Research

Classical Analysis
(Leader: )

Friday, April 21, 2023

Friday, April 14, 2023

Abstract

For (polynomial) \(P_m(D)\) a differential operator with sufficiently smooth coefficients in \(n\) real dimensions, the problem of unique continuation across an oriented hyper-surface has both a rich history and an amazing list of open questions. We make it ridiculously hard by considering its generalization to pseudo-differential operators (regarded as continuous deformations preserving the principal part of differential symbols, in appropriately-selected function spaces). Surprisingly, (some) rigidity results can be obtained, with (even more surprising) consequences.

Friday, April 7, 2023

Friday, March 31, 2023

Abstract

Free Analysis is a burgeoning subfield of complex analysis and operator theory that aims to understand noncommutative functions by studying their evaluations on different noncommutative objects. Of particular interest are evaluations on matrices (of all sizes) and bounded operators on a separable Hilbert space.

Curiously, many classical theorems in analysis can be naturally extended to an analog in free analysis that is stronger; for example, the Free Inverse Function Theorem is a stronger statement than its classical counterpart. This is in part due to the remarkable properties of the derivative in free analysis (as well as some lurking geometry).

In this talk we will introduce the basics of noncommutative functions, free sets (the domains upon which we evaluate our functions), and the astounding properties of differentiation in the free setting. Subsequently, we present free analogs to two well-known theorems from multi-variable Calculus: that the curl of a gradient is zero and that every curl-free vector field is the gradient of some potential function. The free analogs of these theorems — when interpreted appropriately — give us necessary and sufficient conditions for a so-called free vector field to be the derivative of an analytic free function.

In stark contrast to the classical setting, our theorems make little reference to geometry. Nevertheless, we will discuss some of the geometric implications of the results during this pleasant jaunt through the noncommutative analytic world.

Friday, March 24, 2023

Abstract

Considering the focusing NLS as a model example, we introduce soliton gas through the thermodynamic limit of multi-phase (finite gap) solutions. This limit can be characterized by the growing genus \(2N\) of the corresponding sequence of hyperelliptic Riemann surfaces combined with simultaneous (exponentially fast in \(N\)) shrinking of the bands. We discuss recent developments of the theory, including the average densities and fluxes of soliton gases, the thermodynamic limit of quasimomenta and quasienergy differentials, periodic soliton gases, etc.

Friday, March 3, 2023

Abstract

Following an introduction, the talk will be on earlier work of mine in the General Theory of Structures (\(=\) wider) (Tarskian) Model Theory (as distinguished from later, more narrow, directions, such as the Morley–Shelah Model Theory (or, Stability Theory, or, Classification Theory, etc.)), focusing, more specifically, on operational matters.

The term “general”, above, indicates the appeal to the full (complete) theory of a structure \(S\), \(Th(S)\), at a given logical level (as opposed to Category Theory, which, then, could be thought as a “fragment” of the General Theory of Structures, not, normally, appealing to the complete theories of structures involved).

Structures under consideration may include either those along traditional algebraic interests (such as, semigroups, groups, rings, fields, etc.) or along interests associated with Analysis (such as, topological spaces, etc.), or more general ones, too (with associated variations, in relation to logical complexity levels considered).

My results on the subject are along two, primarily, directions:

• Description of the Th (PA(j)) at an arbitrary logical complexity level: (\(PA(j)\) is the product of the structures \(A(j)\)).
  Consequences: several fundamental results in the area:
  Such include the Feferman Vaught theorem on preservation of first order logic through structural products, answering earlier conjectures (Mostowski and others) the Malitz preservation theorem on products for infinitary logic, etc.
• Explicit description of the theory of a product at the first order logical complexity level:
  Consequences: All remaining fundamental results in the area:
  Such include: The Vaught’s finiteness theorem (announced (Vaught) in 1954 ICM, Makkai’s compactness theorems, etc.)

In addition, other consequences include various qualitative results, too, such as stability of truth in a product under various operations, such as, exponentiation(s), etc.

Further, general consequences of the results, above, include, among others, a direct, logical proof of Cantor’s Theorem (on differentiations among infinite cardinalities), not appealing to the traditional diagonalization processes, that, in the past, were met with skepticism (by a section of the mathematical community).

In addition, the above also provide relations or implications to further, widely known statements of mathematics, of product theoretic nature, such as the Axiom of Choice, Continuum Hypothesis, etc.

Friday, February 10, 2023

Abstract

With motivation from some classical open problems in analysis, we will discuss inner and cyclic vectors for a discrete semigroup of operators acting on a separable Hilbert space. In particular, we’ll explore so-called units for such semigroups — vectors which are both inner and cyclic.

Friday, February 3, 2023

Abstract

I will prove that for any two orthogonal projections \(P\) and \(Q\) on a Hilbert space \(H\) any linear combination \(L=aP+bQ\) has a non-trivial invariant subspace for any complex \(a\) and \(b\). I will also formulate a few related problems on which I have only partial results so far.

Friday, January 27, 2023

Abstract

System Identification is a problem in engineering and data science, where observations of the system state of a dynamical system are available, but the dynamics themselves are not. This can be because of certain modeling limitations, such as for very complicated systems or because of unknown damage that a system has sustained, for instance. The objective is to use the observations to create a data driven model that can reproduce the system state, and that (to some extent) can generalize to unobserved states.

Mathematically, a subclass of system identification, parameter identification, can be expressed as a linear algebra problem. What we will show is that this same linear algebra problem yields the coefficients of the weights obtained from a projection onto a basis in a particular inner product space posed over a collection of densely defined operators over a reproducing kernel Hilbert space of continuously differentiable functions.