These questions were used in the trial from which the deposition and court testimony in Appendices F and G were taken. They have been edited for publication, with much of the plaintiff ’s personal information having been removed. The answers contained after the questions were the anticipated answers to Levy’s questions; like any lawyer, Ms. Levy knows the essence of the answer before asking the question. In this context, they are not suggested answers for the reader’s expert witness, but to give an idea of the kind of testimony on the direct examination of the expert witness in this particular trial. The questions are open-ended and permit the entry of several exhibits to help the judge and/or jury make sense of the information being provided.

Appendix D: Questions for Plaintiff Forensic Expert: Direct Examination

In [10], we construct an exotic monotone Lagrangian torus in CP 2 (not Hamiltonian isotopic to the known Clifford and Chekanov tori) using techniques motivated by mirror symmetry. We named it T(1,4,25) because, when following a degeneration of CP 2 to the orbifold CP(1,4,25), it degenerates to the central fiber of the moment map for the standard torus action on CP(1,4,25). Related to each degeneration from CP 2 to CP(a 2 ,b 2 ,c 2 ), for (a,b,c) a Markov triple - see (1.1) - there is a monotone Lagrangian torus, which we call T(a 2 ,b 2 ,c 2 ). We conjectured that no two of them are Hamiltonian isotopic to each other. Here we employ techniques from symplectic field theory to prove that the above conjecture is true.

Infinitely many exotic monotone Lagrangian tori in ℂℙ2

We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we describe surgeries that preserve the ambient Weinstein manifold, but change the skeleton. The surgeries can be iterated to produce more such skeleta --- in many cases, infinitely many more. 
Each skeleton is built around a Lagrangian surface. Passing to the Fukaya category, the skeletal surgeries induce cluster transformations on the spaces of rank one local systems on these surfaces, and noncommutative analogues of cluster transformations on the spaces of higher rank local systems. In particular, the problem of producing and distinguishing such Lagrangians maps to a combination of combinatorial-geometric questions about curve configurations on surfaces and algebraic questions about exchange graphs of cluster algebras. 
Conversely, this expands the dictionary relating the cluster theory of character varieties, positroid strata, and related spaces to the symplectic geometry of Lagrangian fillings of Legendrian knots, by incorporating cluster charts more general than those associated to bicolored surface graphs.

On the combinatorics of exact Lagrangian surfaces

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ and $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in $\mathbb{C}P^2 \# k \overline{\mathbb{C}P^2}$, for k=0,3,4,5,6,7,8. We name these tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ also have infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ . Finally, the Lagrangian tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$ inside a del Pezzo surface $X$ can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus $\Sigma$. We argue that $\Theta^{n_1,n_2,n_3}_{p,q,r}$ give rise to infinitely many exact Lagrangian tori in $X \setminus \Sigma$, even after attaching the positive end of a symplectization to the boundary of $X \setminus \Sigma$.

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $${\mathbb {C}}P^2\# \overline{{\mathbb {C}}P^2}$$
 , $${\mathbb {C}}P^2\# 2\overline{{\mathbb {C}}P^2}$$
 , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in $${\mathbb {C}}P^2\# k\overline{{\mathbb {C}}P^2}$$
 for $$k=0,3,4,5,6,7,8$$
 . We name these tori $$\Theta ^{n_1,n_2,n_3}_{p,q,r}$$
 . Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that $${\mathbb {C}}P^2\# \overline{{\mathbb {C}}P^2}$$
 also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for $${\mathbb {C}}P^2\# 2\overline{{\mathbb {C}}P^2}$$
 . Finally, the Lagrangian tori $$\Theta ^{n_1,n_2,n_3}_{p,q,r} \subset X$$
 can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus $$\Sigma $$
 . We argue that $$\Theta ^{n_1,n_2,n_3}_{p,q,r}$$
 give rise to infinitely many exact Lagrangian tori in $$X \setminus \Sigma $$
 , even after attaching the positive end of a symplectization to $$\partial (X \setminus \Sigma )$$
 .

/pdf/infinitely-many-monotone-lagrangian-tori-in-del-pezzo-4vhay8bygu.pdf

We construct an exotic monotone Lagrangian torus in CP2 using techniques motivated by mirror symmetry. We show that it bounds 10 families of Maslov index 2 holomorphic discs, and it follows that this exotic torus is not Hamiltonian isotopic to the known Cliffordand Chekanov tori.

/pdf/on-exotic-lagrangian-tori-in-cp2-fx9cckd31d.pdf

On Exotic Lagrangian Tori in CP2

Related to each degeneration from CP^2 to CP(a^2,b^2,c^2), for (a,b,c) a Markov triple - positive integers satisfying a^2 + b^2 + c^2 = 3abc - there is a monotone Lagrangian torus, which we call T(a^2,b^2,c^2). We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.

Renato Vianna

Papers

Infinitely many exotic monotone Lagrangian tori in ℂℙ2

On Exotic Lagrangian Tori in CP2

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Infinitely many exotic monotone Lagrangian tori in CP^2