CMU Analysis and Geometry Seminar

Fall 2016 and Spring 2017

Organizers

If you would like to give a talk, please email any one of us!

Time and Place

Fridays, 3:00–4:00pm, in Pearce 223.

Schedule

 Date Speaker Title Sep 2 Olivia Dumitrescu (CMU) On a Conjecture of Gaiotto and Quantum Curves --I Sep 16 Olivia Dumitrescu (CMU) On a Conjecture of Gaiotto and Quantum Curves --II Sep 23 Phil Harrington (Univ of Arkansas) The Diederich-Fornaess Index Oct 14 Mythily Ramaswamy (TIFR CAM, Bengaluru, India) Extensions of Ingham inequality and applications to PDE Control Nov 4 Octavian Mitrea (University of Western Ontario) Polynomial Convexity of Lagrangian Inclusions Nov 11 Luke Edholm (Univ of Michigan) Two decompositions of the Bergman Space Jan 27 Margaret Stawiska-Friedland (Math Reviews) Some approximation problems in semi-algebraic geometry April 7 Biplab Basak (Cornell University) Crystallization and its applications on surfaces Apr 14 Loredana Lanzani (Syracuse University) Special Time: 2:45 to 3:45 TBA

Abstracts

Speaker: Olivia Dumitrescu
Title: On a Conjecture of Gaiotto and Quantum Curves (Part 1 and 2)
Abstract: Gaiotto's conjecture (2014) is a particular construction of $Sl_r( \mathbb{C})$-opers from Higgs bundles in one Hitchin component. The conjecture has been recently solved by a joint paper of Dumitrescu, Fredrickson, Kydonakis, Mazzeo, Mulase, and Neitzke (2016). My first talk will be introductive to familiarize with the Hitchin moduli spaces of Higgs bundles. In the second talk, I will present a holomorphic description of the limiting oper, and its geometry. The importance of this correspondence, in particular the relation with "quantum curve" of Dumitrescu - Mulase (2013) will be discussed.

Speaker: Philip Harrington
Title: The Diederich-Fornaess Index
Abstract: Convexity comes in many different flavors. For example, a ball and a cylinder are both convex, but they have very different geometric properties. In several complex variables, pseudoconvexity is the natural substitute for convexity, and the Diederich-Fornaess Index is a tool for measuring the pseudoconvexity of a domain. We will look at this tool and its applications to partial differential equations.

Speaker: Mythily Ramaswamy
Title: Extensions of Ingham inequality and applications to PDE Control
Abstract: After Ingham proved some trigonometric type inequalities in 1936, they have been applied in Series expansion, Function theory and also in Control theory. Depending on the spectrum of the differential operator, new extensions of these inequalities have been found. I show a few such extensions and application to a coupled system of PDE of hyperbolic and parabolic types.

Speaker: Octavian Mitrea
Title: Polynomial Convexity of Lagrangian Inclusions
Abstract: Polynomial convexity is of key importance in the general theory of approximation of continuous functions, uncovering deep connections to topology, Banach algebras, symplectic geometry, and other areas of mathematics. If $S\subset \mathbb{C}^2$ is a compact real surface, a Lagrangian inclusion of S is a map $S \rightarrow \mathbb{C}^2$ which is a local Lagrangian embedding, except for a finite number of singularities that are either transverse double points or open Whitney umbrellas. In 1986, Givental proved that any such surface $S$, orientable or not, admits a Lagrangian inclusion. In this talk we show that Lagrangian inclusions are locally polynomially convex at every point.

Speaker: Luke Edholm
Title: Two decompositions of the Bergman Space
Abstract: Given $\Omega \in \mathbb{C}^n$, its Bergman projection, $\mathbf{B}_{\Omega}$, is the orthogonal projection from $L^2(\Omega)$ onto its holomorphic subspace, $A^2(\Omega)$. We call $A^2(\Omega)$ the Bergman space. A question of great importance in SCV is to understand the regularity properties of $\mathbf{B}_{\Omega}$ when it acts on other function spaces. For many classes of smoothly bounded $\Omega$, much is known. But when $\Omega$ has boundary singularities, the question is wide open. I will introduce two novel ways to decompose $A^2(\Omega)$ that make this problem viable for a large class of non-smooth domains in $\mathbb{C}^2$. I will discuss new results on this class of model domains, then give an outline of future directions this project is likely to take.

Speaker: Margaret Stawiska-Friedland
Title: Some approximation problems in semi-algebraic geometry
Abstract: Given a semialgebraic norm $\nu$ in $\mathbb{R}^n$, we study best $\nu$-approximation to a closed semialgebraic subset $C \subset \mathbb{R}^n$. We show that when $\nu$ is strictly convex, then the set of points for which best $\nu$-approximation to $C$ is not unique is contained in a semialgebraic set of dimension strictly less than $n$. We also discuss the set of critical points of the distance to an algebraic variety $C$. This is joint work with Shmuel Friedland, motivated by approximation of tensors.

Speaker: Biplab Basak
Title: Crystallization and its applications on surfaces
Abstract: Crystallization is a geometric tool to study topological and combinatorial properties of PL manifolds. Motivated by the theory of crystallizations, we consider an equivalence relation on the class of 3-regular colored graphs. In this talk, I shall show that up to this equivalence (a) there exists a unique contracted 3-regular colored graph if the number of vertices is $4m$ and (b) there are exactly two such graphs if the number of vertices is $4m+2$ for $m\geq 1$. Using this, I shall present a simple proof of the classification of closed surfaces.