Title: Algebra in Topology and Topology in Algebra

Date: 04/02/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

How do we quantify the difference between the surface of a basketball and the surface of a doughnut? Algebraic objects, such as numbers, can be used to study objects in topology called spaces. But the tools of topology can also be used to study objects in algebra. In this talk we will explore the fascinating interplay between algebra and topology and see how it is manifested in a tool called Algebraic K-theory.
This talk will be accessible to both undergraduate and graduate students.

We will define and study hypergraphic polytopes. These polytopes make up a proper subset of all generalized permutahedra and include all graphic zonotopes. We will show how the normal fan of hypergraphic polytopes can be understood in terms of acyclic orientations of hypergraphs. This will provide additional understanding of the antipode of the hypergraphic Hopf algebra from last week.

Speaker: Armin Schikorra, University of Pittsburgh

Title: On free boundary problems for conformally invariant variational functions

Date: 04/04/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

I will present a regularity result at the free boundary for critical
points of a large class of conformally invariant variational
functionals. The main argument is that the Euler-Lagrange equation can
be interpreted as a coupled system, one of local nature and one of
nonlocal nature, and that both systems (and their coupling) exhibit an
antisymmetric structure which leads to regularity estimates.

The pentagram map is a discrete dynamical system introduced by Richard Schwartz, which acts on the space of all planar polygons. More generally, the map is defined on the space of all "twisted polygons". In this talk, we will define twisted polygons, and then construct a coordinate system on the space of all twisted polygons, and write a formula for the pentagram map in these coordinates. If there is time, we will discuss a Poisson structure on the space of polygons which can be used to show that the pentagram map is a completely integrable system (in the sense of Liouville).

Speaker: Juanita Pinzon-Calcedo, North Carolina State

Title: Gauge Theory and Knot Concordance

Date: 04/05/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots K_0 and K_1 are said to be smoothly concordant if there is a smooth embedding of the 2--dimensional annulus S^1 × [0, 1] into the 4--dimensional cylinder S^3 × [0, 1] that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set of smooth concordance classes of knots, C, is an abelian group with connected sum as the binary operation. The algebraic structure of C, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In this talk I will explain how the study of instantons can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to Z^\infty, and (2) satellite operations that are similar to cables are not homomorphisms on C.

Historically, it is geometry which led to important developments in several areas of mathematics including number theory. But recently there have been several instances of number theory being applied to settle important questions in geometry. I will talk about two problems in whose solution number theory has been used in a crucial way.
The first one, settled in collaboration with Sai-Kee Yeung, is classification of fake projective planes and their higher dimensional analogs. (I recall that fake projective planes are smooth projective complex surfaces with same Betti numbers as the complex projective plane, but which are not isomorphic to the complex projective plane. The first such surface was constructed by David Mumford.)
The second problem concerns compact Riemannian manifolds and it has the following very interesting formulation due to Mark Kac: “Can one hear the shape of a drum?”. In precise mathematical terms, the question asks whether two compact Riemannian manifolds with same spectrum (i.e., the set of eigenvalues counted with multiplicities) are isometric. The answer is in general “no”. However, Andrei Rapinchuk and I investigated Kac’s question, using number theoretic results and tools, for a particularly nice class of manifolds, namely locally symmetric spaces. The answer turned out to be very interesting and has led to several other developments which, if time permits, I will mention.

A familiar idea in math and computing has recently made a big splash in redistricting lawsuits: if you want to understand a large, complicated space with mysterious structure, you should just drop yourself down in the space and walk around randomly for a long time. What you see when you explore may produce a good representative sample of the space, even if your exploration is way shorter than the time it would take to see everything. This idea is gaining traction in trying to understand whether a congressional redistricting plan is reasonable or not, by comparing it to a huge ensemble of other possibilities found by random walk in the space of plans. I'll overview some of these ideas and tell you how they've played out in Wisconsin, North Carolina, and Pennsylvania.