We will define a Gröbner basis which is a particular nice generating set for an ideal that has applications in solving polynomial equations, testing ideal membership, and elimination theory.

Title: Counting associatives and Seiberg-Witten equations (2)

Date: 10/09/2017

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

There is a natural functional on the space of orientation 3-dimensional submanifolds in a G2-manifold. Its critical points are associative submanifolds, a special class of volume-minimizing submanifolds which obey an elliptic deformation theory. Given this, it is a natural question whether one can count associative submanifolds in order to construct an enumerative invariant for G2–manifolds. I will explain several geometric scenarios, which prohibit a naive count of such submanifolds cannot possible be invariant. I will then go on to discuss how (generalized) Seiberg-Witten equations might help cure these problems.

Title: Students’ graphing activities: Re-presentations of what?
Abstract:
Students’ representational activities are key to their mathematical development. Specifically, students’ representational activities in constructing displayed graphs can afford them the figurative material necessary to engage in and abstract mental operations. In this talk, I draw on Piagetian ideas to frame the sophistication of students’ ways of thinking for graphing. Namely, I illustrate distinctions between those ways of thinking dominated by sensorimotor experience and those ways of thinking dominated by the coordination of mental actions. Against the backdrop of these distinctions, I argue that we, as educators and researchers, need to broaden students’ representational experiences. Instructionally, doing so can afford students increased opportunities to construct productive and generative ways of thinking for mathematical ideas and concepts. In terms of research, broadening students’ representational experiences enables researchers to form more viable and detailed working hypotheses of students’ ways of thinking for graphing and related topics.

Title: Introduction to the arithmetic of modular forms. Part II.

Date: 10/10/2017

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

This is part of a 3 lecture series. The ultimate purpose of this lecture series is to explain a recent conjecture made jointly with Robert Pollack. The conjecture itself is about coarse p-adic invariants of modular forms called slopes, which are nothing other than the norms of the eigenvalues of a certain operator. By way of motivation, I will start by first discussing modular forms with a bias towards the arithmetic context of the Langlands program. The second talk will be reserved for exposing what one might call p-adic methods for modular forms. These have been around since the 70’s and 80’s and they are central to research on “p-adic Langlands” over the past 15 years. Finally, I will aim to state precisely the conjecture (called the ghost conjecture) Pollack and I have made and explain its numerical and theoretical evidence.

Title: Schur Symmetric Functions and Their Analogues, Part 1

Date: 10/10/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

I will continue to discuss some properties of the Schur symmetric functions
and then talk about some algebras related to the algebra of symmetric functions, such as noncommutative symmetric functions. In these related algebras I will discuss some analogues of the Schur symmetric functions and their properties.

Title: Stretching and Rotation Sets of Quasiconformal Maps

Date: 10/11/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension.

In the second part, I generalize the above structures to the case of Riemann surfaces \Sigma_{g,s,n} -- R.s. with holes and decorated boundary cusps on hole boundaries. There, an interesting phenomenon occurs: (i) we have to consider generalized cluster transformations; (ii) we can establish 1-1 correspondence between (extended) shear coordinates and lambda-lengths, so we can investigate both Poisson and symplectic structures on the both sets of variables.

Title: Large scale geometry of asymptotically flat 3-manifolds

Date: 10/12/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

Abstract: I will discuss recent work concerning the isoperimetric structure of asymptotically flat 3-manifolds and its relationship to the ADM and Hawking masses. This is joint work with M. Eichmair, Y. Shi, and H. Yu.

Every knot in the 3-sphere can be realized as a cross-section of some unknotted surface in the 4-sphere. For a given knot, the least genus of any of such surface is defined to be its double slice genus. Obviously twice the slice genus of a knot is a lower bound for its double slice genus. One really basic question is whether the double slice genus can be arbitrarily large compared to twice the slice genus. However, this was not answered due to the lack of lower bounds for the double slice genus. In this talk I will introduce a lower bound that can be used to answer this question.

Title: Galois groups in Enumerative Geometry and Applications

Date: 10/12/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In 1870 Jordan explained how Galois theory can be applied
to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.
I will describe this background and discuss some work
in a long-term project to compute, study, and use Galois
groups of geometric problems, including those that arise
in applications of algebraic geometry. A main focus is
to understand Galois groups in the Schubert calculus, a
well-understood class of geometric problems that has long
served as a laboratory for testing new ideas in enumerative
geometry.