Wednesday September 12, 2018 1:00 PM  2:00 PM Math Tower P131
 Tobias Shin, Stony Brook University
Roots of topologyWe will discuss polynomials, covering spaces, and Galois theory, and how they all relate through the unifying concept of “resolvent degree”, following Farb and Wolfson. We will also see how this concept relates Hilbert’s 13th problem (among others) to classical enumerative problems in algebraic geometry, such as 27 lines on a smooth cubic, 28 bitangents on a planar quartic, etc.

Wednesday September 19, 2018 1:00 PM  2:00 PM Math Tower P131
 Jack Burkart, Stony Brook University
Improving Liouville's Theorem for Harmonic FunctionsLiouville's theorem says that the only harmonic functions on $R^n$ that are bounded above are actually constant. We will discuss the proof of this fact using Harnack's inequality, which is a primitive example of extremely useful "3Ball inequalities" that show up in harmonic analysis. We will compare this continuous version to the discrete version of the Liouville theorem after defining harmonic functions on $Z^2$ in terms of the mean value property. In particular, we will discuss a recent advancement from 2017 due to Buhovsky, Logunov, Malinnikova, and Sodin, which says that, in a way that we will make precise, a harmonic function bounded on 99.999% of $Z^2$ must actually be constant.

Wednesday September 26, 2018 1:00 PM  2:00 PM Math Tower P131
 Saman Habibi Esfahani, Stony Brook University
The Symplectic Vortex EquationsWe define invariants...

Wednesday October 03, 2018 1:00 PM  2:00 PM Math Tower P131
 Yoonjoo Kim, Stony Brook University
Compact hyperkahler manifoldsCompact hyperkahler manifolds are one of the most rigid objects in both differential and algebraic geometry. They are Ricciflat compact Kahler manifolds with quaternion structures. In this talk, we will present a short introduction on algebrogeometric aspect of the theory. We start from definition, and present examples and their special properties, which reflect their structural rigidity.

Wednesday October 10, 2018 1:00 PM  2:00 PM Math Tower P131
 Lisandra Hernandez, Stony Brook University
Hyperbolic 3Manifolds and Their ClassificationWe’ll begin with an excursion into the geometry and topology of hyperbolic space. This will be followed by a brief introduction to some tools in geometric group theory that will help us understand the fundamental group of hyperbolic 3manifolds a little better. We’ll end the talk with a discussion of Thurston’s Hyperbolization conjecture.

Wednesday October 17, 2018 1:00 PM  2:00 PM Math Tower P131
 JinCheng Guu, Stony Brook University
A Generalization of Fourier AnalysisLet’s see how a group G (compact, connected) smoothly acts on a complex vector space. A fundamental case is when G is abelian (Fourier series), whose applications can be found in PDE, ergodic problems, number theory,... etc. We will then look at the easiest nonabelian case and a basic application to Quantum Mechanics if time permits. This talk is an invitation to an upcoming miniseminar on representations of compact Lie groups.

Wednesday October 24, 2018 1:00 PM  2:00 PM Math Tower P131
 Ben Wu, Stony Brook University
Two Spaces Associated to a Closed Riemann SurfaceIn this talk, we will construct two spaces that are naturally associated to a closed Riemann surface. These spaces can be seen as a toy model for a much more general theory. One of the spaces is related to the topology of the Riemann surface, while the other is related to the smooth structure. Even in our toy model, the geometry of these spaces interesting. Although these two spaces consist of different objects, we will show that they are actually equivalent. Studying generalizations of this equivalence lead to the Riemann Hilbert correspondence which provide strong links between topology and algebra.

Wednesday October 31, 2018 1:00 PM  2:00 PM Math Tower P131
 Jacob Mazor, Stony Brook University
Kakeya Sets and the Finite Field Kakeya ConjectureIf K is a set in the plane which contains a unit line segment pointing in every direction, how large must it be? It turns out that we may construct such a set with zero Lebesgue measure. (!) However any such set must be "rather large" for a measure zero set. Does this result extend to such sets in higher dimensions? We will discuss a formulation for vector spaces over finite fields: what is the minimum number of points that must be contained in some $K⊂ F^n_q$ which contains a line in every direction?

Wednesday November 07, 2018 1:00 PM  2:00 PM Math Tower P131
 JeanFrancois Arbour, Stony Brook University
TBATBA

Wednesday November 14, 2018 1:00 PM  2:00 PM Math Tower P131
 Runjie Hu, Stony Brook University
K Theory and Division AlgebrasThe only dimensions of division algebras over the real numbers are only 1, 2, 4 and 8. We would introduce the topological approach for this problem without assumption of norm preserving. This result is equivalent to the Hspace structures on spheres, which is to require the hopf invariant of the spheres should be one. I would briefly introduce the K theory and Adams operation and use this powerful tool to prove the problem of division algebras.

Wednesday November 28, 2018 1:00 PM  2:00 PM Math Tower P131
 John Sheridan, Stony Brook University
A glimpse of modular formsModular forms are holomorphic functions on the upper half plane satisfying a particular symmetry property. Despite this fairly innocent definition their theory often has a reputation for inaccessibility. We will do a shallow dive into the theory of modular forms, comparing them with polynomials and investigating classical results such as their finite dimensionality and some of the beautiful numerical identities they encode. Overall, we will try and get a first sense of how and why these functions pop up in some of the various places that they do.

Wednesday December 05, 2018 1:00 PM  2:00 PM Math Tower P131
 Lisa Marquand, Stony Brook University
Journey to the TropicsTropical geometry is a relatively new field at the interface of algebraic geometry and combinatorics. We will explore the basics, discussing tropical plane curves and observing results that still hold tropically, such as Bezout's theorem. Through this friendly introduction, we will discuss the breakthrough that brought tropical methods to the attention of geometers  GromovWitten invariants of the plane.

