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The summer went by so quickly, and I still can’t believe I am finally done with grad school! Somehow I managed to write a 230 page long PhD thesis. Am I going insane? Well, that’s not the question I would like to answer just yet. If whoever is reading this is interested in my research and is intimidated by the page count, you are in the right place! Below, I will try to summarize my PhD thesis omitting most of the technical details.
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(It’s summer 2024 and I haven’t been writing anything meaningful in this blog in the last two years. I would like to think that graduate school and research in general has been keeping me busy for a while… but now that my graduate school era is over, there is no more excuse!) I have been thinking about Teichmüller spaces. What exactly are they? How are they relevant? Why should I care about it? Well, there are many questions you can ask, but let me just try to answer the first one here.
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Hartogs extension theorem is one of the few fundamental results in analytic continuation that distinguishes the study of several complex variables and that of one complex variable. While holomorphic functions in one variable can have poles and isolated singularities, the set of singularities of a holomorphic function in several complex variables cannot be isolated and must satisfy some topological conditions.
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In dynamical systems, the entropy $h_{top}(f)$ of a continuous map $f : X \to X$ on a compact space $X$ is a way to measure the complexity of $f$. Computing the exact value of entropy often requires $f$ to be highly regular (e.g. holomorphic). In the case where $X$ is a smooth orientable manifold and $f$ is continuously differentiable ($C^1$), a theorem by Misiurewicz and Przytycki gives a lower bound for $h_{top}(f)$.
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I am by no means an expert in complex geometry. This post is an attempt to capture the basics of Kähler geometry from the point of view of a non-expert, and is written for those of you who have limited background in such and wonder what makes Kähler manifolds worth studying. Most of the content here is based on a talk I gave in a student differential grometry seminar in Stony Brook.
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In simple terms, an iterated function system (IFS) is a set of contractions of a metric space. Each IFS admits a unique set invariant under iterations of these contractions, called the attractor of the IFS. IFS provides a robust method of generating fractals, and conversely the Collage Theorem states that any set can always be approxiated by the attractor of some IFS. In this post, I will describe the elementary theory of IFS and how it works.
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In statistical mechanics, Boltzmann brought up the ‘ergodic hypothesis’ which states that over time, a system of $N$ particles will eventually find all its way to admit all accessible (constant energy) states (position and momentum). The hypothesis is known to be false. The property the system needs to satisfy in order to get the result that Boltzmann originally wanted (time averages = space averages) is now called ergodicity. Ergodic theory has been one of the key measure theoretic studies in the field of dynamical systems. In this post, I aim to introduce the modern formulation of ergodicity in discrete dynamical systems, outline a variety of ways of defining it, and provide simple examples.
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The word entropy refers to a measure of chaos or uncertainty of a system. In the field of dynamical systems, topological entropy is perhaps the most important topological invariant which measures the exponential growth of ‘distinct’ orbits (the ‘complexity’) of a dynamical system. In this post, we will discuss some of the basic properties and examples of topological entropy.
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I have been dealing with a lot of quasicircles lately, and I was thinking it would be practical to write a post about its many equivalent definitions. Quasicircles are Jordan curves which satisfy certain nice analytic and geometric conditions. While conformal maps of the Riemann sphere (aka Möbius transformations) map circles to circles, quasiconformal maps map quasicircles to quasicircles.
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Given a holomorphic endomorphism $f: X \to X$ of a Riemann surface $x \in X$, one often studies the dynamics of $f$ by splitting $X$ into the Fatou set $F(f)$ and the Julia set $J(f)$. In this post, we discuss the classification of Fatou components based on the dynamical behaviour of $f$ within.
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Given a dynamical system $f: X \to X$ with a fixed point $x \in X$, what can we say about the dynamical behaviour of $f$ near $x$? In my previous post, we discussed that for analytic functions of 1 variable, if the derivative $\lambda =f’(x)$ at $x$ is non-zero and $\vert \lambda \vert \neq 1$, then $f$ is locally conjugate to its linear part $z \mapsto \lambda z$. What happens when $\lambda$ lies on the unit circle? In this case, the fixed point $x$ is called neutral and the problem of linearisability near $x$ becomes much more complicated than the non-neutral case.
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We saw in the previous post that the group of automorphisms of a compact Riemann surface $X$ is always finite and can be bounded in terms of its genus. Throwing away the bijective assumption, it turns out that in most situations the group of endomorphisms is still very rigid. We shall classify all holomorphic endomorphisms of hyperbolic Riemann surfaces, which are not necessarily compact, according to their dynamics.
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I recently discovered that the number of conformal automorphisms of a compact genus $g\geq 2$ Riemann surface is finite and it has an upper bound of $84(g-1)$. This is known as the Hurwitz’s Automorphism Theorem. There are a number of ways to prove this theorem, but we shall do so using the language of orbifolds. The standard approach is to see that the automorphism group $\text{Aut}(X)$ is finite by showing that it is discrete. Then, we can apply the Riemann-Hurwitz formula to count and obtain the upper bound.
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The Dirichlet problem is one of the classical problems in the field of PDEs and it is the problem of finding a harmonic function on a domain that takes prescribed values on the boundary. The Dirichlet problem is closely related to the notion of harmonic measure of a domain, which is a probability measure on the boundary of the domain. We will first discuss harmonic measure for nice planar domains and its relevance in solving the Dirichlet problem in 2 dimensions.
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Given a conformal isomorphism $f: \mathbb{D} \to U$ from the unit disk $\mathbb{D}$ to a simply connected domain $U$ embedded in the Riemann sphere $\mathbb{P}^1$, one may ask whether or not $f$ can be continuously extended to the boundary of $U$. The answer depends solely on the topology of the domain $U$.
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Given a dynamical system $f: X \to X$ with a fixed point $x \in X$, what can we say about the dynamical behaviour of $f$ near $x$? For many sufficiently smooth functions $f$, the behaviour of $f$ near $x$ is governed by its linear approximation $Df_x$. More precisely, $f$ is conjugate to $Df_x$ when the derivative $Df_x$ is hyperbolic, i.e. has no eigenvalue with absolute value $1$. Proving local linearisability can be a rather daunting task, requiring a rather involved theory of hyperbolic systems, I would like to discuss the baby problem of local linearisability of analytic functions of 1 variable in this post.
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This post is a continuation of the previous post in which we discuss about circle homeomorphisms with rational rotation number. To complete the story of Poincaré’s classification of circle homeomorphisms, I would like to cover the irrational case, which is split into two: either $f$ is conjugate to the irrational rotation $x+\tau(f)$ and the $\omega$-limit set of every point is $S^1$, or $f$ has an invariant Cantor set $C$ and the $\omega$-limit set of every point is $C$.
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Circle rotations provide easy and simple examples of low-dimensional dynamical systems. A natural generalisation to rotations is circle homeomorphisms. In this post, we will delve into an important invariant of circle homeomorphisms, namely the rotation number, and discuss the dynamics of circle homeomorphisms of rational rotation number.
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The famous Koebe one-quarter theorem gives a sharp bound on the size of the image of univalent functions locally. The standard proof of this theorem which can be found in most complex analysis books (e.g. Rudin1 chapter 14) relies on the area theorem and a rather unnatural conformal change of variables. I found another proof of the theorem from the book of Hubbard2 which is, in my opinion, more fascinating and geometric, relying on covering spaces and conformal moduli.
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A discrete topological dynamical system is a continuous self map $f: X \to X$ on a topological space $X$. The main interest in topological dynamics is to observe how different values of $x$ behave under iteration $f^n(x)$ for $n\geq 0$ without any further regularity assumptions on $f$ and $X$. In this post, I aim to summarise the very basic definitions and propositions in topological dynamics.
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Now that we know orbifolds (see this post), we would like to make an attempt to classify the geometries of all 2 dimensional connected orbifolds. I would like to start off with the theory of covering spaces of orbifolds and then make a classification statement according to the universal cover of these spaces, similar to the uniformisation theorem and the classical classification of surfaces.
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Earlier today, I gave a talk about orbifolds in SBU’s graduate student math seminar and I was thinking it’d be a good time for me to write about them too. Orbifolds are a rather natural generalisation of manifolds especially if you are dealing with a lot of quotient spaces. Let’s have a look at what they really are and what we can do with them.
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For any rational map $f(z)$ on the Riemann sphere $\mathbb{P}^1$ onto itself, can its Julia set $J(f)$ be the whole Riemann sphere? When it is the case, the Fatou set $F(f)$ is empty and it tells us that $f$ is chaotic throughout the whole $\mathbb{P}^1$. The answer is yes. In this post, I will discuss one standard way of constructing such rational maps, often called Lattès maps.
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Classical complex analysis tells us that every non-constant holomorphic map from the Riemann sphere to itself is a rational map $p(z)/q(z)$ for some coprime polynomials $p$ and $q$. Let’s increase the genus and replace the Riemann sphere with a complex torus. It turns out that the space of holomorphic maps from a complex torus to itself is much simpler that what we can imagine.
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Irrational numbers are typically defined upon constructing the set $\mathbb{R}$ of real numbers from the set $\mathbb{Q}$ of rational numbers via either Cauchy sequences or Dedekind cuts. Decimal expansion has been used as the standard way of representing and approximating irrational numbers. However, it turns out that the best way to approximate irrational numbers is through continued fractions.
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In Stony Brook’s maths department, before embarking on any serious research, a graduate student must complete an oral qualifying examination. I took mine in December 1st 2020. I was assessed on Holomorphic Dynamics (major topic) and Several Complex Variables (minor topic) by Dzmitry (Dima) Dudko, my major advisor, Eric Bedford, my minor advisor, and Mikhail (Misha) Lyubich. Here are my thoughts on how it went and some tips on how you can ace your orals.
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The measurable Riemann mapping theorem by Ahlfors & Bers remains one of the most fundamental results in the theory of quasiconformal maps. It states that there is a biholomorphic correspondence between quasiconformal maps (modulo post-composition with conformal isomorphisms) and their complex dilatations. I aim to discuss the theorem firstly on the unit disk and then on general Riemann surfaces.
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Riemann mapping theorem states that every simply connected domain on the complex plane $\mathbb{C}$ is either $\mathbb{C}$ or biholomorphic to the unit disk $\mathbb{D}$. This theorem is like a miracle and its natural generalisation to domains in higher dimensions miserably fails. In higher dimensions, simply connected domains have a much richer variety of biholomorphic invariants, such as higher cohomology groups and automorphism groups. Without using any complicated invariants, I’d like to bring up a classical counterexample by Poincaré, showing that the unit ball $\mathbb{B} \subset \mathbb{C}^n$ and the unit polydisk $\mathbb{D}^n = \{ \vert z_j \vert <1 \text{ for }j=1,\ldots n \} \subset \mathbb{C}^n$ for $n \geq 2$ are not biholomorphically equivalent.
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In several complex variables, a function $f: D \to \mathbb{C}$ on an open subset $D \subset \mathbb{C}^n$ is holomorphic if it is separately holomorphic on each variable. A theorem by Hartogs says that this is equivalent to $f$ having a local power series representation in $z = (z_1,\ldots z_n)$ at every point in $D$, i.e. the usual definition of analyticity. A similar result by Forelli states that holomorphicity about a point $a \in \mathbb{C}$ is also satisfied if the function is $C^\infty$ near $a$ and is holomorphic on every one-dimensional complex disk centered at $a$. To me, both are equally interesting results on their own!
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Extremal length measures the size of curve families in a way that is invariant under conformal mappings. The extremal length also gives a way to define quasiconformal maps, i.e. orientation preserving homoeomorphisms which distort angles locally up to some bounded constant, and equivalently, distort extremal lengths up to some bounded constant. In this post, I aim to discuss the basic concepts of extremal length.
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Conformal mappings are often too rigid for us to use. We know that Riemann mappings, for example, are unique up to conformal automorphisms of the unit disk – the set of which forms a topological group homeomorphic to $S^1 \times \mathbb{D}$. In complex analysis, we often need maps which are more flexible and in many cases, what we are looking for are quasiconformal maps. While conformal maps preserve angle locally, quasiconformal maps distort angles locally up to some factor.
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Consider the family of quadratics $f_c(z) = z^2 + c$ parametrised by $c \in \mathbb{C}$. Recall from here that the filled Julia set $K(f_c)$ of $f_c$ is the complement of the basin of infinity
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As smooth two dimensional smooth real manifolds, Riemann surfaces admit Riemannian metrics. In the study of Riemann surfaces, it is more interesting to look at those Riemannian metrics which behave nicely under conformal maps between Riemann surfaces. This gives rise to the study of conformal metrics. I aim to introduce what conformal metrics are and provide some examples of commonly used conformal metrics in different Riemann surfaces.
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The uniformisation theorem states that every simply connected Riemann surface $X$ is conformally isomorphic to either the unit disk $\mathbb{D}$, the complex plane $\mathbb{C}$ or the Riemann sphere $\mathbb{P}^1$. The classical proof relies heavily on techniques in potential theory including the Dirichlet principle and the Perron method. (Refer to Forster’s book3.) Rather than discussing the proof in detail, I am more interested in some of the immediate applications of the uniformisation theorem. This includes classification of Riemann surfaces in general.
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Intricate images of the Mandelbrot set are ubiquitous. You might have seen them randomly featured on your school math textbooks for God knows what reason. They’re probably on the posters of some of math professors in your college. You’ve probably seen some people wearing Mandelbrot set T-shirts or even having Mandelbrot tattoos. Rather than trying to explain its fame, I would like to firstly define this object and state its basic properties.
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Complex dynamics is the study of iterations of a holomorphic map $f: X \to X$ where $X$ is a Riemann surface. Dynamical objects of interest are the Fatou set $F(f)$, i.e. the subset of $X$ on which the iterations are stable, and the Julia set, which is the unstable region. Below, I am to summarise the basic concepts of Fatou and Julia sets in one-dimensional complex dynamics. I will focus on the case where $X$ is an open subset of the Riemann sphere $\mathbb{P}^1 = \mathbb{C} \cup {\infty}$. Proofs will be skipped and I would personally recommend referring to Beardon1 or Milnor2 for more details.
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Riemann mapping theorem states that every simply connected open proper subset $U$ of the complex plane $\mathbb{C}$ is conformally isomorphic to the unit disk $\mathbb{D}$. In other words, there exists a conformal isomorphism $f: \mathbb{D} \to U$. What I would like to discuss is a way to topologise the set of simply connected open proper subsets $U$ of $\mathbb{C}$ and to naturally normalise the corresponding conformal isomorphisms $f$ in order to establish a natural topological correspondence $U \mapsto (f: \mathbb{D} \to U)$.
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Manifolds are automatically assumed to be second-countable by definition. Without this assumption, horrible topological objects like the long line, i.e. the concatenation of uncountably many copies of $[0,1)$, would be manifolds and paracompactness as well as the existence of partitions of unity would not come for granted. However, even without this assumption, Riemann surfaces are still second-countable.