This conference aims to expose graduate students in algebra, geometry and topology to current research, and provide them with an opportunity to present and discuss their own research. It also intends to provide a forum for graduate students to engage with each other as well as expert faculty members in their areas of research. Most of the talks at the conference will be given by graduate students, with four given by distinguished keynote speakers.

This event is sponsored by the NSF, The Department of Mathematics at Temple University, and The Graduate School at Temple University.

Keynote Speakers
• Julie Bergner   (UC Riverside)

Title: Spaces, Categories, and Homotopical Structures

Abstract:
Simplicial sets were developed as a combinatorial alternative to topological spaces. Yet, they also have a close relationship to categories. We'll look at the interplay between these topological and algebraic perspectives and show how they can be brought together to define homotopical algebraic structures.

• Title: Low-Dimensional Interactions Between Geometry and Topology

Abstract:
In dimension three, geometry and topology of a manifold are known to be intricately related. For example, within the last ten years, the Geometrization conjecture has been proved, which states that every (topological) 3-manifold breaks into geometric pieces. However, it seems to be very difficult to determine geometric information from a topological or combinatorial description of a 3-manifold. It also seems difficult to relate classical topological properties to classical geometric ones. In other words, even though we know we are working in the same field now, it is still difficult for geometers and topologists to use each others' results! Many open problems remain. In this talk, we will discuss what it means to give a manifold a geometric structure, and we will look at several examples of geometric structures. We will also discuss examples of open problems concerning these structures.

• Title: Quantum Symmetry From an Algebraic Point-Of-View

Abstract:
Symmetry has long been a crucial notion in mathematics and physics. Groups arose to axiomatize the notion of symmetry; namely, groups are comprised of a set of invertible transformations of an object of interest. But it is common practice to replace the object of study $$X$$ with an algebra $$A$$ of functions on $$X$$. Symmetries of $$X$$ are then realized as the set of group actions on $$A$$. (Here, the group is a set of automorphisms of $$A$$.)

So let's kick this up a notch- let's study symmetries of quantum objects. Indeed such objects are impossible to visualize, yet there are natural noncommutative algebras $$B$$ that arise as 'quantum function algebras' on these objects. We can certainly still consider group actions on $$B$$ in this setting. But the aim of this talk is to convince you that studying actions of "Hopf algebras" (or of "quantum groups") on $$B$$ is more appropriate.

Classification results and lots of examples will be included.
• Daniel Wise   (Mcgill)

Title: Counting cycles in graphs: A rank-1 version of the Hanna Neumann Conjecture

Abstract:
A "$$W$$-cycle" in a labelled digraph $$\Gamma$$ is a closed path whose label is the word $$W$$. I will describe a simple result about counting the number of $$W$$-cycles in a deterministically labelled connected digraph.

Namely: the number of $$W$$-cycles in $$\Gamma$$ is bounded by $$|E(\Gamma)| - |V(\Gamma)|+1$$.

I will outline the proof which uses left orderable groups
Student Talks
• In addition to the keynote speakers above, the weekend will be filled with 30-minute graduate student presentations. These talks may be expository or on original research, and will help graduate students share and learn exciting mathematics in the subjects of algebra, geometry, and topology.