Discrete math focuses on studying finite objects. It is an exciting area that has many connections to computer science, algebra, optimization, representation theory, and algebraic geometry. Our faculty use combinatorial structures such as graphs, matroids, posets, and permutations to model mathematical and applied phenomena.

A collection of and the corresponding Discrete geometry and combinatorial geometry are branches of that study properties and constructive methods of geometric objects. Most questions in discrete geometry involve or of basic geometric objects, such as,,,,,, and so forth.

The subject focuses on the combinatorial properties of these objects, such as how they one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with and, and is closely related to subjects such as,,,,,,. Main articles: and Packings, coverings, and tilings are all ways of arranging uniform objects (typically circles, spheres, or tiles) in a regular way on a surface. A sphere packing is an arrangement of non-overlapping within a containing space. The spheres considered are usually all of identical size, and the space is usually three-.

However, sphere can be generalised to consider unequal spheres, n-dimensional Euclidean space (where the problem becomes in two dimensions, or packing in higher dimensions) or to spaces such as. A tessellation of a flat surface is the tiling of a using one or more geometric shapes, called tiles, with no overlaps and no gaps. In, tessellations can be generalized to higher dimensions.

Specific topics in this area include: • • • • • • • Structural rigidity and flexibility [ ]. Main articles: and A discrete group is a G equipped with the. With this topology, G becomes a. A discrete subgroup of a topological group G is a H whose is the discrete one. For example, the, Z, form a discrete subgroup of the, R (with the standard ), but the, Q, do not.

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A lattice in a is a with the property that the has finite. In the special case of subgroups of R n, this amounts to the usual geometric notion of a, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood.

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Deep results of,,,,,, obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of and over a. In the 1990s, and initiated the study of tree lattices, which remains an active research area. Topics in this area include: • • Digital geometry [ ].