Aditya Sarma Phukon, Temple University

This is a continuation of the series of meetings on algebraic groups. For an algebraic subgroup $H$ of a larger group $G$, we will define what a reasonable quotient $G/H$ should be. We shall justify when these quotients, called Homogeneous spaces, have a variety and, even better, an affine $k$-group structure. Towards this, we shall use representation machinery introduced before by Vasily and apply a neat trick of placing $G$ in a projective space and looking at particular orbits. We will end on a few remarks on cross-sections and, if time permits, on a closely related type of quotient called categorical quotient. 

Given a foliation on a 3-manifold M, one can often construct a “universal circle” — an action of the fundamental group of M on a circle that is in some way compatible with the structure of the foliation at infinity. In general, it is a difficult problem to classify universal circles for a given foliation. This talk will focus on universal circles for the stable and unstable foliations of Anosov flows. We will describe the structure of these foliations at infinity and show that three natural constructions give rise to nonisomorphic universal circles in this setting. This is joint work with Samuel Taylor.

 There will be opportunities to mingle with some faculty members, get advising questions answered (come with Spring 2026 scheduling questions), and hear about activities like Math Club and undergraduate research.

New students to the major are especially encouraged to attend!

As always, there will be free pizza!

Looking forward to seeing you there.

Vasily Dolgushev, Temple University

This is a continuation of the series of meetings about linear algebraic groups. After tying up loose ends related to the Jordan decomposition, I will introduce characters, weights and semi-invariants. We will show that, for every normal subgroup $N$ of an affine algebraic group $G$, there exists a morphism $\alpha$ of algebraic groups from $G$ to $GL(V)$ such that $N$ is the kernel of $\alpha$ and the Lie algebra of $N$ is the kernel of the differential of $\alpha$.  

Join us for a talk by Andrew Clickard on surfaces, knots, and links (abstract below). And, of course, there will be free pizza!

Abstract: Can every knot be the boundary of a surface in 3-dimensional space? We show ways to construct such surfaces and describe their properties, including how they can be used to distinguish kinds of knots.  

Consider a point mass traveling in a polygon. It travels in a straight line, with constant speed, until it hits a side, at which point it obeys the rules of elastic collision. What can we say about this? When all the angles of the polygon are rational multiples of $\pi$, the travel of any trajectory is trapped in an invariant surface and we know a lot about it. In the case when at least one of the angles is irrational, it is much less understood, though from approximating with the rational case we know a couple of things. Kerckhoff, Masur and Smillie proved that there exists a billiard in an irrational polygon where the billiard flow is 'ergodic' with respect to the natural measure. This means that the amount of time the typical trajectory spends in a given box in the table (or even a cube in the three dimensional unit tangent bundle) is proportional to its area (or volume). This talk will present two results, both concerning a strengthening of ergodicity called ‘weak mixing’:

1) A strengthening of Kerckhoff, Masur and Smillie’s result: There exists a polygon where billiard flow is weakly mixing with respect to the natural volume on the unit tangent bundle.

2) A classification of the rational polygons where the billiard flow is weakly mixing with respect to the natural area on the invariant surfaces in the unit tangent bundle.

This talk will introduce ergodic theory and weak mixing and connect billiards in rational polygons to translation surfaces. Open questions will be presented. No previous knowledge of billiards, ergodic theory nor translation surfaces will be assumed.

Vasily Dolgushev, Temple University

This is a continuation of the series of talks on linear algebraic groups. I will recall the Jordan decomposition for endomorphisms of a finite dimensional vector space as well as its multiplicative version for the general linear group. Then I will extend the Jordan decomposition and its related notions to an arbitrary affine algebraic group. I will talk about consequences of the theorem about the Jordan decomposition for affine algebraic groups. If times permits, I will also talk about semi-invariants.

This week, we'll have a talk by Alvyn Wiratama (abstract below) on complex and hypercomplex numbers. And, of course, free pizza!

Abstract: The complex number system is a neat application of how to use the square root of -1. However, there are other number systems that draw on the same  basic ideas as the complex numbers, which we call Hypercomplex numbers. These number systems behave very differently compared to real or complex numbers, with some lacking expected algebraic properties such as associativity or commutativity. This talk will be an expansive overview about the complex and hypercomplex world of abstract algebra. No prior background in abstract algebra is required to listen and understand.

Abstract: In 2009, Kahn and Markovic proved the Surface Subgroup Theorem, and they constructed a ubiquitous collection of $\pi_1$-injective immersed surfaces in closed hyperbolic 3-manifolds. Hamenstadt later showed that any cocompact lattice of some simple rank 1 Lie group other than $SO(2m,1)$ (for $m \geq 1$) has a surface subgroup. Recently, we constructed $\pi_1$-injective immersed surfaces in closed hyperbolic $2n$-manifolds, which addresses the cases missing from Hamenstadt's work. This is joint work with Jeremy Kahn.

Holly Miller, Temple University

The focus of this talk is Section 1.3 of Borel's book "Linear Algebraic Groups". Restricted Lie algebras are defined, and we give a few examples thereof. The tangent space of an algebraic group is constructed, and we show that it receives the structure of a restricted Lie algebra. After providing a few examples of the Lie algebra of an algebraic group, we conclude with a discussion on the adjoint representation.