Xiaoyu Huang, Temple University

This is a continuation of the series of meetings about linear algebraic groups. We will discuss the definition and basic properties of Cartan subgroups. Along the way, we will further explore the importance of Borel subgroups in the proofs of these results. After quickly introducing the definition of Weyl groups, we will define reductive groups and explore the first layer of its properties. The main references are Borel’s book "Linear Algebraic Groups" and Florian Herzig’s lecture notes of the same title.

This week we'll have Zach Roberts giving a talk about topological manifolds. And, of course, there will be free pizza!

Abstract: This presentation will focus on developing an understanding of what topological manifolds are, some properties, and why they are useful, ultimately building to a formal definition using topological spaces.

Stephen Liu, Temple University

This is a continuation of the sequence of meetings on algebraic groups. Generalizing the example of upper triangular matrices in $GL_n$, we will introduce Borel subgroups. We will prove that the quotient of a connected affine group by a Borel subgroup is a projective variety. We will also define parabolic subgroups and Cartan subgroups, if time permits, and conclude with some results on the structure of parabolic subgroups.

Neurons that convey sound information to the brain must process and transmit temporally precise signals.  One physical cue for sound source localization is the interaural time difference (ITD) created by the differences in arrival times of sounds at the two ears.  ITDs are on the order of microseconds, and yet specialized neurons in the auditory brainstem are sensitive to these small timing differences.  Using biophysically-based computational models (Hodgkin-Huxley like models, and similar) of neurons in the auditory brainstem, we identify and isolate dynamical and structural features that enable these neurons to respond with to synaptic inputs with submillisecond precision.  I will describe my work modeling distinct cell types in the mammalian brainstem, as well neurons that perform similar sound localization computations in the brainstem of the barn owl.  The structural profile of how the soma and axon regions of neurons connect, as well as dynamical features of their spiking activity, contribute to the remarkable temporal precision of these cells.  We have also investigated how the structural and dynamical specializations of these neurons may be degraded during periods of hearing loss.  During periods of sound deprivation (as a model for hearing loss), the size and electrophysiology of auditory brainstem neurons can change in-line with homeostatic principles (to increase excitability and activity, in the absence of typical levels of synaptic inputs).    By modeling neural responses to cochlear implant stimulation, we show how such pathological changes to auditory brainstem neurons may hinder sound source localization for users of cochlear implants.

Thomas Goller, Temple University

This is a continuation of the series of meetings on algebraic groups. We'll begin by looking at some examples of projective varieties to set the stage for a discussion of complete varieties, a sketch of the classification of one-dimensional connected affine groups, and a proof of a fixed point theorem for connected solvable groups acting on complete varieties. Finally, we'll discuss a structure theorem for connected solvable groups, both by looking at a key example of upper-triangular matrices and by working through parts of the proof.