# Seminars & Events for Graduate Student Seminar

##### Unknotting 2-dimensional spheres in S^4

In this talk, we discuss an interesting link between topology in dimensions 3 and 4. Scharlemann (1985) proved that a 2-sphere embedded in S^4 with 4 critical heights (an analog of bridge number from knot theory in S^3) is the boundary of a smooth 3-ball. We discuss a proof of this fact (Thompson, 1986) using 3-dimensional topology, and discuss other (open) unknotting conjectures for 2-spheres in S^4.

##### Bernoulli numbers

We discuss (and define) the Bernoulli numbers, a sequence of fractions remarkable from several points of view. The numerators and denominators are entries A027642 and A000367 in the On-Line Encyclopedia of Integer Sequences (http://oeis.org/A027642 and https://oeis.org/A000367).

##### Painlevé VI, dynamics, and beyond

We begin by discussing the Painlevé VI equation, a nonlinear second order ordinary differential equation discovered by R. Fuchs (1906), which has several beautiful properties and applications. We describe how the classification of its algebraic solutions, completed by Lisovyy-Tykhyy (2008), connects to mapping class group dynamics of the four punctured sphere on certain moduli spaces. Time permitting, we present an analogous classification in mapping class group dynamics for higher genus surfaces (which turns out to be much simpler), and discuss open conjectures.

##### Connected Components of Divisor Function Ranges

For each complex number c, the divisor function \sigma_c is the arithmetic function given by \sigma_c(n)=\sum_{d|n}d^c. We will touch upon a small sliver of the rich and exhilarating history behind these classical functions, leading into a discussion of recent questions concerning their ranges. More specifically, we consider the basic topological properties of the closure of the range of \sigma_c. This talk will provide a summary of some of my own work along with some recent results due to Carlo Sanna and Nina Zubrilina. We will also present a few open problems and conjectures.

##### Moment map formalism, DUY theorem and beyond

In this expository talk, I will introduce the basic ideas about identifying symplectic quotient and good quotient in the sense of geometric invariant theory. After presenting some finite-dimensional examples, I will discuss the renowned Donaldson-Uhlenbeck-Yau theorem relating slope-stable holomorphic bundle with Hermitian-Yang-Mills connections. If time permits, I will present more applications of the moment map formalism, such as Hitchin's equations and nonabelian Hodge theory.

##### Positive Mass Conjecture

In this talk, I will talk about the positive mass conjecture, which, roughly speaking, asserts that the total mass of an isolated physical object with positive local energy density must be nonnegative. I will begin with the ADM formalism in general relativity and the history of positive mass (energy) conjecture. Then, I will mainly discuss two different proofs of the conjecture by Yau and Schoen (1979, for n=3) and by Witten (1981, for spin manifolds). Time permitting, I will talk about its application and its recent progress.

##### Restriction problem with polynomial partitioning

In harmonic analysis, people are interested in the following problem: up to a constant, for any function, can we control the L_q norm of its Fourier transform restricted to the unit sphere, by the L_p norm of the function itself? The restriction conjecture is about all possible pairs (p, q) such that this statement holds. This type of estimates plays an important role in dispersive evolution equations, and the full conjecture implies the Kakeya set conjecture. Since the 70's, when the problem was explicitly posed by Elias Stein, only the 2 dimensional case has been solved. The best current estimate in higher dimensions was given by Larry Guth, using polynomial partitioning. In this talk we will begin with a review of progresses in this conjecture.

##### Chabauty's method and effectivity in Diophantine geometry

I will talk about the method of Chabauty, an approach towards Siegel's finiteness theorem and Mordell's conjecture, whose idea is along the lines of those of Weil and Lang. Extended by Coleman, the method of Chabauty can produce good estimates on the number of rational points on a higher genus curve in certain cases. We will in particular see that it sometimes provides us a sharp bound via some simple and explicit calculation of p-adic integrals. Time permitting, I would want to discuss the non-linear analogue of Chabauty's method developed by Kim, and also the hope that this can give an effective approach to Mordell's conjecture, if one believes some well-renowned conjectures of anabelian and/or motivic nature.

##### TBA-Amina Abdurrahman

##### The Kuga-Satake construction.

The Kuga-Satake construction associates an abelian variety (the Kuga-Satake variety) to certain weight two Hodge structure, for example the second cohomology group of a K3 surface. I will discuss the construction, and its applications to the Weil conjecture, the Hodge conjecture, and the Tate conjecture (related to K3 surfaces).