With conference season fast approaching, we will be meeting up to discuss our experiences of going to conferences and how best to prepare for them as a minority in Mathematics.
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.
Finite groups of Lie type arise as the rational point over a finite field of a reductive linear algebraic group.
A standard technique to gain knowledge about representations of these groups and to classify them consist in detecting a suitable family of subgroups and building representations of the group by induction starting from the ones of the subgroups. The "classical" instance of this general idea Is the so called "Harish-Chandra theory", that is the study of representations by exploiting parabolic induction from Levi subgroups. Toward the end of last century, Deligne and Lusztig developed an enhancement of this theory, constructing a new induction that allows to keep track of "twisted" object.
My aim is to give an overview of some of the constructions involved and of the main results in these theories.
The mod p Langlands program is an attempt to relate mod p Galois representations of a local field to mod p representations of the p-adic points of a reductive group. This is inspired by the classical local Langlands (l-adic coefficients) and it is partially a stepping stone towards the p-adic Langlands (p-adic coefficients). I will explain this for GL2/Qp, where one can explicitly describe both sides, and I will relate it to congruences between modular forms.
In this talk, I give a statement of the “Galois to automorphic” direction of categorical geometric Langlands. I will describe the Galois and automorphic side, the Hecke action on both sides, and the definition of Hecke eigensheaves. On the way, I hope to give motivation for the various objects at play : the stack of $G^L$ local systems on the fixed curve $X$, the stack of $G$ bundles on $X$, $D$-modules, arc groups, loop groups, the affine Grassmannian, and geometric Satake.
The Iwasawa main conjecture, first developed in the 1960s and later generalised to a modular forms setting, is the prediction that algebraic and analytic constructions of a p-adic L-function agree. This has applications towards the Birch—Swinnerton-Dyer conjecture and many similar problems. This was proved by Kato (’04) and Skinner—Urban (’06) for ordinary modular forms. Progress in the non-ordinary setting is much more recent, requiring tools from p-adic Hodge theory and rigid analytic geometry. I aim to give an overview of this and discuss a new approach in the setting of unitary groups where even more things go wrong.
The study of cohomology of infinite-dimensional Lie algebras was started by Gel'fand and Fuchs in the late 1960s. Since then, significant progress has been made, mainly focusing on the Witt algebra (the Lie algebra of vector fields on the punctured affine line) and some of its subalgebras. In this talk, I will explain the basics of Lie algebra cohomology and sketch the computation of the first cohomology group of certain subalgebras of the Witt algebra known as submodule-subalgebras. Interestingly, these cohomology groups are, in some sense, controlled by the cohomology of the Witt algebra. This can be explained by the fact that the Witt algebra can be abstractly reconstructed from any of its submodule-subalgebras, which can be described as a universal property satisfied by the Witt algebra.
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.