, and also below.

Finiteness of totally geodesic hypersurfaces
with
David Fisher and
Ben Lowe
[
arXiv
]
Abstract ±
We prove that a negatively curved analytic Riemannian manifold that contains infinitely many totally geodesic hypersurfaces is isometric to an arithmetic hyperbolic manifold.

The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds
with
J. Lesieutre,
V. Tosatti
[
arXiv
]
Abstract ±
We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain CalabiYau complete intersections known as Wehler Nfolds. We find that the volume function exhibits a pathological behavior when \(N\geq 3\), we obtain examples of a pseudoeffective \(\mathbb{R}\)divisor D for which the volume of \(D+sA\), with s small and A ample, oscillates between two powers of s, and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that \(h^0(X,\lfloor mD\rfloor+A)\) displays a similar oscillatory behavior as m increases, showing that several notions of numerical dimensions of D do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold.

Gaps in the support of canonical currents on projective K3 surfaces
with
V. Tosatti
Journal of Geometric Analysis
(2024), vol. 34, article nr. 76
[
arXiv

journal
]
Abstract ±
We construct examples of canonical closed positive currents
on projective K3 surfaces that are not fully supported on the complex
points.
The currents are the unique positive representatives in their
cohomology classes and have vanishing selfintersection.
The only previously known such examples were due to McMullen on nonprojective K3 surfaces and were constructed using positive entropy automorphisms with a Siegel disk.
Our construction is based on a Zassenhaustype estimate for commutators of automorphisms.

Translation Surfaces: Dynamics and Hodge Theory
EMS Surv. Math. Sci. 11 (2024), no. 1, pp. 63–151
[
pdf

journal
]
Abstract ±
A translation surface is a multifaceted object that can be studied with the tools of dynamics, analysis, or algebraic geometry.
Moduli spaces of translation surfaces exhibit equally rich features.
This survey provides an introduction to the subject and describes some developments that make use of Hodge theory to establish algebraization and finiteness statements in moduli spaces of translation surfaces.

Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents
[
arXiv
]
Abstract ±
We develop a class of uniformizations for certain weight 3 variations of Hodge structure (VHS).
The analytic properties of the VHS are used to establish a conjecture of Eskin, Kontsevich, Möller, and Zorich on Lyapunov exponents.
Additionally, we prove that the monodromy representations are logAnosov, a dynamical property that has a number of global consequences for the VHS. We establish a strong Torelli theorem for the VHS and describe appropriate domains of discontinuity.
Additionally, we classify the hypergeometric differential equations that satisfy our assumptions.
We obtain several multiparameter families of equations, which include the mirror quintic as well as the six other thin cases of DoranMorgan and BravThomas.

A cyclotomic family of thin hypergeometric monodromy groups in
\(\operatorname{Sp}_4(\mathbb{R})\)
with
C. Fougeron
Geometriae Dedicata (2024),
vol. 218,
article nr. 44
[
arXiv

journal
]
Abstract ±
We exhibit an infinite family of discrete subgroups of \(\operatorname{Sp}_4(\mathbb{R})\) which have a number of remarkable properties.
Our results are established by showing that each group plays pingpong on an appropriate set of cones.
The groups arise as the monodromy of hypergeometric differential equations with parameters
\(\left(\tfrac{N3}{2N},\tfrac{N1}{2N},\tfrac{N+1}{2N},\tfrac{N+3}{2N}\right)\)
at infinity and maximal unipotent monodromy at zero, for any integer \(N\geq 4\).
Additionally, we relate the cones used for pingpong in \(\mathbb{R}^4\) with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian.

Canonical currents and heights for K3 surfaces
with V. Tosatti
Cambridge Journal of Mathematics, (2023), vol. 11(3), pp. 699 – 794
[
arXiv

journal
]
Abstract ±
We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.

Asymptotic shifting numbers in triangulated categories
with Y.W. Fan
Advances in Mathematics (2023), vol. 428
[
arXiv

journal
]
Abstract ±
We introduce invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangulated category translates inside the category.
The invariants are analogous to Poincare translation numbers that are widely used in dynamical systems.
We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences.
Additionally, we relate our shifting numbers to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich.

On pseudoAnosov autoequivalences
with Y.W. Fan,
F. Haiden,
L. Katzarkov,
Y. Liu
Advances in Mathematics (2021), vol. 384
[
arXiv

journal
]
Abstract ±
Motivated by results of Thurston, we prove that any autoequivalence on a triangulated category induces a canonical filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. We then propose a new definition of pseudoAnosov autoequivalences, and prove that our definition is more general than the one previously proposed by Dimitrov, Haiden, Katzarkov, and Kontsevich. We construct new examples of pseudoAnosov autoequivalences on the derived categories of quintic CalabiYau threefolds and quiver CalabiYau categories. Finally, we prove that these new examples of pseudoAnosov autoequivalences on quiver 3CalabiYau categories act hyperbolically on the space of Bridgeland stability conditions.

Tropical Dynamics of areapreserving maps
Journal of Modern Dynamics
14 (2019), pp. 179–226
[
arXiv

journal
]
Abstract ±
We consider a class of areapreserving, piecewise affine maps on the 2sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.

Kummer rigidity for K3 surface automorphisms via Ricciflat metrics
with V. Tosatti
American Journal of Mathematics
(2021)
vol. 143, no. 5, pp. 1431–1462
[
arXiv

journal
]
Abstract ±
We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricciflat metrics on K3s and also covers the nonprojective case.

Smooth and Rough Positive Currents
with V. Tosatti
Ann. Inst. Fourier, Grenoble
(2018), vol. 68, no. 7, pp. 29812999
[
arXiv

journal
]
Abstract ±
We study the different notions of semipositivity for (1,1) cohomology classes on K3 surfaces.
We first show that every big and nef class (and every nef and rational class) is semiample, and in particular it contains a smooth semipositive representative.
By contrast, a result of Cantat and Dupont implies that there exist irrational nef classes with no closed positive current representative which is smooth outside a proper analytic subset. We use this to answer negatively two questions of the secondnamed author.

The algebraic hull of the KontsevichZorich cocycle
with A. Eskin
and A. Wright
Ann. of Math. (2) 188 (2018), no. 1, 281–313
[
arXiv

journal
]
Abstract ±
We compute the algebraic hull of the KontsevichZorich cocycle over any \(GL^+_2(R)\) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

Counting special Lagrangian fibrations in twistor families of K3 surfaces
with an appendix by
N. Bergeron
and C. Matheus
Ann. Sci. Éc. Norm. Supér. (4), 53(3):713–750, 2020
[
arXiv

journal
]
Abstract ±
The number of closed billiard trajectories in a rationalangled polygon grows quadratically in the length.
This paper gives an analogue on K3 surfaces, by considering special Lagrangian tori.
The analogue of the angle of a billiard trajectory is a point on a twistor sphere, and the number of directions admitting a special Lagrangian torus fibration with volume bounded by V grows like \(V^{20}\).

Families of K3 surfaces and Lyapunov exponents
Israel J. Math. 226 (2018), no. 1, 29–69
[
arXiv

journal
]
Abstract ±
Consider a family of K3 surfaces over a hyperbolic curve (i.e. Riemann surface).
Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number.
One proof uses the KugaSatake construction, which reduces the question to Hodge structures of weight 1.
A second proof uses integration by parts.
The case of maximal Lyapunov exponent corresponds to modular families, given by the Kummer construction on a product of isogenous elliptic curves.

Quaternionic covers and monodromy of the KontsevichZorich cocycle in orthogonal groups
with G. Forni
and C. Matheus
J. Eur. Math. Soc. (JEMS) 20 (2018), no. 1, 165–198
[
arXiv

journal
]
Abstract ±
We give an example of a Teichmüller curve which contains, in a factor of its monodromy, a group which was not observed before.
Namely, it has Zariski closure equal to the group \(\operatorname{SO}^*(6) \) in its standard representation;
up to finite index, this is the same as \(\operatorname{SU}(3,1)\) in its second exterior power representation.
The example is constructed using origamis (i.e. squaretiled surfaces).
It can be generalized to give monodromy inside the group \( \operatorname{SO}^*(2n)\) for all n, but in the general case the monodromy might split further inside the group.
Also, we take the opportunity to compute the multiplicities of representations in the (0,1) part of the cohomology of regular origamis, answering a question of MatheusYoccozZmiaikou.

Zero Lyapunov exponents and monodromy of the KontsevichZorich cocycle
Duke Math. J. 166 (2017), no. 4, 657–706
[
arXiv

journal
]
Abstract ±
We describe the situations in which the KontsevichZorich cocycle has zero Lyapunov exponents.
Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints.
We also describe the real Lie groups which can appear in the monodromy of the KontsevichZorich cocycle.
The number of zero exponents is then as small as possible, given its monodromy.

Splitting mixed Hodge structures over affine invariant manifolds
Ann. of Math.
(2) 183 (2016), no. 2, 681–713
[
arXiv

journal

Blog post
]
Abstract ±
We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties.
The result is deduced from a generalization of a theorem of Möller.
Namely, we prove that the image of a certain twisted AbelJacobi map lands in the torsion of a factor of the Jacobians.
This statement can be viewed as a splitting of certain mixed Hodge structures.

Semisimplicity and rigidity of the KontsevichZorich Cocycle
Invent. Math. 205 (2016), no. 3, 617–670
[
arXiv

journal

Blog post
]
Abstract ±
We prove that invariant subbundles of the KontsevichZorich cocycle respect the Hodge structure.
In particular, we establish a version of Deligne semisimplicity in this context.
This implies that invariant subbundles must vary polynomially on affine manifolds.
All results apply to tensor powers of the cocycle and this implies that the measurable and realanalytic algebraic hulls coincide.
We also prove that affine manifolds parametrize Jacobians with nontrivial endomorphisms.
Typically a factor has real multiplication.
The tools involve curvature properties of the Hodge bundles and estimates from random walks.
In the appendix, we explain how methods from ergodic theory imply some of the global consequences of Schmid's work on variations of Hodge structures.
We also derive the KontsevichForni formula using differential geometry.

On Höldercontinuity of Oseledets subspaces
with V. Araujo
and A. Bufetov
J. Lond. Math. Soc. (2) 93 (2016), no. 1, 194–218
[
arXiv

journal
]
Abstract ±
For Hölder cocycles over a Lipschitz base transformation, possibly noninvertible, we show that the subbundles given by the Oseledets Theorem are Hoeldercontinuous on compact sets of measure arbitrarily close to 1.
The results extend to vector bundle automorphisms, as well as to the KontsevichZorich cocycle over the Teichmüller flow on the moduli space of abelian differentials.
Following a recent result of ChaikaEskin, our results also extend to any given Teichmüller disk.

Translation Surfaces: Dynamics and Hodge Theory
[
pdf
]
Abstract ±
A translation surface is a multifaceted object that can be studied with the tools of dynamics, analysis, or algebraic geometry.
Moduli spaces of translation surfaces exhibit equally rich features.
This survey provides an introduction to the subject and describes some developments that make use of Hodge theory to establish algebraization and finiteness statements in moduli spaces of translation surfaces.

Global properties of some weight 3 variations of Hodge structure
Proceedings of the 8th European Congress of Mathematics (2023), 553568
[
journal
]
Abstract ±
We survey results on the global geometry of variations of Hodge structure with Hodge numbers (1, 1, 1, 1). Included are uniformization results of domains in flag manifolds, a strong Torelli theorem, as well as the formula for the sum of Lyapunov exponents conjectured by Eskin, Kontsevich, Möller, and Zorich. Additionally, we establish the Anosov property of the monodromy representation, using gradient estimates of certain functions derived from the Hodge structure.

Geometry and dynamics on Riemann and K3 surfaces
Eur. Math. Soc. Mag.
119 (2021), 17–22
[
journal
]
Abstract ±
Surfaces are some of the simplest yet geometrically rich manifolds. Geometric structures on surfaces illuminate their topology and are useful for studying dynamical systems on surfaces. We illustrate below how some of these concepts blend together, and relate them to algebraic geometry.

An introduction to K3 surfaces and their dynamics
[ pdf
]
Abstract ±
These notes provide an introduction to the geometry of K3 surfaces and the dynamics of their automorphisms. The notes are based on lectures delivered in Grenoble in July 2018, and in Beijing in July 2019.
Panoramas & Synthèses
(2022) Teichmüller theory and dynamics, 143

Lectures on the Oseledets Multiplicative Ergodic Theorem
Ergodic Theory and Dynamical Systems
(2019) vol. 39(5), 11531189
[ pdf

arxiv

journal version
]
Abstract ±
The Oseledets Multiplicative Ergodic theorem is a basic result with numerous applications throughout dynamical systems.
These notes provide an introduction to this theorem, as well as subsequent generalizations.
They are based on lectures at summer schools in Brazil and France.

An introduction to hypergeometric equations, via Dmodules
Abstract ±
In progress, available upon request.
we have been organizing the BiSTRO seminar that meets on the last Wednesday of every other month on Zoom.