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Events matching "K3 surfaces: a crash course"

The Mechanics of Nanoscale Devices
15:10 Fri 10 Oct, 2008 :: G03 Napier Building University of Adelaide :: Associate Prof. John Sader :: Department of Mathematics and Statistics, The University of Melbourne

Nanomechanical sensors are often used to measure environmental changes with extreme sensitivity. Controlling the effects of surfaces and fluid dissipation presents significant challenges to achieving the ultimate sensitivity in these devices. In this talk, I will give an overview of theoretical/experimental work we are undertaking to explore the underlying physical processes in these systems. The talk will be general and aimed at introducing some recent developments in the field of nanomechanical sensors.
Lagrangian fibrations on holomorphic symplectic manifolds I: Holomorphic Lagrangian fibrations
13:10 Fri 5 Jun, 2009 :: School Board Room :: Dr Justin Sawon :: Colorado State University

A compact K{\"a}hler manifold $X$ is a holomorphic symplectic manifold if it admits a non-degenerate holomorphic two-form $\sigma$. According to a theorem of Matsushita, fibrations on $X$ must be of a very restricted type: the fibres must be Lagrangian with respect to $\sigma$ and the generic fibre must be a complex torus. Moreover, it is expected that the base of the fibration must be complex projective space, and this has been proved by Hwang when $X$ is projective. The simplest example of these {\em Lagrangian fibrations\/} are elliptic K3 surfaces. In this talk we will explain the role of elliptic K3s in the classification of K3 surfaces, and the (conjectural) generalization to higher dimensions.
Eynard-Orantin invariants and enumerative geometry
13:10 Fri 6 Aug, 2010 :: Ingkarni Wardli B20 (Suite 4) :: Dr Paul Norbury :: University of Melbourne

As a tool for studying enumerative problems in geometry Eynard and Orantin associate multilinear differentials to any plane curve. Their work comes from matrix models but does not require matrix models (for understanding or calculations). In some sense they describe deformations of complex structures of a curve and conjectural relationships to deformations of Kahler structures of an associated object. I will give an introduction to their invariants via explicit examples, mainly to do with the moduli space of Riemann surfaces, in which the plane curve has genus zero.
Counting lattice points in polytopes and geometry
15:10 Fri 6 Aug, 2010 :: Napier G04 :: Dr Paul Norbury :: University of Melbourne

Counting lattice points in polytopes arises in many areas of pure and applied mathematics. A basic counting problem is this: how many different ways can one give change of 1 dollar into 5,10, 20 and 50 cent coins? This problem counts lattice points in a tetrahedron, and if there also must be exactly 10 coins then it counts lattice points in a triangle. The number of lattice points in polytopes can be used to measure the robustness of a computer network, or in statistics to test independence of characteristics of samples. I will describe the general structure of lattice point counts and the difficulty of calculations. I will then describe a particular lattice point count in which the structure simplifies considerably allowing one to calculate easily. I will spend a brief time at the end describing how this is related to the moduli space of Riemann surfaces.
Slippery issues in nano- and microscale fluid flows
11:10 Tue 30 Nov, 2010 :: Innova teaching suite B21 :: Dr Shaun C. Hendy :: Victoria University of Wellington

The no-slip boundary condition was considered to have been experimentally established for the flow of simple liquids over solid surfaces in the early 20th century. Nonetheless the refinement of a number of measurement techniques has recently led to the observation of nano- and microscale violations of the no-slip boundary condition by simple fluids flowing over non-wetting surfaces. However it is important to distinguish between intrinsic slip, which arises solely from the chemical interaction between the liquid and a homogeneous, atomically flat surface and effective slip, typically measured in macroscopic experiments, which emerges from the interaction of microscopic chemical heterogeneity, roughness and contaminants. Here we consider the role of both intrinsic and effective slip boundary conditions in nanoscale and microscale fluid flows using a theoretical approach, complemented by molecular dynamics simulations, and experimental evidence where available. Firstly, we consider nanoscale flows in small capillaries, including carbon nanotubes, where we have developed and solved a generalised Lucas-Washburn equation that incorporates slip to describe the uptake of droplets. We then consider the general problem of relating effective slip to microscopic intrinsic slip and roughness, and discuss several cases where we have been able to solve this problem analytically. Finally, we look at applications of these results to carbon nanotube growth, self-cleaning surfaces, catalysis, and putting insulation in your roof.
Surface quotients of hyperbolic buildings
13:10 Fri 18 Mar, 2011 :: Mawson 208 :: Dr Anne Thomas :: University of Sydney

Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons, and the link at each vertex is the complete bipartite graph K_{v,v}. We investigate and mostly determine the set of triples (p,v,g) for which there is a discrete group acting on I(p,v) so that the quotient is a compact orientable surface of genus g. Surprisingly, the existence of such a quotient depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. We use elementary group theory, combinatorics, algebraic topology and number theory. This is joint work with David Futer.
A strong Oka principle for embeddings of some planar domains into CxC*, I
13:10 Fri 6 May, 2011 :: Mawson 208 :: Mr Tyson Ritter :: University of Adelaide

The Oka principle refers to a collection of results in complex analysis which state that there are only topological obstructions to solving certain holomorphically defined problems involving Stein manifolds. For example, a basic version of Gromov's Oka principle states that every continuous map from a Stein manifold into an elliptic complex manifold is homotopic to a holomorphic map. In these two talks I will discuss a new result showing that if we restrict the class of source manifolds to circular domains and fix the target as CxC* we can obtain a much stronger Oka principle: every continuous map from a circular domain S into CxC* is homotopic to a proper holomorphic embedding. This result has close links with the long-standing and difficult problem of finding proper holomorphic embeddings of Riemann surfaces into C^2, with additional motivation from other sources.
A strong Oka principle for embeddings of some planar domains into CxC*, II
13:10 Fri 13 May, 2011 :: Mawson 208 :: Mr Tyson Ritter :: University of Adelaide

The Oka principle refers to a collection of results in complex analysis which state that there are only topological obstructions to solving certain holomorphically defined problems involving Stein manifolds. For example, a basic version of Gromov's Oka principle states that every continuous map from a Stein manifold into an elliptic complex manifold is homotopic to a holomorphic map. In these two talks I will discuss a new result showing that if we restrict the class of source manifolds to circular domains and fix the target as CxC* we can obtain a much stronger Oka principle: every continuous map from a circular domain S into CxC* is homotopic to a proper holomorphic embedding. This result has close links with the long-standing and difficult problem of finding proper holomorphic embeddings of Riemann surfaces into C^2, with additional motivation from other sources.
K3 surfaces: a crash course
13:10 Fri 12 Aug, 2011 :: B.19 Ingkarni Wardli :: A/Prof Nicholas Buchdahl :: University of Adelaide

Everything you have ever wanted to know about K3 surfaces! Two talks: 1:10 pm to 3:00 pm.
Noncritical holomorphic functions of finite growth on algebraic Riemann surfaces
13:10 Fri 3 Feb, 2012 :: B.20 Ingkarni Wardli :: Prof Franc Forstneric :: University of Ljubljana

Given a compact Riemann surface X and a point p in X, we construct a holomorphic function without critical points on the punctured (algebraic) Riemann surface R=X-p which is of finite order at the point p. In the case at hand this improves the 1967 theorem of Gunning and Rossi to the effect that every open Riemann surface admits a noncritical holomorphic function, but without any particular growth condition. (Joint work with Takeo Ohsawa.)
Mathematical modelling of the surface adsorption for methane on carbon nanostructures
12:10 Mon 30 Apr, 2012 :: 5.57 Ingkarni Wardli :: Mr Olumide Adisa :: University of Adelaide

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In this talk, methane (CH4) adsorption is investigated on both graphite and in the region between two aligned single-walled carbon nanotubes, which we refer to as the groove site. The Lennard–Jones potential function and the continuous approximation is exploited to determine surface binding energies between a single CH4 molecule and graphite and between a single CH4 and two aligned single-walled carbon nanotubes. The modelling indicates that for a CH4 molecule interacting with graphite, the binding energy of the system is minimized when the CH4 carbon is 3.83 angstroms above the surface of the graphitic carbon, while the binding energy of the CH4–groove site system is minimized when the CH4 carbon is 5.17 angstroms away from the common axis shared by the two aligned single-walled carbon nanotubes. These results confirm the current view that for larger groove sites, CH4 molecules in grooves are likely to move towards the outer surfaces of one of the single-walled carbon nanotubes. The results presented in this talk are computationally efficient and are in good agreement with experiments and molecular dynamics simulations, and show that CH4 adsorption on graphite and groove surfaces is more favourable at lower temperatures and higher pressures.
Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds
13:10 Fri 4 May, 2012 :: Napier LG28 :: Dr Tyson Ritter :: University of Adelaide

In complex geometry a manifold is Stein if there are, in a certain sense, "many" holomorphic maps from the manifold into C^n. While this has long been well understood, a fruitful definition of the dual notion has until recently been elusive. In Oka theory, a manifold is Oka if it satisfies several equivalent definitions, each stating that the manifold has "many" holomorphic maps into it from C^n. Related to this is the geometric condition of ellipticity due to Gromov, who showed that it implies a complex manifold is Oka. We present recent contributions to three open questions involving elliptic and Oka manifolds. We show that affine quotients of C^n are elliptic, and combine this with an example of Margulis to construct new elliptic manifolds of interesting homotopy types. It follows that every open Riemann surface properly acyclically embeds into an elliptic manifold, extending an existing result for open Riemann surfaces with abelian fundamental group.
Differential topology 101
13:10 Fri 17 Aug, 2012 :: Engineering North 218 :: Dr Nicholas Buchdahl :: University of Adelaide

Much of my recent research been directed at a problem in the theory of compact complex surfaces---trying to fill in a gap in the Enriques-Kodaira classification. Attempting to classify some collection of mathematical objects is a very common activity for pure mathematicians, and there are many well-known examples of successful classification schemes; for example, the classification of finite simple groups, and the classification of simply connected topological 4-manifolds. The aim of this talk will be to illustrate how techniques from differential geometry can be used to classify compact surfaces. The level of the talk will be very elementary, and the material is all very well known, but it is sometimes instructive to look back over simple cases of a general problem with the benefit of experience to gain greater insight into the more general and difficult cases.
Boundary-layer transition and separation over asymmetrically textured spherical surfaces
12:30 Mon 27 Aug, 2012 :: B.21 Ingkarni Wardli :: Mr Adam Tunney :: University of Adelaide

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The game of cricket is unique among ball sports by the ignorant exploitation of \thetitle in the practice of swing bowling, often referred to as a "mysterious art". I will talk a bit about the Magnus effect exploited in inferior sports, the properties of a cricket ball that allow swing bowling, and the explanation of three modes of swing (conventional, contrast and reverse). Following that there will be some discussion on how I plan to use mathematics to turn this "art" into science.
Holomorphic flexibility properties of compact complex surfaces
13:10 Fri 31 Aug, 2012 :: Engineering North 218 :: A/Prof Finnur Larusson :: University of Adelaide

I will describe recent joint work with Franc Forstneric (arXiv, July 2012). We introduce a new property, called the stratified Oka property, which fits into a hierarchy of anti-hyperbolicity properties that includes the Oka property. We show that stratified Oka manifolds are strongly dominable by affine spaces. It follows that Kummer surfaces are strongly dominable. We determine which minimal surfaces of class VII are Oka (assuming the global spherical shell conjecture). We deduce that the Oka property and several other anti-hyperbolicity properties are in general not closed in families of compact complex manifolds. I will summarise what is known about how the Oka property fits into the Enriques-Kodaira classification of surfaces.
Complex analysis in low Reynolds number hydrodynamics
15:10 Fri 12 Oct, 2012 :: B.20 Ingkarni Wardli :: Prof Darren Crowdy :: Imperial College London

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It is a well-known fact that the methods of complex analysis provide great advantage in studying physical problems involving a harmonic field satisfying Laplace's equation. One example is in ideal fluid mechanics (infinite Reynolds number) where the absence of viscosity, and the assumption of zero vorticity, mean that it is possible to introduce a so-called complex potential -- an analytic function from which all physical quantities of interest can be inferred. In the opposite limit of zero Reynolds number flows which are slow and viscous and the governing fields are not harmonic it is much less common to employ the methods of complex analysis even though they continue to be relevant in certain circumstances. This talk will give an overview of a variety of problems involving slow viscous Stokes flows where complex analysis can be usefully employed to gain theoretical insights. A number of example problems will be considered including the locomotion of low-Reynolds-number micro-organisms and micro-robots, the friction properties of superhydrophobic surfaces in microfluidics and problems of viscous sintering and the manufacture of microstructured optic fibres (MOFs).
Twistor space for rolling bodies
12:10 Fri 15 Mar, 2013 :: Ingkarni Wardli B19 :: Prof Pawel Nurowski :: University of Warsaw

We consider a configuration space of two solids rolling on each other without slipping or twisting, and identify it with an open subset U of R^5, equipped with a generic distribution D of 2-planes. We will discuss symmetry properties of the pair (U,D) and will mention that, in the case of the two solids being balls, when changing the ratio of their radii, the dimension of the group of local symmetries unexpectedly jumps from 6 to 14. This occurs for only one such ratio, and in such case the local group of symmetries of the pair (U,D) is maximal. It is maximal not only among the balls with various radii, but more generally among all (U,D)s corresponding to configuration spaces of two solids rolling on each other without slipping or twisting. This maximal group is isomorphic to the split real form of the exceptional Lie group G2. In the remaining part of the talk we argue how to identify the space U from the pair (U,D) defined above with the bundle T of totally null real 2-planes over a 4-manifold equipped with a split signature metric. We call T the twistor bundle for rolling bodies. We show that the rolling distribution D, can be naturally identified with an appropriately defined twistor distribution on T. We use this formulation of the rolling system to find more surfaces which, when rigidly rolling on each other without slipping or twisting, have the local group of symmetries isomorphic to the exceptional group G2.
Geometry of moduli spaces
12:10 Fri 30 Aug, 2013 :: Ingkarni Wardli B19 :: Prof Georg Schumacher :: University of Marburg

We discuss the concept of moduli spaces in complex geometry. The main examples are moduli of compact Riemann surfaces, moduli of compact projective varieties and moduli of holomorphic vector bundles, whose points correspond to isomorphism classes of the given objects. Moduli spaces carry a natural topology, whereas a complex structure that reflects the variation of the structure in a family exists in general only under extra conditions. In a similar way, a natural hermitian metric (Weil-Petersson metric) on moduli spaces that induces a symplectic structure can be constructed from the variation of distinguished metrics on the fibers. In this way, various questions concerning the underlying symplectic structure, the curvature of the Weil-Petersson metric, hyperbolicity of moduli spaces, and construction of positive/ample line bundles on compactified moduli spaces can be answered.
The geometry of rolling surfaces and non-holonomic mechanics
15:10 Fri 1 Nov, 2013 :: B.18 Ingkarni Wardli :: Prof Robert Bryant :: Duke University

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In mechanics, the system of a sphere rolling over a plane without slipping or twisting is a fundamental example of what is called a non-holonomic mechanical system, the study of which belongs to the subject of control theory. The more general case of one surface rolling over another without slipping or twisting is, similarly, of great interest for both practical and theoretical reasons. In this talk, which is intended for a general mathematical audience (i.e., no familiarity with control theory or differential geometry will be assumed), I will describe some of the basic features of this problem, a bit of its history, and some of the surprising developments that its study reveals, such as the unexpected appearance of the exceptional group G_2.
Holomorphic null curves and the conformal Calabi-Yau problem
12:10 Tue 28 Jan, 2014 :: Ingkarni Wardli B20 :: Prof Franc Forstneric :: University of Ljubljana

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I shall describe how methods of complex analysis can be used to give new results on the conformal Calabi-Yau problem concerning the existence of bounded metrically complete minimal surfaces in real Euclidean 3-space R^3. We shall see in particular that every bordered Riemann surface admits a proper complete holomorphic immersion into the ball of C^2, and a proper complete embedding as a holomorphic null curve into the ball of C^3. Since the real and the imaginary parts of a holomorphic null curve in C^3 are conformally immersed minimal surfaces in R^3, we obtain a bounded complete conformal minimal immersion of any bordered Riemann surface into R^3. The main advantage of our methods, when compared to the existing ones in the literature, is that we do not need to change the conformal type of the Riemann surface. (Joint work with A. Alarcon, University of Granada.)
Quasimodes that do not Equidistribute
13:10 Tue 19 Aug, 2014 :: Ingkarni Wardli B17 :: Shimon Brooks :: Bar-Ilan University

The QUE Conjecture of Rudnick-Sarnak asserts that eigenfunctions of the Laplacian on Riemannian manifolds of negative curvature should equidistribute in the large eigenvalue limit. For a number of reasons, it is expected that this property may be related to the (conjectured) small multiplicities in the spectrum. One way to study this relationship is to ask about equidistribution for "quasimodes"-or approximate eigenfunctions- in place of highly-degenerate eigenspaces. We will discuss the case of surfaces of constant negative curvature; in particular, we will explain how to construct some examples of sufficiently weak quasimodes that do not satisfy QUE, and show how they fit into the larger theory.
Translating solitons for mean curvature flow
12:10 Fri 19 Sep, 2014 :: Ingkarni Wardli B20 :: Julie Clutterbuck :: Monash University

Mean curvature flow gives a deformation of a submanifold in the direction of its mean curvature vector. Singularities may arise, and can be modelled by special solutions of the flow. I will describe the special solutions that move by only a translation under the flow, and give some explicit constructions of such surfaces. This is based on joint work with Oliver Schnuerer and Felix Schulze.
To Complex Analysis... and beyond!
12:10 Mon 29 Sep, 2014 :: B.19 Ingkarni Wardli :: Brett Chenoweth :: University of Adelaide

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In the undergraduate complex analysis course students learn about complex valued functions on domains in C (the complex plane). Several interesting and surprising results come about from this study. In my talk I will introduce a more general setting where complex analysis can be done, namely Riemann surfaces (complex manifolds of dimension 1). I will then prove that all non-compact Riemann surfaces are Stein; which loosely speaking means that their function theory is similar to that of C.
Minimal Surfaces and their Application to Soap Films
12:10 Mon 13 Apr, 2015 :: Napier LG29 :: Jonathon Pantelis :: University of Adelaide

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We all have some idea about what a surface is. We can classify surfaces depending on a range of properties or characteristics. Discussed in this seminar are Minimal Surfaces, a particular class of surface. We will find out what it means for a surface to be minimal and take a look at what these things look like. We will also see how to create them, and also how they relate to soap films.
Gromov's method of convex integration and applications to minimal surfaces
12:10 Fri 7 Aug, 2015 :: Ingkarni Wardli B17 :: Finnur Larusson :: The University of Adelaide

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We start by considering an applied problem. You are interested in buying a used car. The price is tempting, but the car has a curious defect, so it is not clear whether you can even take it for a test drive. This problem illustrates the key idea of Gromov's method of convex integration. We introduce the method and some of its many applications, including new applications in the theory of minimal surfaces, and end with a sketch of ongoing joint work with Franc Forstneric.
Quantisation of Hitchin's moduli space
12:10 Fri 22 Jan, 2016 :: Engineering North N132 :: Siye Wu :: National Tsing Hua Univeristy

In this talk, I construct prequantum line bundles on Hitchin's moduli spaces of orientable and non-orientable surfaces and study the geometric quantisation and quantisation via branes by complexification of the moduli spaces.
The parametric h-principle for minimal surfaces in R^n and null curves in C^n
12:10 Fri 11 Mar, 2016 :: Ingkarni Wardli B17 :: Finnur Larusson :: University of Adelaide

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I will describe new joint work with Franc Forstneric (arXiv:1602.01529). This work brings together four diverse topics from differential geometry, holomorphic geometry, and topology; namely the theory of minimal surfaces, Oka theory, convex integration theory, and the theory of absolute neighborhood retracts. Our goal is to determine the rough shape of several infinite-dimensional spaces of maps of geometric interest. It turns out that they all have the same rough shape.
Counting periodic points of plane Cremona maps
12:10 Fri 1 Apr, 2016 :: Eng & Maths EM205 :: Tuyen Truong :: University of Adelaide

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In this talk, I will present recent results, join with Tien-Cuong Dinh and Viet-Anh Nguyen, on counting periodic points of plane Cremona maps (i.e. birational maps of P^2). The tools used include a Lefschetz fixed point formula of Saito, Iwasaki and Uehara for birational maps of surface whose fixed point set may contain curves; a bound on the arithmetic genus of curves of periodic points by Diller, Jackson and Sommerse; a result by Diller, Dujardin and Guedj on invariant (1,1) currents of meromorphic maps of compact Kahler surfaces; and a theory developed recently by Dinh and Sibony for non proper intersections of varieties. Among new results in the paper, we give a complete characterisation of when two positive closed (1,1) currents on a compact Kahler surface behave nicely in the view of Dinh and Sibony’s theory, even if their wedge intersection may not be well-defined with respect to the classical pluripotential theory. Time allows, I will present some generalisations to meromorphic maps (including an upper bound for the number of isolated periodic points which is sometimes overlooked in the literature) and open questions.
Algebraic structures associated to Brownian motion on Lie groups
13:10 Thu 16 Jun, 2016 :: Ingkarni Wardli B17 :: Steve Rosenberg :: University of Adelaide / Boston University

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In (1+1)-d TQFT, products and coproducts are associated to pairs of pants decompositions of Riemann surfaces. We consider a toy model in dimension (0+1) consisting of specific broken paths in a Lie group. The products and coproducts are constructed by a Brownian motion average of holonomy along these paths with respect to a connection on an auxiliary bundle. In the trivial case over the torus, we (seem to) recover the Hopf algebra structure on the symmetric algebra. In the general case, we (seem to) get deformations of this Hopf algebra. This is a preliminary report on joint work with Michael Murray and Raymond Vozzo.
Hilbert schemes of points of some surfaces and quiver representations
12:10 Fri 23 Sep, 2016 :: Ingkarni Wardli B17 :: Ugo Bruzzo :: International School for Advanced Studies, Trieste

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Hilbert schemes of points on the total spaces of the line bundles O(-n) on P1 (desingularizations of toric singularities of type (1/n)(1,1)) can be given an ADHM description, and as a result, they can be realized as varieties of quiver representations.
Energy quantisation for the Willmore functional
11:10 Fri 7 Oct, 2016 :: Ligertwood 314 Flinders Room :: Yann Bernard :: Monash University

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We prove a bubble-neck decomposition and an energy quantisation result for sequences of Willmore surfaces immersed into R^(m>=3) with uniformly bounded energy and non-degenerating conformal structure. We deduce the strong compactness (modulo the action of the Moebius group) of closed Willmore surfaces of a given genus below some energy threshold. This is joint-work with Tristan Riviere (ETH Zuerich).
Minimal surfaces and complex analysis
12:10 Fri 24 Mar, 2017 :: Napier 209 :: Antonio Alarcon :: University of Granada

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A surface in the Euclidean space R^3 is said to be minimal if it is locally area-minimizing, meaning that every point in the surface admits a compact neighborhood with the least area among all the surfaces with the same boundary. Although the origin of minimal surfaces is in physics, since they can be realized locally as soap films, this family of surfaces lies in the intersection of many fields of mathematics. In particular, complex analysis in one and several variables plays a fundamental role in the theory. In this lecture we will discuss the influence of complex analysis in the study of minimal surfaces.
Geometric structures on moduli spaces
12:10 Fri 31 Mar, 2017 :: Napier 209 :: Nicholas Buchdahl :: University of Adelaide

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Moduli spaces are used to classify various kinds of objects, often arising from solutions of certain differential equations on manifolds; for example, the complex structures on a compact surface or the anti-self-dual Yang-Mills equations on an oriented smooth 4-manifold. Sometimes these moduli spaces carry important information about the underlying manifold, manifested most clearly in the results of Donaldson and others on the topology of smooth 4-manifolds. It is also the case that these moduli spaces themselves carry interesting geometric structures; for example, the Weil-Petersson metric on moduli spaces of compact Riemann surfaces, exploited to great effect by Maryam Mirzakhani. In this talk, I shall elaborate on the theme of geometric structures on moduli spaces, with particular focus on some recent-ish work done in conjunction with Georg Schumacher.
Hodge theory on the moduli space of Riemann surfaces
12:10 Fri 5 May, 2017 :: Napier 209 :: Jesse Gell-Redman :: University of Melbourne

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The Hodge theorem on a closed Riemannian manifold identifies the deRham cohomology with the space of harmonic differential forms. Although there are various extensions of the Hodge theorem to singular or complete but non-compact spaces, when there is an identification of L^2 Harmonic forms with a topological feature of the underlying space, it is highly dependent on the nature of infinity (in the non-compact case) or the locus of incompleteness; no unifying theorem treats all cases. We will discuss work toward extending the Hodge theorem to singular Riemannian manifolds where the singular locus is an incomplete cusp edge. These can be pictured locally as a bundle of horns, and they provide a model for the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Joint with J. Swoboda and R. Melrose.
Curvature contraction of axially symmetric hypersurfaces in the sphere
12:10 Fri 4 Aug, 2017 :: Engineering Sth S111 :: James McCoy :: University of Wollongong

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We show that convex surfaces in an ambient three-sphere contract to round points in finite time under fully nonlinear, degree one homogeneous curvature flows, with no concavity condition on the speed. The result extends to convex axially symmetric hypersurfaces of S^{n+1}. Using a different pinching function we also obtain the analogous results for contraction by Gauss curvature.
Conway's Rational Tangle
12:10 Tue 15 Aug, 2017 :: Inkgarni Wardli 5.57 :: Dr Hang Wang :: School of Mathematical Sciences

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Many researches in mathematics essentially feature some classification problems. In this context, invariants are created in order to associate algebraic quantities, such as numbers and groups, to elements of interested classes of geometric objects, such as surfaces. A key property of an invariant is that it does not change under ``allowable moves'' which can be specified in various geometric contexts. We demonstrate these lines of ideas by rational tangles, a notion in knot theory. A tangle is analogous to a link except that it has free ends. Conway's rational tangles are the simplest tangles that can be ``unwound'' under a finite sequence of two simple moves, and they arise as building blocks for knots. A numerical invariant will be introduced for Conway's rational tangles and it provides the only known example of a complete invariant in knot theory.
Measuring the World's Biggest Bubble
13:10 Tue 19 Sep, 2017 :: Napier LG23 :: Prof Matt Roughan :: School of Mathematical Sciences

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Recently I had a bit of fun helping Graeme Denton measure his Guinness World Record (GWR) Largest (Indoor) Soap Bubble. It was a lot harder than I initially thought it would be. Soap films are interesting mathematically -- in principle they form minimal surfaces, and have constant curvature. So it should have been fairly easy. But really big bubbles aren't ideal, so measuring the GWR bubble required a mix of maths and pragmatism. It's a good example of mathematical modeling in general, so I thought it was worth a few words. I'll tell you what we did, and how we estimated how big the bubble actually was. Some links: http://www.9news.com.au/good-news/2017/08/02/13/44/adelaide-man-wins-world-record-for-largest-bubble http://www.abc.net.au/news/2017-08-03/science-performer-creates-worlds-largest-indoor-soap-bubble/8770260
Twisted K-theory of compact Lie groups and extended Verlinde algebras
11:10 Fri 12 Oct, 2018 :: Barr Smith South Polygon Lecture theatre :: Chi-Kwong Fok :: University of Adelaide

In a series of recent papers, Freed, Hopkins and Teleman put forth a deep result which identifies the twisted K -theory of a compact Lie group G with the representation theory of its loop group LG. Under suitable conditions, both objects can be enhanced to the Verlinde algebra, which appears in mathematical physics as the Frobenius algebra of a certain topological quantum field theory, and in algebraic geometry as the algebra encoding information of moduli spaces of G-bundles over Riemann surfaces. The Verlinde algebra for G with nice connectedness properties have been well-known. However, explicit descriptions of such for disconnected G are lacking. In this talk, I will discuss the various aspects of the Freed-Hopkins-Teleman Theorem and partial results on an extension of the Verlinde algebra arising from a disconnected G. The talk is based on work in progress joint with David Baraglia and Varghese Mathai.

Publications matching "K3 surfaces: a crash course"

Publications
Topological chaos in flows on surfaces of arbitrary genus
Finn, Matthew; Thiffeault, J, XXII International Congress of Theoretical and Applied Mechanics, Adelaide 24/08/08
Algebraic deformations of compact kahler surfaces II
Buchdahl, Nicholas, Mathematische Zeitschrift 258 (493–498) 2008
Holomorphic classification of four-dimensional surfaces in C3
Beloshapka, V; Ezhov, Vladimir; Schmalz, G, Izvestiya Mathematics 72 (413–427) 2008
Algebraic deformations of compact Khler surfaces
Buchdahl, Nicholas, Mathematische Zeitschrift 253 (453–459) 2006
Lifting surfaces with circular planforms
Tuck, Ernest; Lazauskas, Leo, Journal of Ship Research 49 (274–278) 2005
Predicting the off-site deposition of spray drift from horticultural spraying through porous barriers on soil and plant surfaces.
Mercer, G; Roberts, Anthony John, 22nd Mathematics-In -Industry Study Group, Auckland, New Zealand 24/01/05
Monads and bundles on rational surfaces
Buchdahl, Nicholas, Rocky Mountain Journal of Mathematics 34 (513–540) 2004
Compact Khler surfaces with trivial canonical bundle
Buchdahl, Nicholas, Annals of Global Analysis and Geometry 23 (189–204) 2003
Ruled cubic surfaces in PG(4, q), Baer subplanes of PG(2, q2) and Hermitian curves
Casse, Rey; Quinn, Catherine, Discrete Mathematics 248 (17–25) 2002
Reanalysis of Travelling Speed and the Risk of Crash Involvement in Adelaide South Australia
Kloeden, Craig; McLean, Alexander; Glonek, Garique,
A Nakai-Moishezon criterion for non-Khler surfaces
Buchdahl, Nicholas, Annales de L Institut Fourier 50 (1533–1538) 2000
Numerical design tools for thermal replication of optical-quality surfaces
Stokes, Yvonne, Computers & Fluids 29 (401–414) 2000

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