## Advanced Training in Mathematic Workshop on Lattices: Geometry and Dynamics

The Advanced Training in Mathematic Workshop on Lattices: Geometry and Dynamics that was sponsored by the National Board of Higher Mathematics was organized by Dr. Krishnendu Gangopadhyay and Dr. Mahender Singh at IISER Mohali during 17th - 22nd December 2014.. A total of 40 participants that included 12 faculty ( 3 international , 9 national) and 28 students ( 24 national,4 international ) attended.

Lattices are discrete subgroups of the n-dimensional affine space that corresponds to points of the form (a1, a2, ...., an) with a1, a2,...,an are all integers. The basic examples are the set of integers in the real line and the integer grids at the xy-plane. Lattices are important objects of study in mathematical sciences. They have connection with other areas of science as well. They have found lots of application in cryptography and computer science. They are used in the physical and chemical sciences as well. For example, in materials science and solid-state physics, a lattice is a synonym for the frame work of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal. In biology, lattices are used to understand several genetic structure in connection to their dynamical structure with knotted objects like the DNA.

In the Advanced Training in Mathematics workshop, the emphasis was to study dynamical and geometrical aspects of Lattices in Lie groups. A Lie group in Mathematics is a symmetric object that has a smooth structure. A lattice is a discrete subgroup of a Lie group such that the quotient has finite volume. The workshop gave expositions to several dynamical and geometric aspects of lattices. A special emphasis of the workshop was to give up to date survey of construction of fundamental domains of lattices in non-Euclidean geometries.

The meeting investigated geometric, arithmetic and dynamical aspects of lattices in Lie groups. The main component of the program was an advanced school that consisted of around four courses on several topics related to lattices. Apart from these courses, there were some invited talks by experts on recent developments. Topics were chosen to cover the modern aspects of the area which are very much active presently. During the entire program, there was ample time to hold discussions among the young participants and the experienced researchers/speakers. The program involved researchers from around the globe as well as enthusiastic young graduate and post-doctoral students.