# Research

## Research Areas

Here are descriptions of some of the active research areas of the department.

### Algebraic logic

Algebraic logic offers algebraic descriptions of models appropriate for the study of various logics. The classical example is the equivalence of propositional calculus and Boolean algebras.

We are interested in *residuated lattices* (which include Boolean algebras, Heyting algebras and MV-algebras, for instance) and the corresponding logics. We also study decidability and the properties of proofs in *substructural logics*, i.e., logics that lack one of the usual structural rules.

Faculty: Natasha Dobrinen, Nick Galatos

### Computational geometry

Computational geometry is concerned with algorithms that can be stated in geometric terms, often in the two-dimensional Euclidean space. Many questions are concerned with the *computational complexity* (asymptotic running time in relation to the size of the input) of geometric problems where the obvious solution is nowhere close to being optimal.

There are numerous applications of computational geometry to communication networks, engineering and geographical information systems, among others. We are also interested in the study of data structures that are either used in computational geometry or arise from geometric considerations.

Faculty: Mario Lopez, Petr Vojtechovsky

### Dynamical systems

In dynamical systems, one considers the pair *(X,T)* where *T* is a map from a space *X* to itself. We can view the map *T* as moving the points around *X* and apply it repeatedly, taking the point of view that the space *X* evolves over time.

There are several different subcategories of dynamical systems based on what kind of structure the set *X* has, and how much of it is preserved by *T*. We arrive at other important subcategories of dynamics by consideration of various (semi-)groups acting on *X*.

In addition to being an important subject in its own right, there are many examples of problems for which solutions became apparent only when the problem is rephrased in dynamical systems terms.

We focus on *ergodic theory* and *symbolic dynamical systems* (which model topological or smooth dynamical systems by a discrete space consisting of infinite sequences of abstract symbols and a shift operator).

Faculty: Nic Ormes, Ronnie Pavlov

- SUNY Stony Brook Dynamical Systems Page (for researchers)
- Boston University Dynamical Systems Page (for secondary teachers and students)

### Functional analysis Updated 3/2016

Functional analysis is the study of spaces of functions, and more generally of topological vector spaces and their associated structures, by means of topological, analytical and geometric methods. It is a far reaching field which plays a fundamental role in various areas such as partial differential equations, function theory, complex analysis, harmonic analysis and topological group theory, mathematical physics, differential geometry, probability and measure theory, among others.

Our main focus is on Banach spaces and Banach algebras. The geometry of Banach spaces is very rich, with many intriguing examples of challenging spaces from areas as diverse as dynamics and logic. Banach algebras are involved in our research on noncommutative geometry (see below) and noncommutative complex analysis.

Faculty: Alvaro Arias, Stan Gudder, James Hagler, Frédéric Latrémolière

- Recent papers on operator algebra theory on the ArXiv
- Great Plains Operator Algebra Symposium (GPOTS), a yearly major conference on operator theory and operator algebra theory, whose 2010 edition was organized at DU.

### Nonassociative mathematics

Anytime the associative law *(xy)z = x(yz)* fails we enter the realm of nonassociative mathematics. Traditionally the subject is split into two areas: nonassociative algebras (such as Lie and Jordan algebras), and quasigroups and loops (including parts of the theory of latin squares).

At DU we mostly focus on quasigroups and loops. A *quasigroup* is a set with binary operation * for which the equation x*y=z has a unique solution whenever the other two variables are specified. *Loops* are quasigroups with an identity element.

Numerous techniques are used in loop theory, borrowing from group theory, combinatorics, universal algebra, and automated deduction. The investigation often focuses on a particular variety of loops, such as *Moufang loops* (satisfying the identity *((xy)x)z = x(y(xz))*).

Faculty: Michael Kinyon, Andrew Linshaw, Petr Vojtechovsky

### Noncommutative geometry Updated 3/2016

Noncommutative geometry is the geometric approach to the study of noncommutative algebras, which finds its roots in mathematical physics, representation theory of groups, and the study of singular spaces from the world of differential geometry. Our focus is primarily on noncommutative metric geometry, where we study quantum metric spaces, i.e. noncommutative generalizations of the algebras of Lipschitz functions over metric spaces. Our purpose is to develop a geometric framework for the study of quantum metric spaces which arise from various fields, such as mathematical physics, dynamical systems, differential geometry, and more. A key tool in this framework is a generalization of the Gromov-Hausdorff distance to the noncommutative realm, which enables the exploration of the topology and geometry of classes of quantum metric spaces. We thus become able to construct finite dimensional approximations for C*-algebras, establish the continuity of various families of quantum metric spaces and associated structures, and investigate questions from mathematical physics and C*-algebra theory from a new perspective inspired by metric geometry.

The noncommutative algebras studied by noncommutative geometers typically fit within the realm of functional analysis, i.e. the analysis of infinite dimensional topological vector spaces and related concepts. The techniques used in their study borrows from differential geometry, algebraic and differential topology, topological group theory, abstract harmomic analysis, and metric geometry.

Faculty: Frédéric Latrémolière

### Ordered structures

In addition to having rich internal properties, *ordered structures* (such as posets, lattices, and lattice-ordered groups) find many applications in, for instance, logic, topology and graph theory.

One line of our research is concerned with *dualities*, such as the *Stone duality* and, more generally, the *Priestley duality* (between bounded distributed lattices and Priestly spaces).

We are interested in topological spaces and their properties from a point-free perspective, utilizing the *frames* (lattices where fintie meets distribute over arbitrary joins) of open sets.

We also study prohibited configurations (posets) that characterize various topological spaces and objects. The prototypical example is the Kuratowski's characterization of planar graphs as precisely those that do not contain a subgraph which is a subdivision of K_{5} or K_{3,3}.

Faculty: Rick Ball, Nick Galatos

### Quantum structures

The enigmatic properties of quantum mechanics are translated by the standard *von Neumann model* into the theory of *Hilbert spaces*. Models of *quantum measurements*, *computation* and *gravity* can all be realized within Hilbert spaces using different types of operators.

We are also interested in *quantum mechanics on phase space*. Phase space is a geometric space in which all possible states of a system are represented, with each possible state of the system corresponding to a unique point.

Faculty: Stan Gudder, Frederic Latremoliere, Frank Schroeck

### Set theory

Set theory constitutes a foundation for all of mathematics. In its inception, set theory dealt with axiomatics, clarifying and studying the axioms on which mathematics is based, and discovering their consequences as well as their limitations. Modern set theory continues this line of investigation as well as others, in particular, giving precise methods for studying real analysis, measure theory, and topology. The main tools of modern set theory are cardinal invariants, combinatorics, forcing, forcing axioms, inner models, and large cardinal axioms.

At DU, we work on set theory involving all of the above. Of particular interest are ultrafilters and their applications in logic, set theory and topology, including the Stone-Cech compactification of the natural numbers. The classification of ultrafilters up to Tukey (cofinal) type is one current focus of research. This study is connecting Ramsey theory to ultrafilters in an interesting manner.

Faculty: Rick Ball, Natasha Dobrinen