2024 Summer Course on Computational Conformal Geometry
Computational Conformal Geometry Summer Course【Lecturer】Professor Gu Xianfeng
Tsinghua University Yau Shing-tung Mathematical Sciences Center
State University of New York at Stony Brook [Course Time] 2024/06/29-2024/08/31
【Meeting link】https://meeting.tencent.com/dm/xVg3FxVhZUbW
[Live broadcast at Station B] http://online.conformalgeometry.org
[Koushare Live Broadcast] https://www.koushare.com/lives/room/35633
With the development of the times and the advancement of science and technology, modern topological geometry theory has increasingly penetrated into the industrial and medical fields more widely and deeply. In order to comply with the requirements of the times, Mr. Qiu Chengtong began calling on mathematicians from all over the world to give lectures in China decades ago. Since 2012, the author has taught the course "Computational Conformal Geometry" at the Yau Shing-tung Mathematics Center of Tsinghua University every summer, targeting scholars and students with mathematical and engineering backgrounds. During the epidemic, courses were moved online, and the number of students participating in the courses increased unprecedentedly. To this day, whether the author travels in China, North America or Europe, I will frequently meet young people who have studied the "Computational Conformal Geometry" course. Most of them are developing in academia or high-tech fields. In the past two decades, under the leadership of Mr. Yau Shing-tung, through the joint efforts of many scholars in the fields of basic mathematics and computer science, the theory of computational conformal geometry has gradually developed from its prototype to maturity, and the algorithm has become increasingly practical after being tempered by various applications. We look forward to the next ten years when computational conformal geometry will truly enter the industry on a large scale, substantially promote the development of engineering and medical fields, and contribute to the country and society.
In recent years, the new energy automobile industry is rising, and generative artificial intelligence is also developing rapidly. Our curriculum has also been adjusted to respond to the needs of the times. The automotive industry relies on industrial software, and the core of industrial software requires a lot of in-depth modern topological geometry theory. The author has cooperated deeply with Cadence's R&D department to overcome many basic problems in the field of multi-physics numerical simulation. Mr. Lu Chuanlin, Cadence R&D Director of Multiphysics System Analysis, believes that the key to multiphysics numerical simulation lies in the geometry rather than the numerical solution. In particular, how to generate high-quality meshes is the core of the entire system. Dr. Si Hang, a Chinese leader in the field of international mesh generation, believes that the difficulty of mesh generation lies in the generation of high-quality surface meshes. Volume mesh generation will encounter various topological obstacles, and its theoretical foundation is not yet mature. The generation of unstructured meshes for curved surfaces requires the use of surface singularization theorem and Ricci flow theory, while the generation of structured meshes for curved surfaces relies on the Abel-Jacobi theory and Riemann-Roch theory on Riemann surfaces, and the generation of volume meshes requires the use of Thurston's three-dimensional manifold geometry theory. Generative artificial intelligence requires the use of optimal transmission theory, Minkowski-Alexandrov theory in convex differential geometry. In this year's course, we will cover these basic theories. In order to apply these theories to engineering practice, we will also introduce the generalization of these theories to discrete manifolds and the corresponding algorithms and direct applications.
Modern geometric theory mainly studies various types of manifolds.structure, and between various structuresrelationship. The main research tools include combinatorial, algebraic, analytical and geometric methods. Various geometric realities can be defined on each structure, corresponding invariants can be calculated, and are restricted by other structures and also affect other structures. For different structures, humans have invented different theories and calculation tools, as well as different research techniques, and have different direct applications. Fixed the underlying structure, and the set of all possible high-level structures of the same type constitutes the so-called modulus space. The structure of the modulus space is crucial. The course "Computational Conformal Geometry" mainly explains the topological structure, differential structure, conformal structure, Riemannian metric structure, and isometric embedding of surfaces. It also involves the topological structure and Riemannian metric structure of three-dimensional manifolds.
Topology
Topological structure is an invariant structure under the action of topological homeomorphism group. The surface case is very simple, and the three-dimensional manifold case is extremely complex. The core difficulty comes from the fact that humans are creatures in three-dimensional Euclidean space and cannot directly experience and imagine geometry in higher-dimensional spaces. Most three-dimensional manifolds cannot be realized (equidistance) in three-dimensional Euclidean space. The topological invariants of the entire system of surfaces are orientability, genus and number of boundaries. The topological invariants of three-dimensional manifolds are difficult to describe in one word. The most important invariant is the standard metric (that is, using geometry to express and study topology).
Combination methodThere are two main types: one is decomposition and the other is surgery. The decomposition method divides the entire manifold into simple units, expresses each unit, and records the pasting methods between different units. Topological surgery is to remove some areas from the current manifold, then select another manifold whose boundary submanifold has the same topology as the boundary submanifold of the removed area, then select a homeomorphism between the two boundary manifolds, and paste the second manifold to the first manifold along the boundary. Common decomposition methods include triangulation of surfaces and tetrahedron decomposition of three-dimensional manifolds. The trouser decomposition of the surface is also a common method, that is, the surface is decomposed into trouser legs (zero genus surface with three boundaries). A three-manifold with edges is generally decomposed into a truncated tetrahedron (that is, a tetrahedron with four vertices cut off). The orientable surface can also be decomposed into the topological sum of topological torus, that is, the surface is divided along some loops, and each connected branch is a surface with genus 1. Similarly, non-orientable surfaces can be decomposed into the topological sum of projective surfaces. We can decompose the three-manifold along some essential spheres into the topological sum of so-called prime manifolds. The so-called prime manifolds are branches that cannot be divided by spheres to obtain simpler topological branches. Milnor proved that this decomposition is unique. However, the prime three-dimensional manifold is still very complex. We need to further decompose the prime manifold along the essential torus, and finally obtain that each branch is a three-manifold bounded by (multiple) topological torus. This is the so-called JSJ decomposition (William Jaco, Peter Shalen and Klaus Johannson decomposition). Thurston's geometrization theorem means that each connected branch finally obtained can be matched with one of the eight standard geometries. The cutting surface here is often incompressible, that is, if a circle on the surface can be reduced to a point in the three-manifold, then the circle can also be reduced to a point on the surface. If there is an orientable, regularly embedded, incompressible surface inside a three-manifold, the manifold is called Haken. We can decompose the Haken manifold along the incompressible surface, and each connected branch can be decomposed recursively until it cannot be decomposed, thus obtaining the so-called Haken decomposition (Haken). Another commonly used decomposition method is CW-cell decomposition, that is, starting from discrete points to obtain a 0-dimensional skeleton, and pasting a (k+1)-dimensional topological disk on the k-dimensional skeleton at each step, so that the boundary of the topological disk is on the k-dimensional skeleton. Similarly, we define the direct product of the (k-1)-dimensional sphere and the (n-k)-dimensional topological disk as a k-ring handle, and the topological ring handle is pasted into a so-called handle body through topological surgery. A closed three-manifold can be divided into two handles by cutting a curved surface. This decomposition method is called Heegaard Splitting.
On the contrary, we can paste the two handle bodies along the boundary surface to obtain a closed 3-manifold. In this case, the homeomorphic Tonglun class between the boundaries determines the topology of the 3-manifold. Furthermore, according to Mostow's rigidity theorem, the geometric structure of a 3-manifold is generally determined by the geometric structure on its boundary surface, and the geometric structure of a closed 3-manifold is determined by its topological structure. This means that once we determine the homotopy class of two handles and their boundary mappings, then the standard metric on the boundary surface has been determined, and this Riemannian metric is preserved by the boundary mapping. This is actually Thurston's surface mapping classification theory, which requires the use of the concept of surface mapping groups.
The 3-manifold Dehn operation refers to digging out a solid tire inside the 3-manifold, then cutting the solid tire into a solid cylinder, twisting the cylinder along the central axis several times, and then gluing it back into a solid tire (similar to the process of frying twists), and then gluing the solid tire back to the original 3-manifold. In this way we change the topology of the original 3-manifold. We start from the 3-dimensional sphere and dig out some knots. These knots are connected to each other to form links. We perform Dhen surgery along each knot. The distortion is represented by a rational number, and all 3-manifolds can be constructed. And this chain link, together with the twisted representation on each knot, forms the so-called Kirby Diagram. This is a universal representation of 3-dimensional manifolds. This transformation between representations is performed by the so-called Kirby Calculus.
It can be seen that the combined decomposition of manifolds is flexible and changeable, and different applications require different decomposition methods. The key to JSJ decomposition and Haken decomposition is to find incompressible surfaces. Heegaad Splitting needs to find invariant metrics, and the algorithm complexity involved is often very high. This is exactly where the technical bottleneck of current industrial software lies.
algebraic methodIt is to define various groups or rings on the manifold. The topological properties of the manifold are reflected in the structure of the group, that is, various functors are established between the topological category and the algebraic category.
The most critical algebraic topology method is to establish the basic group, that is, the non-commutative group formed by the homotopy class of all closed curves on the manifold, connected end to end. The basic group is represented by its generators and relational elements, which completely reflects the structure of the surface and also determines the topology of most 3-manifolds. For example, Poincaré's theorem states that if the fundamental group of a compact 3-manifold is mediocre, then the manifold is a 3-dimensional sphere; and if two knots are of the same trace, they must be isomorphic if and only if their complementary space fundamental groups are isomorphic. The incompressible surface inside the 3-manifold corresponds to the subgroup of the fundamental group of the 3-manifold. For example, if there is an incompressible torus inside a 3-manifold, then there is a two-dimensional Abelian subgroup in its fundamental group. The subgroup conjugate class of a manifold corresponds to the isomorphism class (Galois correspondence) of the manifold covering space. In particular, the identity element corresponds to the universal covering space of the manifold. All subhomeomorphisms that keep the projection mapping unchanged in the universal overlay space form the so-called deck mapping group. The topological meaning of the universal overlapping space is the space composed of all road homotopy classes. By lifting the manifold problems to the universal overlapping space, many topological calculations can be simplified. The decomposition and operation of manifolds correspond to the transformation of the fundamental group, which is characterized by the Seifert-van Kampen theorem. Given the CW-cell decomposition of a manifold, a representation of the fundamental group can be written directly. For the complementary space of knots, there are also simple algorithms that directly write the representation of the basic group. However, the same basic group may have infinitely many different representations, and it is exponentially complex to verify that two representations are equivalent to each other through symbolic calculations. Therefore, in practical applications, geometric methods are still relied on.
The one-dimensional homology group is the Abelianization of the fundamental group, so it can be calculated using linear algebra methods. The intuitive meaning of hypohomology group is "the difference between circle and edge", that is, the difference between proper chain and closed chain. If we have a triangulation of a manifold, we can use edge operators and integer matrix operations to obtain hypohomology groups in different dimensions. On the surface, the homology group and the fundamental group give the same information, but on the 3-manifold, the homology group is much weaker than the fundamental group. In applications, we often consider that the surface is embedded inside the 3-manifold, so that the selection of the base of the homology group of the surface is related to the background 3-manifold, such as the ring handle ring and the tunnel ring on the surface in Euclidean space. These can be calculated using persistent coherence methods.
In order to study the topology of 3-manifolds, we need to study the surface mapping class. The generator of the surface mapping group is Dehn Twist: select a closed curve on the surface, cut the surface along the closed curve, fix one port, twist the other port counterclockwise, and then glue it back. Thurston proved that all surface maps can be divided into three categories: periodic maps, Pseudo-Anosov maps and reducible maps. The finite power of the periodic map is equal to the identity map, which maintains the unique hyperbolic metric on the surface; the Pseudo-Anosov map maintains the two families of leaf-like structures on the surface; the reducible map maintains a set of closed circles on the surface (possibly rearranging these circles), and is limited to the complementary space of these circles, which can be classified into the first two categories. When gluing 3-manifolds, we need to consider the classification of boundary gluing maps and the corresponding invariant metrics.
geometric methodIt is to study topological manifolds with standard Riemannian metrics. The singularization theorem of the surface asserts that we can equip the surface with a measure of constant curvature, so that its universal covering space can be equivalently embedded in the unit sphere (zero genus), the Euclidean plane (one genus) or the hyperbolic disk (high genus) according to the genus of the surface. The deck mapping group of the universal covering space is expressed as a subgroup of the equivalent transformation groups of the three standard spaces, thereby converting the topological problem into a group theory problem. Similarly, Thurston's geometrization theorem asserts that the manifold obtained after JSJ decomposition can be equipped with one of the eight standard geometries, and the same picture exists.
differential structure
A differential structure of a manifold can be regarded as an atlas, and all local coordinate transformations are smooth, so that traditional calculus concepts can be extended to manifolds. The tangent vector field of the manifold is defined on the differential structure. The isolated zero point of the smooth tangent vector field can define the index, that is, draw a small circle around the zero point and rotate the tangent vector several times along the small circle. The Poincare-Hopf theorem states that the total index of a smooth tangent vector field is equal to the Euler characteristic number of the surface, which gives the intrinsic relationship between differential structure and topological structure.
Similarly, we can define the so-called Morse function on the manifold, that is, a second-order smooth function with an isolated singular point. The number of negative eigenroots of the Hessain matrix at the singular point is defined as an indicator of the singular point. Smale-Morse theory points out that according to the indicators of singular points, we can determine the topological type of the level set on the manifold. This also gives the intrinsic relationship between differential structure and topological structure.
Based on the differential structure, we can define the dual of the tangent vector field: the differential form field, thereby establishing the so-called external differential operator. External differential operators are direct high-dimensional generalizations of classic differential operators, such as gradient, spinor and divergence operators. The image of the external differential operator is called the proper differential form, and the core of the external differential operator is called the closed differential form. The difference between the closed differential form and the proper differential form gives the de Rham cohomology group. Although cohomology is more intuitive and easy to understand, de Rham cohomology is more suitable for calculating various differential invariants, and external differentials are a powerful tool for studying differential geometry.
We regard the union of the tangent spaces at each point of the manifold as a whole and give it a natural topology. In this way, we get the concept of fiber bundles, and the tangent vector field is an integral cross-section of the tangent bundle. If we examine the fiber bundle composed of unit tangent vectors, then we obtain the unit tangent bundle of the manifold. The zero point of the tangent vector field becomes an obstacle to the existence of the overall section of the unit tangent bundle. Mr. Chen Shengshen's theory of representative classes points out that each fiber bundle class corresponds to a top-dimensional cohomology class on the bottom manifold, which is the so-called Chen class. If the fundamental group of fibers is Abelian, then Chern class theory can be applied directly.
The tangent spaces at two different points on the manifold are different spaces, and there is no direct connection between them. We draw a path connecting these two points, and then define a so-called connection structure, which can move the tangent vector parallel along the path, thereby establishing a connection between the two tangent spaces. Contacts can be expressed in differential form, and parallel movement boils down to solving ordinary differential equations. A path is geodesic if its tangent vector fields are parallel along the path. For one parallel movement along a closed curve, the difference between the initial tangent vector and the final tangent vector is one rotation, which gives the intrinsic curvature of the surface. The external differentiation of the associated differential forms gives the curvature differential form, which gives the descriptive class of unit tangent bundles. Generalizing this picture to 3-manifolds, we have the Chern-Simons theory.
In engineering, industrial software is based on spline theory, which is based on the affine structure of curved surfaces. If there is an affine structure on the surface, we can construct a special connection so that its curvature is zero everywhere, and thus the closed surface must be a torus. In other words, generally closed surfaces do not have an affine structure, so it is inherently difficult to form a smooth spline everywhere. This is also one of the inherent difficulties in industrial software.
conformal structure
The conformal structure of a surface can be viewed as an atlas, and the transformation between local coordinates is holomorphic (complex analytic). Conformal structures do not have the concept of length, but they have the concept of angle. A surface with a conformal structure is called a Riemann surface. The basic concepts of complex functions can be directly extended to the Riemann surface, such as holomorphic and meromorphic functions, holomorphic and meromorphic differentials, etc. The concepts of harmonic function and harmonic differential in harmonic analysis can also be defined on the Riemannian surface. These concepts all have some kind of global rigidity, that is, the properties of the function are determined by the local parts of several points as a whole. The mathematical characterization of this rigidity is the core of the Riemann surface theory.
For example, under conformal structures we can define the concept of harmonic differential forms. Hodge's theory asserts that in each de Rham cohomology class, there is a unique harmonic differential form. Hodge theory connects topological structures with conformal structures (or more broadly, Riemannian metric structures). Its further generalization is the so-called Atiya-Singer index theorem: the dimensionality of the solution space of elliptical partial differential equations on a Riemannian manifold is closely related to the topology of the manifold. Our previous work extended the Hodge decomposition theory on smooth manifolds to discrete manifolds and developed an algorithm for discrete Hodge decomposition, which essentially extended the finite element method to the field of external differentials.
For example, according to the maximum principle, holomorphic functions on the compact Riemannian surface must be constants; on the complex plane, bounded holomorphic functions must be constants, etc. There are infinitely many meromorphic functions on the Riemann surface. Meromorphic functions are determined by their zero points and poles. The positions of the zero and pole points are not arbitrary, but are determined by the Abel-Jacobi theorem. Given the positions of some zeros and poles, all meromorphic functions that satisfy the conditions form a linear space, and the dimension of the space is determined by the Riemann-Roch theorem. The Riemann-Roch theorem can be regarded as the intrinsic relationship between the conformal structure and the topological structure of the manifold.
All meromorphic functions on the compact Riemannian surface form an algebraic domain, and two compact Riemannian surfaces are conformally equivalent if and only if their meromorphic function domains are isomorphic to each other. We can also compute the conformal invariants of compact Riemann surfaces if the conformal invariants of two Riemann surfaces are the same if and only if they are conformally equivalent to each other. The more common conformal invariants are the periodic matrix of the Riemann surface and its conformal module. Between the conformally inequivalent Riemannian surfaces, there is a unique Teichmuller mapping for the Tonglun class of fixed mappings, which minimizes the angular distortion. The Teichmuller map determines the two holomorphic quadratic differentials on the source surface and the target surface. It maps the zero point of the holomorphic quadratic differential to the zero point, and the horizontal and vertical trajectories to the horizontal and vertical trajectories. At the same time, if we use the flatness measurement induced by the holomorphic quadratic differential to measure, the Teichmuller map is stretching in the horizontal direction and compressing in the vertical direction. The stretching ratio is exactly equal to the compression ratio.
In the sense of conformal equivalence, there are only three types of simply connected Kaiyenne Riemannian surfaces: unit sphere, complex plane and complex unit disk, which correspond to the universal overlapping spaces of 0 genus, 1 genus and high genus compact Riemann surfaces. This is the unique value theorem in the conformal sense. This is equivalent to the fact that we can equip the compact Riemann surface with three standard constant curvature measures.
Standard metrics also give different structures of surfaces, for example the hyperbolic metric of highly genus surfaces gives complex projective structures. The real projective structure on the surface is determined by the holomorphic cubic differential of the surface. In many applications, projective structures play a crucial role.
We fix the topology of the Riemann surface and examine all possible conformal structures. All conformal structures constitute a finite-dimensional space: Teichmuller space. There is a unique Teichmuller map between any two conformal structures, which is determined by the only holomorphic quadratic differential. According to the Riemann-Roche theorem, the space composed of all holomorphic quadratic differentials is 6g-6 dimensions, where g is the genus of the surface and is simply connected, that is, a topological disk. We consider a ray in the space of holomorphic quadratic differentials and treat the ray as a point on the boundary of Teichmuller space, thus obtaining a compact Teichmuller space. Any automorphism of the Riemann surface will map the conformal structure of the Riemann surface to another conformal structure (pull back the conformal structure), that is, the surface mapping group acts on the compact Teichmuller space. According to Brouwer's fixed point theorem, the surface sub-mapping must have fixed points. If the fixed point is inside the Teichmuller space, the mapping is periodic; if the fixed point is on the boundary, the mapping is Pseudo-Anosov. Holomorphic quadratic differentials on the same ray, their horizontal and vertical trajectories induce the same horizontal and vertical foliation. Therefore, the Pseduo-Anosov mapping keeps the two sets of leaf structures unchanged. Each point inside the Teichmuller space uniquely corresponds to a standard Riemannian metric, that is, each conformal structure has a unique hyperbolic metric, which can be guaranteed by Yamabe equation theory or Ricci flow theory.
Riemannian metric structure
The so-called Riemannian metric defines an inner product on the tangent space of each point of the manifold. Given a connection and a metric, we say that they are compatible with each other if "parallel movement preserves the inner product", that is, we move two tangent vectors in parallel along a path at the same time. At every moment during the movement, the inner product of the two vectors remains unchanged. Such a connection is called a Levi-Civita connection and is uniquely determined by the metric, that is, the first-order differential of the metric gives the Levi-Civita connection. From this, we can determine curvature, geodesics, etc. from measurements. The famous Gauss-Bonnet-Chern theorem asserts that the total Gauss curvature induced by any Riemannian metric on a surface is a topological invariant, that is, the indicator number of the tangent bundle of the surface. This theorem gives the intrinsic connection between Riemannian metric structure and topological structure.
In turn, given the curvature, how do we get the corresponding Riemannian metric. On the surface, under the premise of maintaining the conformal structure, metric exists, which can be proved by the surface Ricci flow theory. In the 1980s, Hamilton proposed the Ricci flow method to prove the Poincaré conjecture and Thurston's geometrization conjecture, which is to deform the Riemannian metric so that its rate of change is proportional to the current Ricci curvature. Then, under appropriate conditions, the Riemannian metric will converge to a constant curvature metric. Obviously, the surface Ricci flow theory can derive the classic surface singularization theorem.
Our previous work extended the surface Ricci flow theory from the smooth surface case to the discrete polyhedral mesh, thus proving that the single-valued theorem still holds true in the discrete case. This algorithm can design discrete metrics based on discrete Gaussian curvature, thereby mapping all discrete surfaces conformally into three standard spaces, and converting 3D geometric calculation problems into plane problems. Discrete Ricci flow theory guarantees the existence, uniqueness and convergence of the solution. At the same time, the algorithm is based on a convex geometric variation problem and can be optimized by Newton's method. This is currently the only algorithm that can calculate the Riemannian metric from curvature.
Unlike surfaces, 3-manifolds have no hyperbolic structure. According to Mostow's rigidity theorem, the hyperbolic metric of a hyperbolic 3-manifold is uniquely determined by its topological structure (the same is true for high-dimensional cases). The proof of Mostow's rigidity theorem relies on the concept of simple volume proposed by Gromov. Simple volume comes from simple cohomology and is a topological quantity, but its value is equal to the hyperbolic volume of the manifold, that is, a geometric quantity. Simple volume is therefore seen as a natural extension of the Gauss-Bonnet-Chern theory.
After many years of hard work by many mathematicians, Hamilton, Perelman, and Mr. Qiu’s team, Thurston’s geometrization conjecture (which contains the Poincaré conjecture) was finally proved by the Ricci flow method. 3-After the manifold is decomposed by JSJ, each piece is a manifold bounded by a torus, and the most difficult one is the hyperbolic case. We once gave an algorithm to deal with the case where the boundary is a fully geodesic hyperbolic 3-manifold. For the complementary space of knots, Thurston gave perfect theoretical results.
Thurston proved that if the knot is not a satellite knot or a torus knot, then the complement space of the knot must be hyperbolic. We divide the knot complement space into truncated tetrahedrons, and the hyperbolic geometry of each tetrahedron is represented by a complex parameter. These tetrahedrons are perfectly spliced together, so that the sum of the dihedral angles around each edge must be the circumferential angle (curvature condition). At the same time, the boundaries of these truncated tetrahedrons form a topological torus with a flat metric, and the deck mapping of the curved surface is a translation. In this way, the flat ring conditions are obtained. Through these conditions, the complex parameters of the tetrahedron can be solved, thereby obtaining the hyperbolic metric of the three-manifold. From this, after Dehn's operation, we can obtain a wider range of hyperbolic three-manifolds.
For convex surfaces, the Riemannian metric induces isometric embedding in Euclidean space. The famous Minkowski-Alexandrov theory gives embedded theoretical results on how to reconstruct a surface given a Gaussian curvature defined on a sphere. From a computational perspective, these theorems are equivalent to the optimal transmission problem in Euclidean space and spherical space, and the optimization of Hilbert-Einstein functionals in metric space.
The discrete Ricci flow theory relies on optimizing the Hiltert-Einstein functional in the hyperbolic metric space of a three-dimensional manifold. We use the input triangular mesh to construct a hyperbolic three-manifold. This 3-manifold is divided by hyperbolic triangular prisms. Generally, it cannot be embedded in a three-dimensional hyperbolic space. After optimization, this 3-manifold can be embedded equidistantly in a three-dimensional hyperbolic space, and the result is a single-valued hyperbolic metric. Therefore, we see that computing conformal geometry requires the use of hyperbolic 3-manifold theory.
Summary
The topology and geometric theories of surfaces and 3-manifolds are not independent of each other, but constitute an organic whole. Many concepts and theorems depend on each other, confirm each other, support each other, and cannot be separated. We hope that this course can help everyone weave a knowledge network of low-dimensional topology and geometry, open their "eyes in the sky", improve their spatial imagination, enhance their geometric intuition, and reserve modern geometric and topological algorithms.
This year, the author plans to continue the online "Computational Conformal Geometry Course". The main reference textbooks for the course are "Computational Conformal Geometry - Theory" written by Mr. Qiu and the author and published by Higher Education Press, and a monograph on algorithms. The course starts on June 29th, every Monday, Wednesday, and Saturday evening, starting at 8:30 Beijing time, and lasts about one and a half hours. The Tencent conference number is 536-4137-8835, and the conference password is 467783. The live broadcast URL of Station B is: http://online.conformalgeometry.org. I look forward to learning and appreciating modern mathematics with all my classmates and preparing for the coming industrial revolution.
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Machine-translated from the Chinese original 2024年夏季《计算共形几何》课程预告 (published 2024-06-29). WeChat original ↗