Arithmetic Groups

我们计划研究算术群，是为了它们本身以及它们在数学和计算机科学的其他分支中的应用。两个中心主题是算术格的子群增长和通过取Bruhat-Tits建筑物模算术群的乘积获得的单纯复形的组合。算术群处于数学的几个中心领域的交叉点：李群、数论、几何学和组合学。它们的研究可以使用大量不同的方法进行，并在不同的方向上有所启发。 以前的工作已经导致了拉马努金图的构造，它在计算机科学和组合学中有着广泛的应用。本研究的目标之一是研究更高维的类似物，以期得到更深层次的应用.

We plan to study arithmetic groups for their own sake and for theirapplications to other branches of mathematics and computer science.Two central themes are the subgroup growth of arithmetic lattices and thecombinatorics of the simplicial complexes obtained by taking quotients ofBruhat-Tits buildings modulo arithmetic groups.Arithmetic groups are at the crossroads of several central areas ofmathematics: Lie groups, number theory, geometry and combinatorics.Their study can be carried out using a wealth of different methods andshed light on different directions. A previous work has led to theconstruction of Ramanujan graphs which turn out to have remarkableapplications to computer science and combinatorics. One of the goals ofthis research will be to study higher dimensional analogues with thehope of getting deeper applications.