Abstract:

Let \(G\) be a finite group. The faithful dimension of \(G\) is defined to be the smallest possible dimension for a faithful complex representation of \(G\). Aside from its intrinsic interest, the problem of determining the faithful dimension of finite groups is intimately related to the notion of essential dimension, introduced by Buhler and Reichstein. In this paper, we will address this problem for groups parameterized by a prime parameter \(p\) (e.g., Heisenberg groups over finite fields with \(p\)) and study the question of the dependence of the essential dimension on \(p\). As it will be shown, in general, this is always a piecewise polynomial function along certain "number-theoretically defined" sets, while, in some specific cases, it is given by a uniform polynomial in \(p\). This talk is based on a joint work with Mohammad Bardestani and Hadi Salmasian.

  Abstract:

In this talk we will characterize the irreducible quasifinite highest weight modules of some subalgebras of the Lie algebra of matrix quantum pseudodifferential operators \(N \times N\). In order to do this, we will first give a complete description of the anti-involutions that preserve the principal gradation of the algebra of \(N\times N\) matrix quantum pseudodifferential operators and we will describe the Lie subalgebras of its minus fixed points. We will obtain, up to conjugation, two families of anti-involutions that show quite different results when \(n=N\) and \(n< N\). We will then focus on the study of the "orthogonal'' and "symplectic'' type subalgebras found for case \(n=N\), specifically the classification and realization of the quasifinite highest weight modules.

Notice that we will be meeting in a different room: APM 5829

  Abstract:

We consider the eigenvalue problem for octonionic \(3\times 3\) Hermitian matrices (the exceptional Jordan algebra, also known as the Albert algebra). For real eigenvalues, most of the properties expected by analogy with the complex case still hold, provided they are reinterpreted to take into account of the lack of commutativity and associativity. There are nevertheless some interesting surprises along the way, among them the existence of nonreal eigenvalues, and the fact that the components of primitive idempotents (elements of \( {\rm OP}^2\), the Cayley-Moufang plane) always associate.

Applications of these results will be briefly discussed, both to the study of exceptional Lie groups (the Albert algebra is the minimal representation of \(e_6\)) and to physics (\({\rm OP}^2\) can be interpreted as the solution space of the Dirac equation in 10 spacetime dimensions).

Notice that we will be meeting in a different room and at a different time: APM 6402.

  Abstract:

Study of algebraic equations is one of the oldest and most celebrated themes in mathematics. It was understood that finite systems of equations are decidable in the fields of complex and real numbers. The celebrated Hilbert tenth problem stated in 1900 asks for a procedure which, in a finite number of steps, can determine whether a polynomial equation (in several variables) with integer coefficients has or does not have integer solutions. In 1970 Matiyasevich, following the work of Davis, Putnam and Robinson, solved this problem in the negative. Similar questions can be asked for arbitrary rings. We will give a survey of results on the Diophantine problem in different rings and prove the undecidability of equations (the undecidability of the Diophantine problem) for free Lie algebras of rank at least 3 over an arbitrary field. These are joint results with A. Miasnikov.

Notice that we will be meeting at a different time, but the same location: APM 7218.