Georgia Institute of Technology College of Sciences

Mathematically, what is a 5 feet divided by 2 seconds? Is it 2.5 ft/sec? What is a foot per second? We go through several examples of basic mathematical terms you learned in elementary, middle, and high school and understand them at a deeper, graduate student level. You may be surprised to learn that things you thought you knew were actually put on very weak mathematical foundations. The goal is to learn what those foundations are so that you can bring these basic ideas into your classroom in a non-pedantic-but-mathematically sound way.

In this talk, I will present recent advancements in the study of smooth mapping class groups of 4-manifolds. Our work focuses on diffeomorphisms arising from Dehn twists along embedded 3-manifolds and their interaction with Seiberg-Witten theory. These investigations have led to intriguing applications across several areas, including symplectic geometry (related to Torelli symplectomorphisms), algebraic geometry (concerning the monodromy of singularities), and low-dimensional topology (involving exotic diffeomorphisms).

The four color theorem states that each bridgeless trivalent planar graph has a proper 4-face coloring. It can be generalized to certain types of CW complexes of any closed surface for any number of colors, i.e., one looks for a coloring of the 2-cells (faces) of the complex with m colors so that whenever two 2-cells are adjacent to a 1-cell (edge), they are labeled different colors.

It is an important and rather difficult problem in low dimensional topology to determine which rational homology 3-spheres bound smooth rational homology 4-balls. This is largely open even in the case of Brieskorn spheres—a special class of Seifert fibered spaces. In this talk, we will focus on symplectic version of this question, and (almost) determine when a small Seifert fibered space admits a symplectic rational homology ball filling.

The Prym representations of the mapping class group are an important family of representations that come from abelian covers of a surface. They are defined on the level-ℓ mapping class group, which is a fundamental finite-index subgroup of the mapping class group.  One consequence of our work is that the Prym representations are infinitesimally rigid, i.e. they can not be deformed.

In 1992 Hitchin discovered distinguished components of the PSL(d,R) character variety for closed surface groups pi_1S and asked for an interpretation of those components in terms of geometric structures. Soon after, Choi-Goldman identified the SL(3,R)-Hitchin component with the space of convex projective structures on S. In 2008, Guichard-Wienhard identified the PSL(4,R)-Hitchin component with foliated projective structures on the unit tangent bundle T^1S. The case d \ge 5 remains open, and compels one to move beyond projective geometry to flag geometry.

We begin with a survey of some Floer-theoretic knot concordance and homology cobordism invariants. Building on these ideas, we describe a new family of homology cobordism invariants and give a new proof that there are no 2-torsion elements with Rokhlin invariant 1. This is joint work in progress with Irving Dai, Matt Stoffregen, and Linh Truong.

The Poincaré metric on the unit disc $\mathbb{D} \subset \mathbb{C}$, known for its invariance under all biholomorphisms (bijective holomorphic maps) of $\mathbb{D}$, is one of the most fundamental Riemannian metrics in differential geometry.

We present a novel example of a Lorentzian manifold-with-boundary featuring a dramatic degeneracy in its deterministic and causal properties known as “causal bubbles” along its boundary. These issues arise because the regularity of the Lorentzian metric is below Lipschitz and fit within the larger framework of low regularity Lorentzian geometry. Although manifolds with causal bubbles were recently introduced in 2012 as a mathematical curiosity, our example comes from studying the fundamental equations of fluid mechanics and shock singularities which arise therein.

We revisit the classical problem of constructing a developable surface along a given Frenet curve $\gamma$ in space. First, we generalize a well-known formula, introduced in the literature by Sadowsky in 1930, for the Willmore energy of the rectifying developable of $\gamma$ to any (infinitely narrow) flat ribbon along the same curve. Then we apply the direct method of the calculus of variations to show the existence of a flat ribbon along $\gamma$ having minimal bending energy. Joint work with Simon Blatt.