Abstract: In this talk, we will quickly review some basic facts about test functions. Then we will introduce definitions of distribution and differentiation of distributions. Besides the abstract part, we will also calculate some concrete examples, in particular, the Gauss--Green formula, showing the power of the notion of distribution.

Reference: Lars Hörmander, The Analysis of Linear Partial Differential Operators I, Ch. I-III.

Abstract: We will discuss a classical algebro-geometric result, namely: there are exactly 27 projective lines lying on any smooth cubic surfaces in P^3.

Abstract: This topic will cover some interesting fact about homotopy type. Our goal is to find out how critical points influence manifold structure; also we will talk about some interesting result after it.

Abstract: This is the first talk on Morse theory. We will show that how can critical points of a smooth function determine the CW-structure of a manifold.

Abstract: In this topic, we will discuss the singularities of ordinary linear differential equation. Use Frobenius method based on the indicial equation to find possible solutions, and we will talk about singularity at infinity.

Abstract: In this talk we will give an overview of topological groups and compact groups. We will mainly discuss about compact abelian groups and their structure.

Abstract: This is the first of a series of talks. We will outline the Arakelov intersection theory of regular surfaces and give the details in subsequent talks.

Abstract: This is the first talk of a series of my talk concerning the intertwining between algebraic topology and smooth manifold. In this talk, we will focus on de Rham theory. We will rapidly introduce the concept of de Rham / compactly supported cohomology and Mayer--Vietoris sequence, then compute some concrete example. We will also generalise the Mayer--Vietoris argument to obtain Poincaré Duality and some other application. If the time permitted, we will also discuss on the cohomology on vector bundle and the Thom isomorphism.

Abstract: The main goal of this talk is to follow [1] to prove the main asymptotic existence theorem. This theorem gives an important reduction for ODEs with irregular singular points to some special cases (which will be dealt in the subsequent talks). If time permits, we shall also follow [2] to investigate the (growth) orders of solutions to ODEs with singular points, as another application of the main asymptotic existence theorem.

References: [1] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Interscience Publishers, 1965. Section 6-12, 14, 18.
[2] Y. Sibuya, Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation, American Mathematical Society, 1990. Chapter 5.

Abstract: In this talk, we will explain Eilenberg and MacLane's acyclic model theorem, which (and the proof of which) can be used to obtain several foundational results in algebraic topology. We will also compute the singular homology of several typical topological spaces, and then go over a proof of the generalized Jordan separation theorem and Brouwer's invariance of domain.

Abstract: In the beginning I will talk about the basic concept of differential operator. Before I talk the pseudo-differential operator, I will talk what Sobolev space is, without proofs. After pseudo-differential operator, I will define and talk about some properties of elliptic differential operator.
If the time is enough, I will precisely explain the proof of some properties of the Sobolev space.

Abstract: In this talk, we will introduce the concept of distributions with compact support, convolution of distributions, and Fourier transform of distributions. Then we will calculate the Fourier transform of Gaussian function. If time still permits, we will prove the central limit theorem as an application.

Abstract: Basics of algebraic groups, comparisons between algebraic groups and Lie groups, and algebraic group structure on coset spaces.

Abstract: In this talk I will talk about some object in Riemannian geometry. We will focus on variation field especially in first and second variation formula. After that we will talk about some interesting result using these two formulas.

Abstract: This is the second talk on Morse theory. We will show that any manifold has a Morse function, and hence has the homotopy type of a CW-complex which can be written down with respect to critical points of the function. Moreover, if the manifold is a complex algebraic variety, the CW-structure can be even finer.

Abstract: In this topic, we will discuss some properties of the hypergeometric equation. Using Riemann's P-Function, we will talk about Kummer's Twenty-four Series and the behavior of hypergeometric equations in some special cases.

Abstract: In this talk we are discussing the locally compact Abelian (LCA) groups. After applying Haar measure on such groups, we will look into the Fourier transform and dual group on them. The main goal is to study the structure of dual groups.

Abstract: We will use Green's functions to define the arithmetic intersection number and prove some proposition of it. We will see that Green's functions assign each Arakelov divisor an admissible bundle structure. After this, we will prove the Faltings' Riemann-Roch theorem.

Abstract: This is the continuation of my last talks. This time, we will discuss the cohomology on vector bundles, and discuss some of their properties. In particular, we will prove Thom isomorphism theorem, and discuss the connection between Poincaré dual and Thom class. Furthermore, we will construct some explicit examples for Thom class of vector bundle of rank 2. If the time is permitted, we will discuss some basic definitions in generalized Mayer-Vietoris sequence and Čech cohmology.

Abstract: There are two goals in this talk. First, we shall follow [1] to prove a structure theorem for solutions to ODEs with irregular singular points, with the aid of the main asymptotic existence theorem discussed in the previous talk of this series. Second, we shall follow [2] to investigate the (growth) orders of solutions to ODEs with singular points, with the aid of cyclic vectors.

References: [1] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Interscience Publishers, 1965. Section 6-12, 14, 18.
[2] Y. Sibuya, Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation, American Mathematical Society, 1990. Chapter 5.

Abstract: This talk will cover the following topics:
1. Existence of faithful linear representations of algebraic groups.
2. Algebraic variety structure on G/H for algebraic groups H≤G --- We shall exhibit Chevalley's theorem.
3. The Jordan Decomposition in an algebraic group --- Some basic notions of algebraic tori will be introduced.
4. Structures of solvable algebraic groups and Borel subgroups --- We will prove that every Borel subgroup is conjugate to each other, and whose quotient space is a projective variety.

References: Onishchik and Vinberg, Lie Groups and Algebraic Groups, Sections 3.1, 3.2.

Abstract: 1. Algebraic part. We will talk about the Künneth formula which relates the homology groups of the tensor product and the tensor product of the homology groups of two given chain complexes. We will also briefly introduce materials of derived functors which are needed.
2. Topological part. We will exhibit the Eilenberg--Zilber theorem, which establishes a functorial chain equivalence between the tensor product of the singular chain complexes of two given topological spaces and the singular chain complex of their product space. We will also compute several examples as an application.

Abstract: In this speech we will talk about some analysis on compact complex manifold. We shall assume that everyone knows what the elliptic differential operator is and I will give some property of the elliptic differential operator which is important to the main theorem of this speech with rough proof.
The following is the main topic: elliptic complex, fundamental form and Hodge operator, harmonic form on complex manifold.
If the time is enough I will prove some of properties of elliptic differential operator.

Abstract: In this talk, our main goal is to prove the stationary phase formula. We first calculate the special case formula for Gaussian function, and then prove the general case. As an application, we will give an elegant proof of Stirling's formula.

Abstract: In this speech, we talk about the index theorem in order to determine the value of the index of energy function. After that we study the topological space by constructing a finite dimensional one approximating it.

Abstract: We will recall the Lefschetz hyperplane theorem and prove the second theorem. The proof will involve usage of the results in Morse theory (which were mentioned in the previous 2 talks), blow-up of varieties, and some techniques in topology.

Reference: The second Lefschetz theorem on hyperplane sections, by A. Andreotti and T. Frankel, in Global Analysis: Papers in Honer of K. Kodaira, University of Tokyo Press. Edited by D.C. Spencer and S. Iyanaga. pp.1-20.

Abstract: In this topic, we will discuss the different type of Lamé equation. Use them to conclude the period of solution function and the relation between different solutions.

Abstract: In this last talk, we will continue our study on Euler class and global angular form (concrete example of Thom Class of rank 2). After that, we will introduce Čech-De Rham complex and generalised Mayer--Vietoris sequence. Then, we will define the concept of presheaves and Čech cohomology. If the time permitted, we will talk about sphere bundle and revisit the global angular form.

Abstract: There are two main goals in this speech:
1. To give an introduction of the dual group and combine the topological structure (locally compactness) and the group structure, which is the continuation of the previous speech.
2. To sketch the proof of the Pontryagin duality theorem (I will give some details if we have enough time).

Main reference: Walter Rudin, Fourier Analysis on groups, section 1.2, 1.3, 1.5-1.7.