清华大学清华学堂数学班2018届本科生毕业设计学术报告分享会

发布时间:2018-07-03

7月,毕业季!在众多学子们即将告别美丽的清华园之际,清华学堂数学班2018届部分毕业生将与大家一起分享他们的毕业设计学术报告及研究内容,并就前沿问题进行深入交流。欢迎您的参与!



1

主题

清华大学清华学堂数学班

2018届本科生毕业设计学术报告分享会

2

时间和地点:

2018年7月 5日(周四)

清华大学数学系A304 (上午8:50-12:00)

清华大学近春园西楼3层报告厅(下午1:30-5:00)

7  月  5  日  上  午

日程安排

第1场(8:50 - 9:20)

1

报告人:徐凯

2

指导教师:李思

3

题目

Factorization Algebras and Index Theorem

4

摘要

We show that local observables of the one-dimensional Chern-Simons theory encode the deformation quantization of a symplectic manifold and local-to-global map gives the trace map of quantum algebra using the homotopical renormalization machinery of [1] [?]. By this construction we are able to compute the algebraic index which is also the partition function of our theory in terms of characteristic classes.


Our approach is in the spirit of deformation quantization via Gelfand-Kazhdan formal geometry: we construct a quantization of one-dimensional Chern-Simons theory with an n-dimensional formal disk as target and glue it over a symplectic manifold via Gelfand-Kazhdan descent. Our argument is a formal version of [2] and can recover it via Gelfand-Kazhdan descent.

5

预备知识

linear algebra, smooth manifolds, cohomology

第2场(9:25 - 9:55)

1

报告人:赵瑞屾

2

指导教师:扶磊

3

题目

A short introduction to Iwasawa theory

4

摘要

We will first give an interesting motivation for Iwasawa theory, concerning about special value and analogy between number field and function field. Then, to make things easy and avoid too many machinery, we will mainly focus on Iwasawa theory for units. Via such way, we will establish p-adic zeta function and prove Iwasawa’ s theorem about units. After that, we discuss the relation with main conjecture. Finally, depend on time, we plan to show other aspect of Iwasawa theory and further development.

5

预备知识

Basic number theory, p-adic analysis, Galois cohomology.

第3场(10:30 - 11:00)

1

报告人:李理

2

指导教师:左怀青

3

题目

3-fold hypersurfaces in weighted projective spaces

4

摘要

In the lecture I will introduce the notions and algorithms of some birational invariants for 3-fold principal hypersurfaces in 4-dim weighted projective spaces with certain conditions. I also give the methods to determine the quotient cyclic singular type for each isolated singular point and give the criterion for terminality of quotient cyclic singular points using toric methods.

Preliminary: Algebraic geometry, mainly including notions of schemes and cohomology. It benefits if you are familiar with toric varieties but that’s not necessary.

第4场(11:05 - 11:35)

1

报告人:吴志翔

2

指导教师:Jean-Franscois Dat

3

题目

Fargues-Fontaine curve and vector bundles

4

摘要

We introduce a pair of adjoint functors: the ring of Witt vectors and the tilting. We show untilts of an algebraically closed perfectoid field of characteristic $p$ form a space which can be identified with a space of divisors of holomorphic functions of variable $p$. The structure can be used to prove the almost purity theorem of fields. We define the algebraic curve which is an analogue of projective lines and we present the classification theorem of vector bundles on the curve.

5

预备知识

abstract algebra

第5场(11:40 - 12:10)

1

报告人:张志宇

2

指导教师:扶磊

3

题目

Milnor K theory and Merkurjev-Suslin theorem

4

摘要

There is a deep connection between Milnor-K theory and Galois cohomology through Brauer groups. In this talk, we discuss Merkurjev-Suslin theorem which involves many different tools and is the first step to understand such connection. Besides, we discuss some applications of this theorem.

5

预备知识

Basic knowledge of Brauer group

7  月  5  日  下  午

日程安排

第6场(13:50 - 14:20)

1

报告人:贾楸烨

2

指导教师:林海

3

题目

Generalized Complete Intersection Calabi-Yau Manifolds

4

摘要

CY(Calabi-Yau) manifolds have many important applications in math and physics. P. Candelas proposed an approach to contruct such manifolds through complete intersection with respect to polynomials in projective spaces. Recently, L.B. Anderson and his cooperators brought up the ideal to replace those sections global over the entire background space by more local data,

involving techniques of sheaves and bundles.

We generalized a vanishing theorem of Bott, Deligne and others, which is a key step in the computation of the Hodgenumbers of CY manifolds. An example of gCICY is exploited with the help of it and other techniques.

In order to compute the change on Hodge numbers brought by blowing up, which is employed to remove singularities caused by group action quotient, we worked on genus of curves as well.

Moreover, we will introduce another approach: spectral sequence.

During construction, we find a way to prove a class identity claimed by P. Candelas.

5

预备知识

A basic knowledge on manifold and algebraic geometry

第7场(14:30 - 15:00)

1

报告人:薛宇皓

2

指导教师:吴云辉

3

题目

Eigenvalues of hyperbolic Riemann surfaces

4

摘要

I will recall some basic knowledge about Riemann surfaces first. And then I will discuss the estimations of the first 2g-2 eigenvalues of Laplacian on closed hyperbolic Riemann surfaces.

5

预备知识

complex analysis, basic Riemann geometry, Riemann surfaces if better

第8场(15:30 - 16:00)

1

报告人:王志涵

2

指导教师:徐国义

3

题目

Polynomial Growth of Minimal Graphs on R^n

4

摘要

One of the most classical problem in minimal surface theory and equation theory concerns the solutions of minimal surface equation in the whole R^n, a classical theorem of Bernstein states that for n = 2, such solution is always linear. While for higher dimension, some positive and negative results have been obtained in 1960-1980, though the general behavior for the minimal graphs (for n > 7) is, as far as the speaker is concerned, still not clear.

In this lecture, we give a precise review of classical Bernstein’s problem and some technical results in geometric measure theory. In particular, we provide an interior interpretation of the splitting of infinite tangent cone of a nontrivial minimal graph. We shall see how the critical dimension for the solution to be trivial arise in this way and which behaviors are still true for higher dimension.

5

预备知识

A basic knowledge on real analysis and differential geometry

第9场(16:10 - 16:40)

1

报告人:甘盛文

2

指导教师:Cyril Imbert

3

题目

Fully nonlinear elliptic equations

4

摘要

A brief discussion of: ABP estimate, Krylov-Safonov estimate, Evans-Krylov estimate, Caffarelli's W^{2, p} estimate, Caffarelli's C^{2, alpha}.

Also,we discuss the regularity theory of Monge-Ampere equations.

5

预备知识

Basic knowledge on real analysis and PDE

清华大学清华学堂数学班2018届本科生毕业设计学术报告分享会

发布时间:2018-07-03

7月,毕业季!在众多学子们即将告别美丽的清华园之际,清华学堂数学班2018届部分毕业生将与大家一起分享他们的毕业设计学术报告及研究内容,并就前沿问题进行深入交流。欢迎您的参与!



1

主题

清华大学清华学堂数学班

2018届本科生毕业设计学术报告分享会

2

时间和地点:

2018年7月 5日(周四)

清华大学数学系A304 (上午8:50-12:00)

清华大学近春园西楼3层报告厅(下午1:30-5:00)

7  月  5  日  上  午

日程安排

第1场(8:50 - 9:20)

1

报告人:徐凯

2

指导教师:李思

3

题目

Factorization Algebras and Index Theorem

4

摘要

We show that local observables of the one-dimensional Chern-Simons theory encode the deformation quantization of a symplectic manifold and local-to-global map gives the trace map of quantum algebra using the homotopical renormalization machinery of [1] [?]. By this construction we are able to compute the algebraic index which is also the partition function of our theory in terms of characteristic classes.


Our approach is in the spirit of deformation quantization via Gelfand-Kazhdan formal geometry: we construct a quantization of one-dimensional Chern-Simons theory with an n-dimensional formal disk as target and glue it over a symplectic manifold via Gelfand-Kazhdan descent. Our argument is a formal version of [2] and can recover it via Gelfand-Kazhdan descent.

5

预备知识

linear algebra, smooth manifolds, cohomology

第2场(9:25 - 9:55)

1

报告人:赵瑞屾

2

指导教师:扶磊

3

题目

A short introduction to Iwasawa theory

4

摘要

We will first give an interesting motivation for Iwasawa theory, concerning about special value and analogy between number field and function field. Then, to make things easy and avoid too many machinery, we will mainly focus on Iwasawa theory for units. Via such way, we will establish p-adic zeta function and prove Iwasawa’ s theorem about units. After that, we discuss the relation with main conjecture. Finally, depend on time, we plan to show other aspect of Iwasawa theory and further development.

5

预备知识

Basic number theory, p-adic analysis, Galois cohomology.

第3场(10:30 - 11:00)

1

报告人:李理

2

指导教师:左怀青

3

题目

3-fold hypersurfaces in weighted projective spaces

4

摘要

In the lecture I will introduce the notions and algorithms of some birational invariants for 3-fold principal hypersurfaces in 4-dim weighted projective spaces with certain conditions. I also give the methods to determine the quotient cyclic singular type for each isolated singular point and give the criterion for terminality of quotient cyclic singular points using toric methods.

Preliminary: Algebraic geometry, mainly including notions of schemes and cohomology. It benefits if you are familiar with toric varieties but that’s not necessary.

第4场(11:05 - 11:35)

1

报告人:吴志翔

2

指导教师:Jean-Franscois Dat

3

题目

Fargues-Fontaine curve and vector bundles

4

摘要

We introduce a pair of adjoint functors: the ring of Witt vectors and the tilting. We show untilts of an algebraically closed perfectoid field of characteristic $p$ form a space which can be identified with a space of divisors of holomorphic functions of variable $p$. The structure can be used to prove the almost purity theorem of fields. We define the algebraic curve which is an analogue of projective lines and we present the classification theorem of vector bundles on the curve.

5

预备知识

abstract algebra

第5场(11:40 - 12:10)

1

报告人:张志宇

2

指导教师:扶磊

3

题目

Milnor K theory and Merkurjev-Suslin theorem

4

摘要

There is a deep connection between Milnor-K theory and Galois cohomology through Brauer groups. In this talk, we discuss Merkurjev-Suslin theorem which involves many different tools and is the first step to understand such connection. Besides, we discuss some applications of this theorem.

5

预备知识

Basic knowledge of Brauer group

7  月  5  日  下  午

日程安排

第6场(13:50 - 14:20)

1

报告人:贾楸烨

2

指导教师:林海

3

题目

Generalized Complete Intersection Calabi-Yau Manifolds

4

摘要

CY(Calabi-Yau) manifolds have many important applications in math and physics. P. Candelas proposed an approach to contruct such manifolds through complete intersection with respect to polynomials in projective spaces. Recently, L.B. Anderson and his cooperators brought up the ideal to replace those sections global over the entire background space by more local data,

involving techniques of sheaves and bundles.

We generalized a vanishing theorem of Bott, Deligne and others, which is a key step in the computation of the Hodgenumbers of CY manifolds. An example of gCICY is exploited with the help of it and other techniques.

In order to compute the change on Hodge numbers brought by blowing up, which is employed to remove singularities caused by group action quotient, we worked on genus of curves as well.

Moreover, we will introduce another approach: spectral sequence.

During construction, we find a way to prove a class identity claimed by P. Candelas.

5

预备知识

A basic knowledge on manifold and algebraic geometry

第7场(14:30 - 15:00)

1

报告人:薛宇皓

2

指导教师:吴云辉

3

题目

Eigenvalues of hyperbolic Riemann surfaces

4

摘要

I will recall some basic knowledge about Riemann surfaces first. And then I will discuss the estimations of the first 2g-2 eigenvalues of Laplacian on closed hyperbolic Riemann surfaces.

5

预备知识

complex analysis, basic Riemann geometry, Riemann surfaces if better

第8场(15:30 - 16:00)

1

报告人:王志涵

2

指导教师:徐国义

3

题目

Polynomial Growth of Minimal Graphs on R^n

4

摘要

One of the most classical problem in minimal surface theory and equation theory concerns the solutions of minimal surface equation in the whole R^n, a classical theorem of Bernstein states that for n = 2, such solution is always linear. While for higher dimension, some positive and negative results have been obtained in 1960-1980, though the general behavior for the minimal graphs (for n > 7) is, as far as the speaker is concerned, still not clear.

In this lecture, we give a precise review of classical Bernstein’s problem and some technical results in geometric measure theory. In particular, we provide an interior interpretation of the splitting of infinite tangent cone of a nontrivial minimal graph. We shall see how the critical dimension for the solution to be trivial arise in this way and which behaviors are still true for higher dimension.

5

预备知识

A basic knowledge on real analysis and differential geometry

第9场(16:10 - 16:40)

1

报告人:甘盛文

2

指导教师:Cyril Imbert

3

题目

Fully nonlinear elliptic equations

4

摘要

A brief discussion of: ABP estimate, Krylov-Safonov estimate, Evans-Krylov estimate, Caffarelli's W^{2, p} estimate, Caffarelli's C^{2, alpha}.

Also,we discuss the regularity theory of Monge-Ampere equations.

5

预备知识

Basic knowledge on real analysis and PDE

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