakayama space. Our main result is a new categorified excision result for parabolic sheaves, which relies on the technology of root stacks.
T-duality and brane correspondence on nilmanifolds
We use topological T-duality and the generalized complex geometry point of view to make a correspondence between branes on T-dual nilmanifolds. We discuss some examples up to real dimension 6.
Shifted Hyperkahler Structures
The notion of a shifted symplectic structure has recently been developed as the analogue of a (holomorphic) symplectic structure in derived algebraic geometry. I will the corresponding notion for hyperkahler manifolds in derived algebraic geometry, and explain how this can provide new constructions of hyperkahler manifolds or derived hyperkahler desingularizations. This talk is based on joint work with Ludmil Kazarkov and Pranav Pandit.
Mirror symmetry for lattice polarized del Pezzo surfaces
Mirror symmetry for lattice polarized K3 surfaces is a well-established correspondence. I will discuss the possibility of extending this theory to log Calabi-Yau surfaces, and in particular to weak del Pezzo surfaces. I will propose a way to endow a weak del Pezzo surface with a lattice polarization and suggest how one might define the mirror notion of a lattice polarized Landau-Ginzburg model. Finally, I will discuss how this notion of mirror symmetry may be thought of as a degeneration of mirror symmetry for lattice polarized K3's.
Picard-Fuchs equations, extensions and monodromy
I will review the setup and status of extended mirror symmetry, which is intermediate between the classical relation and full-blwon homological mirror symmetry. I will focus on two related arithmetic features exhibited by the B-model that are so far lacking a fully convincing A-model explanation.
Nonlinear descent on moduli of local systems
We discuss the Diophantine geometry of moduli spaces for special linear rank two local systems on topological surfaces with fixed boundary traces. We show that they form a rich family of log Calabi-Yau varieties, where a structure theorem for their integral points can be established using mapping class group dynamics. This generalizes a classical Diophantine work of Markoff (1880). Related analysis also yields new results on the arithmetic of curves in these moduli spaces.
The Frobenius structure conjecture in dimension two
The Frobenius structure conjecture is a conjecture about the geometry of rational curves in log Calabi-Yau varieties proposed by Gross-Hacking-Keel. It was motivated by the study of mirror symmetry. It predicts that the enumeration of rational curves in a log Calabi-Yau variety gives rise naturally to a Frobenius algebra satisfying nice properties. In a joint work with S. Keel, we prove the conjecture in dimension two. Our method is based on the enumeration of non-archimedean holomorphic curves developed in my thesis. We construct the structure constants of the Frobenius algebra directly from counting non-archimedean holomorphic disks. If time permits, I will also talk about compactification and extension of the algebra.