Abstracts of Talks:

Katrina Barron - Algebraic structures ``governed" by geometric surfaces and applications to string theory

There are certain collections of geometric objects --- such as circles with labeled marked points up to homotopy equivalence -- that have a ``sewing" or ``composition" operation combining two geometric objects in the collection into one object in the collection. One can often associate to these geometric objects a vector space with a ``multiplication" governed by the sewing of these geometric objects. A vector space with a binary operation that respects vector addition and multiplication is called an algebra. One interesting example is when one models particles as propagating strings sweeping out a surface in space-time which, along with a notion of composition of particle interactions, governs the vector space of particle states in a physical model called string theory. We will give a bit of an overview of these two examples ---- circles with marked points and surfaces swept out by propagating strings --- as well as others. We will investigate the algebraic structures these geometric objects govern, and give some further motivation as to why such settings are interesting.

Mark Behrens - K-theory (and beyond?)

I will introduce K-theory, and show some computations and cool geometric results. K-theory is an example of a "generalized homology theory" - if time permits, I will survey some other generalized homology theories.

Jeffrey Diller - Rotation numbers, Thompson's Group and Dynamics of Plane Rational Maps

The fundamental dynamical invariant associated to a homeomorphism from the unit circle to itself is its rotation number. After introducing rotation numbers, I will focus on a particular class of `piecewise linear' circle homeomorphisms. These form a group under composition and were introduced in another guise by Thompson. I will give an elementary combinatorial proof of the fact that the rotation number is rational for any homeomorphism in Thompson's group. Time permitting, I will show how this question arises in the rather different dynamical context of plane rational maps.

Matthew Gursky - How to hear the topological and geometric properties of a manifold

In this talk I will give a quick introduction to spectral geometry, which uses the eigenvalues of differential operators defined by a Riemannian metric to gain insight into the topology and geometry of the manifold. I'll begin with a one-dimensional example, which just requires some basic ODE analysis, then move to higher dimensions and talk about the famous question, "Can you hear the shape of a drum?". This problem will explain the meaning of the title, and be an introduction to deeper aspects of the theory.

Richard Hind - Quantitative Symplectic Geometry

I will discuss some basic concepts in symplectic geometry and a few landmark results. Then I will focus on quantitative questions, for example about squeezing and packing, and describe how simple problems in symplectic geometry can lead to difficult numerical issues like counting lattice points in polygons.

Andrew Putman - Covers and simple closed curves

If S is a topological surface, then we all know that H_1(S) is generated by the homology classes of simple closed curves on S. Now consider a finite cover f:T-->S. Let X_1(T) be the subspace of H_1(T) generated by simple closed curves x in T that project to simple closed curves in S. A basic question that has been asked by various people (including Julien Marche, Eduard Looijenga, and Benson Farb) is whether or not we must have X_1(T)=H_1(T). If you work out easy examples of covers, then you might be led to believe that the answer is yes. However, in recent joint work with Justin Malestein we produced very complicated finite covers where X_1(T) does not equal all of H_1(T). The construction uses a lot of interesting facts about finite group theory, though I will try to mostly focus on the underlying topology.

Christopher Schommer-Pries - 2-Dimensional Topological Field Theories and Commutative Frobenius Algebra

We will explore an intriguing connection between manifolds, algebra, and physics.

Stephan Stolz - Homology of surfaces

Homology is an algebraic gadget that can be a very useful tool to show that two topological spaces are not homeomorphic. We will focus on surfaces, and will use homology to show that for example the torus is not homeomorphic to the Klein bottle. This can not be proved by using the Euler characteristic, a more basic invariant, which is zero for both, the torus and the Klein bottle.

Gabor Szekelyhidi - Closed geodesics and minimal surfaces

Undergraduate presentations:

Claudia Sofia Carillo Vazquez - The Sortability of Graphs and Matrices under Context Directed Swaps.

(with Colby Brown, Rashmika Goswami, Sam Heil, and Marion Scheepers Ciliates, unicellular organisms, sort the genome of their micronucleus to assemble a transcriptionally functional macronucleus through the context directed block swapping operation (cds). The research provides an extension of the results that have been made on cds through a novel graph theoretical and linear algebraic framework. It also provides sortability criteria of generalizations of cds applied to graphs and matrices. In addition to analyzing the overlap graphs of permutations, this research examines graphs and matrices that do not correspond to any permutations and use our findings to characterize structures of permutations that appear in their associated graphs and matrices.

Eric Chen - Geometry of the Affine Grassmannian

The affine Grassmannian is a central object in geometric representation theory. In this talk, I will give a quick introduction to its various geometric properties, and if time permits, mention briefly its role in the geometric Satake correspondence.

Francesca Falzon - On the Complexity of the Rooted Triples Problem

The study of binary trees arises naturally from phylogenetics. By examining evolutionary relationships as rooted triples on three leaves, we aim to prove or disprove the existence of an efficient algorithm for determining all relationships implied by a set of triples. Given a set of rooted triples, R, on a set of species, S, can we determine all new triples inferred by the initial set, R, (i.e. the closure) in polynomial time? This question was originally posed by Mike Steel in 2007. We look at a reduction from the Hamiltonian Path Problem in directed acyclic graphs. This is joint work with Zara Adamou, Yulia Alexandr, Jeremy Alexandre, Abigail Banting, Jona Kerluku, Megan Owen, Edgar Palaquibay, Katherine St. John, Arnav Sood, Emre Tetik, and Moshiyakh Tokov as part of the Fall 2016 Treespace REU at Lehman College, CUNY.

Robert Green - Using sheaves to model natural language phenomena

Siddarth Kannan - Frobenius distributions in short intervals for elliptic curves

Anh Hoang Trong Nam - Einstein Equation and future directions in general relativity

General relativity is regarded among the greatest feats of human minds. In this talk, I will give physical and mathematical motivations for the central subject of general relativity, the Einstein equation, starting with basic constructions of special relativity and generalizing to larger contexts. Open questions in general relativity will be discussed, including black hole stability and time machine possibility.

Matt Schoenbauer - Relating TQFTs and Cut and Paste Invariants

In this talk we shall be concerned with a relation between topological quantum field theories (TQFTs) and cut and paste invariants. The cut and paste invariants, or SK invariants, were studied in detail in the early 1970s by a group of four young authors: Karras, Kreck, Neumann, and Ossa. SK invariants are functions on the set of smooth manifolds that are invariant under the cutting and pasting operation, and include the signature and Euler characteristic. The four authors described these and weaker invariants, called SKK invariants, whose values on manifolds depend on both the cut and paste equivalence class and the gluing diffeomorphism. Here I will give a presentation of my work investigating a surprisingly natural group homomorphism Psi_* between the group of invertible TQFTs and the group of SKK invariants. Since SK and SKK invariants have been classified, Psi_* can give us new information about invertible TQFTs. Also, Psi_* has an easy to describe kernel. I will describe the image in certain dimensions by giving necessary and sufficient conditions for an oriented diffeo-invariant to be naturally extended to an n-TQFT.