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Alan Weinstein studies the mathematical interface between classical mechanics and quantum mechanics, particularly symplectic geometry. His Berkeley research profile identifies this connection.^[ref]

With Sean Bates, he wrote Lectures on the Geometry of Quantization by Sean Bates and Alan Weinstein. Their treatment of half-densities explains how to integrate products of wave-like objects intrinsically: the product of two half-densities is a density, giving a coordinate-independent inner product and hence a Hilbert space. This is part of the geometric infrastructure needed to relate classical phase spaces to quantum states.^[ref]

Ana Cannas da Silva authored an exposition of symplectic geometry. A nondegenerate closed two-form organizes phase-space geometry and Hamiltonian motion, giving a geometric framework shared by mechanics and global mathematical questions.^[ref]

Contact topology studies global properties of contact structures on odd-dimensional smooth manifolds. Locally such a structure is a hyperplane distribution $\xi =\ker \alpha$ with $\alpha \wedge (d\alpha {)}^n$ nowhere zero in dimension $2n+1$. Multiplying $\alpha$ by a nowhere-zero function leaves the hyperplane distribution unchanged: the structure is the field of hyperplanes, not a preferred one-form.^[ref]

In ${\mathbb{R}}^3$, take $\alpha =dz-ydx$. Then $d\alpha =dx\wedge dy$ and $\alpha \wedge d\alpha =dz\wedge dx\wedge dy\ne 0$. The spanning vector fields ${\partial }_x+y{\partial }_z$ and ${\partial }_y$ have Lie bracket $-{\partial }_z$, outside the distribution. Thus the planes cannot be tangent to a foliation by surfaces. The local normal form is universal, while global topology distinguishes contact structures and constrains their relation to symplectic geometry.

Denis Auroux studies symplectic geometry, with work linking geometric constructions to topology and algebraic geometry. His Harvard research profile identifies the specialization.^[ref]

His lectures with Ivan Smith develop descriptions of symplectic manifolds through Lefschetz pencils and branched covers of the projective plane. The resulting invariants combine the algebra of monodromy with geometric information from pseudoholomorphic curves. This creates a bridge between concrete maps and the classification of manifolds.^[ref]

Kähler geometry studies complex manifolds with a Hermitian metric $g$ whose associated real two-form $\omega (X,Y)=g(JX,Y)$ is closed. Here $J$ is the complex structure. Equivalently, the Levi-Civita connection preserves $J$. Thus the same metric simultaneously supports complex geometry, Riemannian geometry and symplectic geometry.^[ref]

On ${\mathbb{C}}^n$, the Euclidean metric gives $\omega ={\sum }_{j}d{x}_j\wedge d{y}_j$, which is closed and nondegenerate. Locally one can express a Kähler form as $\omega =i\partial \bar{\partial }\phi$ for a real Kähler potential $\phi$, using a stated normalization. On Complex projective space, $\phi =\log (1+{\sum }_j|z_j{|}^2)$ gives the Fubini-Study metric in an affine chart. The potentials need not glue as one global function; their forms glue. This is how locally elementary differentiation can define nontrivial global geometry.

Marco Gualtieri works in differential geometry and mathematical physics, with a focus on generalized geometry. His own research page identifies that focus rather than presenting him only as an instructor.^[ref] The geometric viewpoint concerns structures carried by a smooth manifold, where tangent directions and covectors can be studied together. It connects the study of complex geometry and symplectic geometry with mathematical descriptions of physical fields.

Homepage: www.math.toronto.edu/mgualt/

Instructor for the 2012 Topology I: smooth manifolds (MAT1300HF) course; the course information explicitly identifies his role as instructor.

Mikhail Gromov is a mathematician whose work connects Riemannian geometry, symplectic geometry and geometric group theory. His 1985 paper introduced powerful methods based on J-holomorphic curves and established the ball-embedding obstruction underlying the Gromov non-squeezing theorem.^[ref]

The obstruction says that a standard symplectic ball of radius $R$ cannot fit symplectically into $B^2(r)\times {\mathbb{R}}^{2n-2}$ when $R>r$. In higher dimensions the target has infinite volume, so preserving volume is weaker than preserving the symplectic form. This gives a concrete example of the global rigidity his work uncovered. His IHES profile records his permanent professorship from 1982 to 2015 and subsequent emeritus status.^[ref] He received the 2009 Abel Prize.^[ref]

Let the step be $h$. Forward Euler method gives $x_1=x_0+h{v}_0$, $v_1=v_0+hF(x_0)$. Applying the same map with $-h$ gives $x_2=x_1-h{v}_1=x_0-h^{2}F(x_0)$, which generally differs from $x_0$. It is therefore not its own inverse under time reversal.

A kick–drift–kick leapfrog integrator instead hasThe step $-h$ first recovers $v_{1/2}$, then $x_0$, then $v_0$, so $\boxed{{\Phi }_{-h}={\Phi }_h^{-1}}$. This assumes a position-dependent force and the same fixed step. Newtonian mechanics is reversible; respecting that structure helps avoid artificial directional bias and often improves long-term qualitative behaviour. The usual leapfrog also has a structure preserving symplectic geometry, which supports bounded oscillatory energy error, but reversibility by itself does not imply exact conservation of energy. Adaptive stepping requires special care to preserve these properties.

Robert E. Gompf studies four-dimensional topology, symplectic topology and contact topology. His University of Texas research profile also identifies Stein surfaces as a focus.^[ref]

His symplectic sum construction glues symplectic manifolds along suitable codimension-two submanifolds. Among its consequences, every finitely presented group occurs as the fundamental group of a compact symplectic manifold in each even dimension at least four. This shows how flexible symplectic geometry can be compared with the additional restrictions imposed by a Kähler manifold.^[ref]

Sean Bates coauthored Lectures on the Geometry of Quantization by Sean Bates and Alan Weinstein with Alan Weinstein. The manuscript's title page verifies the byline; its preface traces the notes to Weinstein's courses on symplectic geometry and geometric quantization.^[ref]

The lectures connect Hamiltonian mechanics on a symplectic manifold with unitary evolution on a Hilbert space. Their account of half-densities supplies an intrinsic way to define an inner product, avoiding dependence on a chosen coordinate volume element. The contribution is an exposition of the geometry linking the two mathematical descriptions of motion.^[ref]

A solid ellipsoid is the filled region bounded by an ellipsoid. The open version centered at the origin iswhere $Q$ is a real symmetric positive-definite matrix; replacing $<$ by $\le$ includes the boundary. Its semi-axis lengths are ${\lambda }_i^{-1/2}$, where ${\lambda }_i$ are the eigenvalues of $Q$. In three dimensions its volume is $4\pi /(3\sqrt{\det Q})$, by a change of variables from the unit ball.

In even-dimensional symplectic geometry, a uniform dilation $x\mapsto tx$, $t>0$, multiplies the symplectic capacity by $t^2$. This concerns one common factor in every coordinate. Independent changes in the different semi-axis lengths are not a uniform dilation and have no such common square factor.