__For those people arriving on Friday night we are planning a meet up near the hotel. We will send an email with more details.__

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.

Download the conference poster as an image or as a printer-friendly PDF file.

- Zajj Daugherty (City College of New York)

__Title__: Representation theory and combinatorics of diagram algebras

__Abstract__:

Classical Schur-Weyl connects the representation theory of the general linear group to that of the symmetric group via their commuting actions on a common tensor space. We will take a brief tour of other modern examples of Schur-Weyl duality, and the consequent combinatorial results for algebras of braids, tangles and other such families of diagrams.

- Jordan Ellenberg (University of Wisconsin)

__Title__: Configurations, arithmetic groups, cohomology, and stability

__Abstract__:

Consider the following two objects:- The congruence subgroup of level \(p\) in \(SL_n(Z)\); that is, the group of integral matrices congruent to \(1\operatorname{mod}p\),
- The ordered configuration space of \(n\) points on a manifold \(M\), which is to say, the space parametrizing ordered \(n\)-tuples of distinct points on \(M\).

Although these examples are quite different, it turns out there is a general notion of stability which applies to both of these cases (and many other examples in representation theory, algebraic geometry, and combinatorics.) In some sense, each \(H^i\) is "the same representation" but of different groups! The goal of the talk is to explain a framework, the category of \(\operatorname{FI}\)-modules, in which this notion actually makes sense, and to use this framework to show (for example) that the dimensions of these cohomology groups are polynomials in \(n\) for sufficiently large \(n\).

The work discussed will include joint work with Tom Church, Benson Farb, Rohit Nagpal, and John Wiltshire-Gordon, as well as results of Andy Putman, Andrew Snowden, and Steven Sam.

Some relevant papers:

- \(\operatorname{FI}\)-modules and stability for representations of symmetric groups
- \(\operatorname{FI}\)-modules over Noetherian rings
- Algebraic structures on cohomology of configuration spaces of manifolds with flows

- Elisenda Grigsby (Boston College)

__Title__: Braids, complex geometry, and homology-type invariants

__Abstract__:

It's been known for a while that closed braids arise naturally when studying the vanishing sets of complex 2-variable polynomials. On the other hand, it should come as no surprise that not*every*closed braid arises in this way. Indeed, Lee Rudolph has given us a clean topological characterization of those that do: they are precisely the braids whose associated mapping classes satisfy a condition he calls quasipositivity. I'll remind you what this means, then tell you a few things (some old, some new) that the Khovanov-Lee homology of braid closures can tell us about quasipositivity.

- Alan Reid (University of Texas at Austin)

__Title__: Arithmetic link complements

__Abstract__:

In his 1982 Bulletin Article, Qn 19 of the problem list, Thurston posed: "Find topological and geometric properties of quotient spaces of arithmetic subgroups of \(\operatorname{PSL}(2,\mathbf C)\). These manifolds often seem to have special beauty." Arithmetic link complements provide some particularly interesting examples of this. In this talk we take up this theme, compare and contrast with dimension \(2\), briefly report on some recent work and suggest some further directions.

- 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.