Department of Mathematics

The University of Western Ontario

In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study the equations of motion of a rigid body in R^3 and the equations of ideal hydrodynamics. I will describe how to extend his formalism and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bi-Hamiltonian structure, the presence of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and global existence of solutions.

Consider two domains $D$ and $G$ in $\mathbb{C}^n$, each of which is the product of smoothly bounded domains, and assume that each factor of $D$ satisfies condition R, i.e, the Bergman projection preserves the class of functions smooth up to the boundary. We show that any proper holomorphic map from $D$ to $G$ extends smoothly to the closures, and splits as a product of equidimensional mappings of the factors. We also consider some possible generalization to a class of piecewise smooth domains. This is joint work with Kaushal Verma.

In this talk, I will highlight some differences between the moduli space of Higgs bundles (in the sense of Hitchin) on a curve of positive genus and the the moduli space of "twisted" Higgs bundles at genus 0. The Betti numbers of both spaces can be determined by a localization calculation, with respect to an $S^1$ action. This was exactly Hitchin's method for obtaining the Betti numbers of the rank-2 instance of the usual Higgs moduli space. The $S^1$ fixed points are what are called "holomorphic chains": these are similar to complexes of vector bundles, but the differential (the Higgs field itself) is nilpotent with order equal to the length of the complex. I will show how the localization calculation can be made very combinatorial in the genus 0 case. The appropriate language for organizing this data is that of quivers, which we use to represent (and construct) families of chains.

Discrete symplectic mappings are not called discrete Hamiltonian systems by the cognoscenti for a variety of reasons, and my aim with this project was to explore the problems involved. In this talk I first show how conservation laws for high order Lagrangian systems transfer to their equivalent Hamiltonian systems with a strikingly beautiful formula. This formula transfers mutatis mutandis to the discrete case. However, analogues of other results do not transfer. I present a series of examples that illustrate the various difficulties, and end with some conjectures and possible ways forward that could involve specialist analytic techniques. The talk assumes no specialist knowledge of the topic.

The classical result of H.Poincare states that a local biholomorphic mapping of an open piece of the 3-sphere in $\mathbb{C}^2$ onto another open piece extends analytically to a global holomorphic automorphism of the sphere. This theorem was generalized by H.Alexander to the case of a sphere in an arbitrary $\mathbb{C}^n,\,n\geq 2$, then later by S.Pinchuk for the case of strictly pseudoconvex hypersurface in the preimage and a sphere in the image, and finally by R.Shafikov and D.Hill for the case of an essentially finite hypersurface in the preimage and a quadric in the image. In this joint work with R.Shafikov we consider the - essentially new - case when a hypersurface $M$ in the preimage contains a complex hypersurface. We demonstrate that the above extension results fail in this case, and prove the following analytic continuation phenomenon: a local biholomorphic mapping of $M$ onto a non-degenerate hyperquadric in $\mathbb{CP}^n$ extends to a punctured neighborhood of the complex hypersurface, lying in $M$, as a multiple-valued locally biholomorphic mapping.

We will discuss some recent results on the $L^2$-theory of the $\overline{\partial}$-equation on the domain $\{\vert z_1 \vert < \vert z_2\vert

I shall report about my 2011 results, which is the resolution of one long standing problem in the theory of Darboux transformations. It is known that many Darboux transformations can be constructed using Darboux Wronskian formulas. The only known exceptions have been two transformations of order one - Laplace transformations, which are often used in applications. I shall show that for order one there is no other exceptions and that for order two Wronskian formulas are complete. History of the question as well as an introduction into the area will be provided.

I shall report about my 2011 results, which is the resolution of one long standing problem in the theory of Darboux transformations. It is known that many Darboux transformations can be constructed using Darboux Wronskian formulas. The only known exceptions have been two transformations of order one - Laplace transformations, which are often used in applications. I shall show that for order one there is no other exceptions and that for order two Wronskian formulas are complete. History of the question as well as an introduction into the area will be provided.

After a brief review of classical function theory on $\mathbb C$, we will discuss its extension to functions with values in an algebraic variety, i.e. an entire holomorphic curve, and motivate the fact that such a curve should be constrained by birational invariants of the variety that pertains to "hyperbolicity." We will verity this conjectural fact for varieties of maximal Albanese dimension, itself a birational invariant. We will assume no prior knowledge of birational geometry. This is joint work with Joerg Winkelmann.

We investigate non-degenerate Lagrangians of the form $$ \int f(u_x, u_y, u_t) dx dy dt $$ such that the corresponding Euler-Lagrange equations $ (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0 $ are integrable by the so-called method of hydrodynamic reductions. The integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f, are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the `master-Lagrangian' corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball.

Symplectomorphism groups are one of the classical infinite-dimensional Lie groups that have been studied. Arnold's paper in 1966, where he used methods of infinite-dimensional Lie theory to study the hydrodynamics of a perfect incompressible fluid, has motivated intensive research in infinite-dimensional Lie theory. He showed that the geodesics on the group of volume preserving diffeomorphisms are essentially solutions of the Euler's equations. In a fundamental paper in 1970 Marsden and Ebin studied some infinite-dimensional groups in more details which included symplectomorphism groups. In this talk we study a special class of symplectomorphism groups that resemble compact Lie groups in a particular way. We see there is a similar notion of the so-called maximal tori in the symplectomorphism groups of toric manifolds. As a consequence we see there is an analogue of the Schur-Horn-Kostant convexity theorem in this infinite-dimensional setting. It also should be mentioned that these results are a generalization of results that were obtained by Bao-Ratiu 1997, Bloch-Flaschka-Ratiu 1993 and El-hadrami 1996 for special cases of toric manifolds.

We develop the basic elements of complex function theory within certain subalgebras of holomorphic functions on coverings of complex manifolds (including holomorphic extension from complex submanifolds, properties of divisors, corona type theorem, holomorphic analogue of Peter-Weyl approximation theorem, Hartogs type theorem, characterization of the uniqueness sets, etc). Our model examples are: (1) subalgebra of Bohr's holomorphic almost periodic functions on tube domains (i.e. the uniform limits of exponential polynomials) (2) subalgebra of all fibrewise bounded holomorphic functions (arising in corona problem for $H^\infty$) (3) subalgebra of holomorphic functions having fibrewise limits. Our proofs are based on the analogues of Cartan theorems A and B for coherent type sheaves on the maximal ideal spaces of these subalgebras. This is joint work with Alexander Brudnyi.

This two-session talk will be concerned with holomorphic mappings between complex analytic sets (or more generally, analytic spaces). Local regularity of such a mapping can be measured by uniformity (or lack thereof) of the family of its fibres. In the first part of the talk, we will discuss the general idea of testing local regularity (like openness or flatness) by passing to fibred powers of a given map. The second session will be devoted to a recent joint work with Janusz Adamus: We establish an analytic version of flatness descent to prove a criterion for flatness of a holomorphic mapping with singular target. Previously, the best analogous result had been known only for the case of smooth targets.

This two-session talk will be concerned with holomorphic mappings between complex analytic sets (or more generally, analytic spaces). Local regularity of such a mapping can be measured by uniformity (or lack thereof) of the family of its fibres. In the first part of the talk, we will discuss the general idea of testing local regularity (like openness or flatness) by passing to fibred powers of a given map. The second session will be devoted to a recent joint work with Janusz Adamus: We establish an analytic version of flatness descent to prove a criterion for flatness of a holomorphic mapping with singular target. Previously, the best analogous result had been known only for the case of smooth targets.

We introduce Schatten-class operators in p-adic Hilbert spaces and study their properties and give several examples. We also show that the Trace class operators in p-adic Hilbert spaces strictly contains the class of completely continuous operators.This gives a totally new perspective to the space of compact operators in p-adic Hilbert spaces. Apart from its intrinsic interest as discussed in this paper, p-adic functional analysis is a fast growing research area and has numerous applications in differential equations, statistics, quantum physics, dynamical systems, cognitive sciences, psychology and sociology, to name a few.

This is a continuation of the talk from October 11. Details will be given of the proof that a Lagrangian surface $X\subset \mathbb C^2$ near an isolated singularity which is a Whitney umbrella is locally polynomially convex. In this talk I will discuss Bruno's construction of normal forms for the dynamical system that determines the phase portrait of the characteristic foliation.

This is a continuation of the talk from October 6. Details will be given of the proof that a Lagrangian surface $X\subset \mathbb C^2$ near an isolated singularity which is a Whitney umbrella is locally polynomially convex. In this talk I will discuss the connection between polynomial convexity of surfaces and their characteristic foliation.

One of the main notion introduced in the study of finite dimensional compact Lie groups is the so-called maximal torus. In 1997, Bao and Ratiu discovered an infinite dimensional subgroup in the group of the volume-preserving diffeomorphisms of the 2-dimensional annulus that can potentially play the role of a maximal torus. They showed this subgroup is a path-connected submanifold which is flat and totally geodesic with respect to the hydrodynamic metric. Moreover it is a maximal abelian subgroup (with a finite Weyl group). This suggested that part of finite dimensional Lie group theory may be extended to the volume-preserving diffeomorphisms of the annulus. Indeed, in a later work, Bloch, Flaschka and Ratiu showed that after an appropriate completion of the spaces considered, a version of Schur-Horn-Kostant convexity theorem holds. El-Hadrami extended these results to the case of the unit sphere and CP^{2}, found a candidate for the maximal torus in the symplectomorphism group of symplectic toric manifolds, and then conjectured that some results in previous works can be extended to those groups. However, a gap in El-Hadrami’s arguments was later discovered. In two talks we discuss some possible extensions and corrections to El-Hadrami´s work. We also mention the Schur-Horn-Kostant convexity theorem for the symplectomorphism groups of toric manifolds.

One of the main notion introduced in the study of finite dimensional compact Lie groups is the so-called maximal torus. In 1997, Bao and Ratiu discovered an infinite dimensional subgroup in the group of the volume-preserving diffeomorphisms of the 2-dimensional annulus that can potentially play the role of a maximal torus. They showed this subgroup is a path-connected submanifold which is flat and totally geodesic with respect to the hydrodynamic metric. Moreover it is a maximal abelian subgroup (with a finite Weyl group). This suggested that part of finite dimensional Lie group theory may be extended to the volume-preserving diffeomorphisms of the annulus. Indeed, in a later work, Bloch, Flaschka and Ratiu showed that after an appropriate completion of the spaces considered, a version of Schur-Horn-Kostant convexity theorem holds. El-Hadrami extended these results to the case of the unit sphere and CP^{2}, found a candidate for the maximal torus in the symplectomorphism group of symplectic toric manifolds, and then conjectured that some results in previous works can be extended to those groups. However, a gap in El-Hadrami’s arguments was later discovered. In two talks we discuss some possible extensions and corrections to El-Hadrami´s work. We also mention the Schur-Horn-Kostant convexity theorem for the symplectomorphism groups of toric manifolds.

Real hypersurfaces in a complex space $\mathbb C^N, N \geq 2$, satisfying the Levi non-degeneracy condition, were very well studied in the famous works of Poincare, Cartan, Tanaka, Chern and Moser and in a large number of further papers. The Levi-degenerate case, which is trivial for $N=2$ (all Levi degenerate hypersurfaces in this case are essentially flat), turns out to be absolutely non-trivial for $N=3$. The reason is that a hypersurface in $\mathbb C^3$ can have Levi form of rank $1$ at a generic point, and, in this case, is neither Levi-flat nor Levi non-degenerate. If, in addition, it satisfies some non-degeneracy condition, guaranteeing that it can not be reduced to a product of a hypersurface in $\mathbb C^2$ and a complex line, the hypersurface is called 2-nondegenerate. 2-nondegenerate hypersurfaces in $\mathbb C^3$ were deeply studied in a series of papers by Ebenfelt, Beloshapka, Zaitsev, Merker, Fels and Kaup and many other authors, but a lot of essential questions, concerned with their holomorphic classification and symmetry groups, remained opened. In the present talk we demonstrate a new approach to the study of 2-nondegenerate hypersurfaces, based on the consideration of degenerate quadratic models. This new point of view enables us to give a complete solution for most of the above open questions.

Real hypersurfaces in a complex space $\mathbb C^N, N \geq 2$, satisfying the Levi non-degeneracy condition, were very well studied in the famous works of Poincare, Cartan, Tanaka, Chern and Moser and in a large number of further papers. The Levi-degenerate case, which is trivial for $N=2$ (all Levi degenerate hypersurfaces in this case are essentially flat), turns out to be absolutely non-trivial for $N=3$. The reason is that a hypersurface in $\mathbb C^3$ can have Levi form of rank $1$ at a generic point, and, in this case, is neither Levi-flat nor Levi non-degenerate. If, in addition, it satisfies some non-degeneracy condition, guaranteeing that it can not be reduced to a product of a hypersurface in $\mathbb C^2$ and a complex line, the hypersurface is called 2-nondegenerate. 2-nondegenerate hypersurfaces in $\mathbb C^3$ were deeply studied in a series of papers by Ebenfelt, Beloshapka, Zaitsev, Merker, Fels and Kaup and many other authors, but a lot of essential questions, concerned with their holomorphic classification and symmetry groups, remained opened. In the present talk we demonstrate a new approach to the study of 2-nondegenerate hypersurfaces, based on the consideration of degenerate quadratic models. This new point of view enables us to give a complete solution for most of the above open questions.

We discuss some new results on existence and regularity of the $\overline{\partial}$-problem on product domains. It is shown that although the $\overline{\partial}$-Neumann operator is not compact, we can still obtain regularity estimates for the canonical solution in certain Sobolev spaces. This is joint work with Mei-Chi Shaw.

We discuss a (new) connection between algebraic geometry/representation theory and convex geometry. We explain a basic construction which associates convex bodies to semigroups of integral points. We see how this gives rise to convex bodies associated to algebraic varieties encoding information about their geometry. This far generalizes the notion of Newton polytope of a toric variety. As an application, we give a formula for the number of solutions of an algebraic system of equations (equivalently self-intersection of a divisor/linear system) on any variety, in terms of volumes of these bodies. This has many interesting applications in algebraic geometry, in particular theory of linear systems. We will see how several convex polytopes naturally appearing in representation theory (of Lie groups) are special cases of this geometric construction. The origin of this approach goes back to influential work of A. Okounkov on multiplicities of representations.

The holomorphic closure dimension is one of the possible biholomorphic invariant that gives some information on how much of the complex ambient structure a subset of a complex manifold inherits. I will review a series of results by Rasul, Rasul and Janusz, and Janusz and myself that discuss the behaviour of the holomorphic dimension in a real analytic set on the one hand and in semialgebraic sets on the other hand.

Transformational Methods are known to be one of the most efficient methods for finding exact solutions of Partial Differential Equations. In this talk we shall be concentrated on the differential transformations introduced by Darboux (DT). DT can be defined by so-called (m,n)-transformations which are Linear Partial Differential Operators without mixed derivatives. The (m,n)-transformations have interesting algebraic structure. The (m,n)-transformations can help us to solve the problem of the generality of the Darboux Wronskian formulas. Namely, Darboux stated and different authors proved for different cases that given some number of partial solutions, a DT can be defined via some Wronskians. Darboux believed that the reverse statement will be true "generally speaking". In this talk we show several results on our way to prove this reverse statement and to decide what is "the general case" in this context. The second part of the talk will be devoted to an invariant description of the DT. We start with an idea that in view of the said above it would be more efficient to defined DT in terms of invariants of the pair (L,z), where L is a Linear Partial Differential Operator, and z is an element of its kernel. We show that such invariants is in correspondence with solutions of certain PDE, and that instead of a chain of DT we can consider mappings of invariants.

One of the most impressive phenomena in several complex variables is the phenomenon of forced analytic continuation for holomorphic functions. The biggest domain, to which the family of all holomorphic functions extends, is called the envelope of holomorphy of a domain or of a real submanifold in a complex space. Envelopes of holomorphy have some nice geometric description, making them in a sense similar to convex hulls of domains and submanifolds in a Euclidian space. In the present talk we discuss some classical theorems for domains of holomorphy as well as some new results for real submanifolds in a complex space.

One of the most impressive phenomena in several complex variables is the phenomenon of forced analytic continuation for holomorphic functions. The biggest domain, to which the family of all holomorphic functions extends, is called the envelope of holomorphy of a domain or of a real submanifold in a complex space. Envelopes of holomorphy have some nice geometric description, making them in a sense similar to convex hulls of domains and submanifolds in a Euclidian space. In the present talk we discuss some classical theorems for domains of holomorphy as well as some new results for real submanifolds in a complex space.

Suppose M is a compact complex manifold. Model theory (a branch of mathematical logic) provides at least two approaches to the study of the complex-analytic subsets of Cartesian powers of M, roughly corresponding to whether one focuses on the real or complex structure on M. We can view M as definable in the structure R_an; that is, as a real globally subanalytic manifold. On the other hand, we can work in the Zariski-type structure CCM where M is the universe and there are predicates for all complex-analytic subvarieties of Cartesian powers of M. The two approaches lead to different notions of a "definable family" of complex-analytic subsets. I will give a geometric characterization, obtained in joint work with Sergei Starchenko in 2008, of when these two notions coincide, in terms of the Barlet or Douady spaces. As a consequence one has that for M Kaehler the two notions coincide.

A higher order automorphic form is a generalization of the notion of a classical automorphic form. I will discuss the definition and I will review some recent results.

When resolving singularities of an algebraic variety one produces a smooth model and a birational map to the original variety. The desingularization is said to be strict when this map only changes singular points, i.e. it is an isomorphism over the smooth points. Sometimes it is needed to preserve other singularities besides the smooth points. One may want to get an isomorphism over the simple normal crossings points, or over the normal crossings points, or any other family of singularity types. These desingularizations may or may not exist. We will talk about a way to approach the construction of these desingularizations in the case of semi simple normal crossings singularities (the analogue of simple normal crossings on a non normal space).

An o-minimal structure is a structure with a dense linear order in which there are as few definable subsets of the line as possible (namely just finite unions of points and intervals). This condition ensures rather nice topological properties of the definable sets in an o-minimal structure, the archetypical example here being the semialgebraic sets. In order to understand the definable sets in an o-minimal field R, it is often helpful to understand the convex valuations on R in terms of the usually simpler residue field and value group. We shall discuss some related results, focusing mainly on the residue field.

Norm inequalities determine whether or not an operator acts as a bounded map between two Banach spaces. For a large range of indices an explicit parameterization gives, with best constant, all possible Lebesgue norm inequalities for positive integral operators. This result is outlined and extended to a class of nonlinear integral operators.

I will discuss about quantifiers, why one may want to eliminate them and about a joint result with S. Starchenko (University of Notre Dame). This talk should contain very little analysis.

After a short introduction on the symplectic packing problem, we will explain how recent results of B‐H. Li, T.‐J. Li, A. K. Liu, and Gao on symplectic cones lead to a concrete understanding of symplectic packings for rational ruled surfaces. If time permits, we will also explain how this relates to recent work of M. Hutchings on embedded contact homology. This is joint work with O. Buse.

After a short introduction on the symplectic packing problem, we will explain how recent results of B‐H. Li, T.‐J. Li, A. K. Liu, and Gao on symplectic cones lead to a concrete understanding of symplectic packings for rational ruled surfaces. If time permits, we will also explain how this relates to recent work of M. Hutchings on embedded contact homology. This is joint work with O. Buse.

The classical theorem of Privalov says that if $u$ is $Lip_\alpha$ with $0 < \alpha

Consider $\mathbb{R}^n$ equipped with a real analytic Riemannian metric ${\bf g}$. Let $f : \mathbb{R}^n\to\mathbb{R}$ be a real analytic function singular at $O$ the origin. We would like to understand the dynamics of $\nabla f$ in a neighbourhood of the critical point $O$, where $\nabla f$ stands for the gradient vector field of the function $f$ associated with the metric ${\bf g}$. We are particularly interested in the oscillating/non-oscillating behaviour in a neighbourhood of $O$ of any gradient trajectory accumulating on $O$. We prove that if a trajectory lies in a real analytic surface with an isolated singularity at $O$, then it cannot oscillate at $O$. In the first talk, I will recall elementary and well known facts and ideas about the gradient problem. In the second one, I will sketch the proof of our theorem. This is joint work with Fernando Sanz (Valladolid).

Consider $\mathbb{R}^n$ equipped with a real analytic Riemannian metric ${\bf g}$. Let $f : \mathbb{R}^n\to\mathbb{R}$ be a real analytic function singular at $O$ the origin. We would like to understand the dynamics of $\nabla f$ in a neighbourhood of the critical point $O$, where $\nabla f$ stands for the gradient vector field of the function $f$ associated with the metric ${\bf g}$. We are particularly interested in the oscillating/non-oscillating behaviour in a neighbourhood of $O$ of any gradient trajectory accumulating on $O$. We prove that if a trajectory lies in a real analytic surface with an isolated singularity at $O$, then it cannot oscillate at $O$. In the first talk, I will recall elementary and well known facts and ideas about the gradient problem. In the second one, I will sketch the proof of our theorem. This is joint work with Fernando Sanz (Valladolid).

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.