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个人经历 Personal experience 工作经历 教育经历 2017.8至今 西北工业大学应用数学系 助理教授 2016.11--2017.5 哥伦比亚大学数据实验室 访问 2015.9--2016.9 香港中文大学数学系 博士后 2010.8--2015.7 清华大学数学系 博士2006.8--2010.7 清华大学数理基科班 学士

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教育教学 Education and teaching 教育教学 2020年秋 西北工业大学 本科生3学分课程 概率论与数理统计 选课人数1922019年秋 西北工业大学 本科生3学分课程 概率论与数理统计 选课人数2202018年秋 西北工业大学 本科生3学分课程 概率论与数理统计 选课人数2352016年春 香港中文大学 本科生3学分英文课程 Introductory Probability 选课人数60

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综合介绍 General Introduction Updated 2021/9/13We are happy to announce the launch of our Math101 Seminars in fall 2021. This is a series of seminars aimed at discussing mathematical analysis and advanced algebra of undergraduate level on a weekly basis. While we now have independent topics for each session, in a later stage we might choose to follow some book chapters. The seminar is open to public. Feel free to join us!#Math101 Session 2Time：Sep. 15, 2021, 13:30--15:30Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Xu, Jiaru (Class of 2024)Title：Rolle's theorem and zeros of generalized polynomialsAbstract：The fundamental theorem of algebra is undoubtedly one of the most famous existence theorems which shows that any non-zero polynomial has a number of zeros at most equal to its degree. When it comes to positive zeros, the basic result is known as Descartes' rule of signs. In this talk, we recollect the classical Rolle's theorem by which we derive Descartes' rule with its generalizations. Bounds of number of zeros of Dirichlet polynomials will be obtained in terms of sign changes in the sequence of coefficients and, in a finer degree, the sequence of their partial sums. Snacks and drinks will be provided.#Math101 Session 1Time：Sep. 8, 2021, 13:30--15:30Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Shi, Mingfei (Class of 2024)Title：Continuity and intermediate value property of functionsAbstract：There had been two myths surrounding the essence of continuous functions before the last quarter of 19th century, one being that a continuous function must be differentiable at a large set of points, and the other being that the intermediate value property was, in fact, equivalent to continuity. Weierstrass in 1872 and Darboux in 1875 proved that was not the case and constructed counter examples respectively. In this talk, we present two proofs of Darboux’s theorem, which says that any derivative has the intermediate value property, although the derivative need not be continuous. Snacks and drinks will be provided.AcknowledgmentsThe refreshments in #RAFS and #Math101 are sponsored, respectively, by Prof. Niu, Pengcheng and a math competition mentors project which is led by Prof. Ma, Rong. The support is gratefully acknowledged.AlumniClass of 2021   Mr. Gao, ChaoqunClass of 2022   Mr. Li, Yulong   Mr. Yang, Rui   Mr. Wu, Qingtong Archive for Past Seminars -- 2020-2021 Real Analysis Friday Seminars#RAFS Concluding Session 19 (Faculty Talk)Time：Jun. 18, 2021, 10:00--12:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Prof. Niu, PengchengTitle：The extremum with constraintAbstract：Let f be a smooth function defined on Euclidean space and let S be a smooth surface in the space. It is well known that a necessary condition for a point x belonging to S to be a local extremum of f|S is that the tangent space to the surface S at x is contained in the tangent space to the corresponding level surface. This is the essence of the method of Lagrange multipliers. We will review the powerful result and explore some of its applications.This is the last session of the current two-year series of seminars discussing mathematics at undergraduate level. A new round resumes in September 2021.#RAFS Session 18Time：Jun. 11, 2021, 10:00--12:00Location：434 Qi Xiang Building, Chang'an CampusTitle：The Poincare recurrence theoremAbstract：In his 1890 King Oscar prize-winning memoir, H. Poincare established the basic principle that the typical trajectory of a volume-preserving system with finite volume comes back infinitely often to any neighborhood of its initial point. This is considered to have initiated the modern study of measure-preserving transformations, or ergodic theory. In this talk, by an argument that is essentially the same as the Pigeonhole Principle, we will prove two versions of the recurrence theorem, by which we conclude our yearlong discussion of Measure and Category.#RAFS Session 17Time：May 16, 2021, 15:30--17:30Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Li, Yulong (Class of 2022)Title：Category analogue of Fubini’s theoremAbstract：We discussed last time the usual measure version of Fubini’s theorem, which says that almost all vertical sections of a two-dimensional Lebesgue nullset have one dimensional Lebesgue measure zero. In this talk, we are concerned with a nice category analogue, the Kuratowski-Ulam theorem (1932). Among other things, we will show that the x-section of a plane set of first category is a linear set of first category for all x except a set of first category.#RAFS Session 16 (Featuring a Guest Talk)Time：May 9, 2021, 15:30--17:30Location：434 Qi Xiang Building, Chang'an CampusSpeaker 1：Mr. Zhang, Jinhan (Aeronautics, Class of 2021)Title 1：A quick tour of fluid mechanics: theory and engineeringAbstract 1：All liquids and gases are fluids: they are substances that flow or deform under the application of certain stress. In this talk, we first show how to apply the basic laws of classical physics to fluid mechanics and establish the governing equations. Then we briefly talk about turbulence and computational fluid dynamics, as well as the difference between theory and practice. Finally, cavity flow, a flow that has been studied extensively in theory and engineering and that may cause noise and structural fatigue problems, will be introduced.Speaker 2：Mr. Yang, Rui (Class of 2022)Title 2：Fubini’s TheoremAbstract 2：The most powerful tool for the evaluation of a double integral is Fubini’s theorem, which enables us to only consider iterated integrals. Thus the measure of any plane measurable set can be expressed as the integral of the linear measure of its sections perpendicular to an axis. In this talk, we confine attention to the case of nullset; we show that almost all vertical sections of a nullset have measure zero.#RAFS Session 15Time：Apr. 23, 2021, 10:00--12:00Location：434 Qi Xiang Building, Chang'an CampusTitle：Absolute G-deltaAbstract：We realize the importance of G-delta sets on several occasions—e.g., the set of points of continuity of a real-valued function is a G-delta, and a set E in Baire space is residual if and only if E contains a dense G-delta subset. In this talk, complete metrizability, or topological completeness, proves to serve as a third source. We will show that any non-empty G-delta subset of a complete metric space is topologically complete and, conversely, if a set X of a metric space Z is topologically complete, then X is a G-delta subset of Z, thus identifying the key property of an absolute G-delta, which in turn is a basic notion in descriptive set theory.#RAFS Session 14Time：Apr. 16, 2021, 10:00--12:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Li, Yulong (Class of 2022)Title：Everywhere continuous nowhere differentiable functions, revisitedAbstract：We discussed last year in this seminar (#MATS Session 3) pathological functions which are everywhere continuous nowhere differentiable. In particular, we presented three kinds of explicit examples with this property: the classical Weierstrass function (1872), Takagi function (1903) & van der Waerden function (1930), and Swift function (1961). In this talk, we will be using the category method on which an existence proof is based. We will show that, in the sense of category, almost all continuous functions are nowhere differentiable; in fact, the set of all continuous functions that have a finite one-sided derivative somewhere is of first category.  #RAFS Session 13Time：Mar. 26, 2021, 10:00--12:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Gao, Chaoqun (Class of 2021)Title：Examples of metric spacesAbstract：We will investigate three examples of metric spaces: the set of real-valued continuous functions endowed with the uniform metric, the set R of Riemann integrable functions on an interval with the usual pseudo metric, and the class S of sets of finite measure m with the symmetric difference metric (Fréchet-Nikodym metric). We will discuss whether these spaces are complete or not, and remark that there exists an isometric embedding of R in S, when m is taken to be the 2-dimensional Lebesgue measure in the plane. In particular, it is possible to identify the closure of R in S with the space of Lebesgue integrable functions, thus providing one motivation for enlarging the class of integrable functions.#RAFS Session 12Time：Mar. 19, 2021, 10:00--12:00Location：434 Qi Xiang Building, Chang'an CampusTitle：Metric and topological spacesAbstract：The usefulness of the theory of category only becomes fully apparent in more general spaces, especially metric spaces. In this talk, we first recollect basic notions and properties of metric and topological spaces. Then we prove the Baire category theorem for topologically complete metric space, which by definition is homeomorphic to some complete space (however, the space itself may not be complete). The audience will be reminded that the result is unusual in that the hypothesis involves a uniform property but the conclusion is of a purely topological property. Finally, we introduce the second category everywhere space or Baire space, and present an equivalent condition for its subset being residual (co-meager).#RAFS Session 11Time：Mar. 12, 2021, 10:00--12:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Yang, Rui (Class of 2022)Title：Revisiting Littlewood's principlesAbstract：The workshop resumes this coming Friday morning in continuation to our discussion of measure and category. In this talk, we will revisit two of the three principles proposed by J.E. Littlewood (1885-1977): every (measurable) function is nearly continuous, and every convergent sequence is nearly uniformly convergent, a.k.a. Theorems of Lusin and Egorov. We first show the proof of the theorems with the help of equivalent topological definitions of continuity and measurability, then we discuss their counterparts in the sense of category and try to construct counterexamples when certain conditions are violated.#RAFS Session 10Time：Jan. 8, 2021, 15:00--17:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Gao, Chaoqun (Class of 2021)Title：Lebesgue's criterion for Riemann integrabilityAbstract：As a consequence of Baire’s category theorem, we have found conditions under which the set D of points of discontinuity of a function is of first category. In a parallel manner, it is natural to inquire under what conditions D is a nullset; hence comes the famous integrability criterion due to Lebesgue which says a function is integrable on any finite interval if and only if it is bounded and continuous almost everywhere. We will prove this theorem in light of oscillation of the function on subdivisions of the interval. Snacks and drinks will be provided. #RAFS Session 9Time：Nov. 27, 2020, 16:00--18:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Gao, Chaoqun (Class of 2021)Title：Functions of first class of BaireAbstract：It is well known that the limit of a uniformly convergent sequence of continuous functions is continuous. Yet the pointwise limit of a convergent sequence of continuous functions, or the so-called Baire class 1 function, need not be continuous, as shown by simple examples. However, this kind of function cannot be everywhere discontinuous. We will prove that a Baire class 1 function is continuous except at a set of points of first category. Snacks and drinks will be provided. #RAFS Session 8Time：Nov. 20, 2020, 16:00--18:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Li, Yulong (Class of 2022)Title：The Banach-Mazur GameAbstract：In this talk, we are mainly concerned with a mathematical game that was proposed by S. Mazur in 1935 and first appeared in “Scottish book”, the world-renowned collection of problems. It involves two players and is believed to be a first example of infinite positional game. Surprisingly, the solution to this game is related to Baire category theorem. We will first define the game and understand what is meant by strategy. Then we show an equivalent condition in terms of category for one player to be sure to win, that is, having a strategy. This gives new insight into the sense in which a set of first category is small. Snacks and drinks will be provided.  #RAFS Session 7Time：Oct. 30, 2020, 16:00--18:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Gao, Chaoqun (Class of 2021)Title：The property of BaireAbstract：We have come to the stage where we could make a first comparison of category and measure: the property of Baire is a property analogous to the measurability of a set, sets of first category could play the role of Lebesgue null sets. In this talk, we focus on the class of sets having the property of Baire, and try to study in a parallel manner with the class of measurable sets. Despite the evident difference as illustrated by the example that a null set could be of second category, we remark that both of the two classes include the Borel sets, and each is invariant under translation. Snacks and drinks will be provided.#RAFS Session 6Time：Oct. 23, 2020, 16:00--18:00Location：434 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Gao, Chaoqun (Class of 2021)Title：Lebesgue's one-dimensional density theoremAbstract：The study of densities appears naturally in geometric measure theory, a subject where rigorous proofs are often intricate and difficult. In this talk, we work on the real line and first define the density of a measurable set E at some point x, then we prove the classical Lebesgue density theorem (1904), which tells us how much of the set E, in the sense of Lebesgue measure, is concentrated near x. The main tool is a Vitali-type covering theorem. Snacks and drinks will be provided.#RAFS Session 5Time：Oct. 16, 2020, 16:00--18:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Wu, Qingtong (Class of 2022)Title：Liouville numbersAbstract：The study of Liouville numbers plays an important role in the transcendence theory and diophantine approximation, which in turn is widely recognized as a first window to metric number theory. In this talk, we first define Liouville numbers and prove that every Liouville number is transcendental, then we show that the set of Liouville numbers is of zero measure and its complement is a countable union of nowhere dense sets, thus providing another partition of R which consists of two “small” sets in the sense of measure and category, respectively. Snacks and drinks will be provided.#RAFS Session 4Time：Oct. 4, 2020, 16:00--18:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Wu, Qingtong (Class of 2022)Title：Measure and category on the real lineAbstract：We will be talking about Oxtoby’s book from this week onward. Its main subject is twofold, the Baire category theorem with various applications and the duality between measure and category, both of which are based on the notion of countability, thus making us start with Cantor’s theorem which says no interval of real numbers is countable. We then explore two more existence theorems of Baire and Borel, and finally construct a set on the real line that is small in one sense but large in the other sense. Snacks and drinks will be provided.#RAFS Session 3Time：Sep. 25, 2020, 16:00--18:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Yang, Rui (Class of 2022)Title：Littlewood’s three principlesAbstract：The British mathematician J.E. Littlewood (1885-1977) once proposed three principles as guide for working in real analysis: every (measurable) set is nearly a finite sum of intervals, every (measurable) function is nearly continuous, and every convergent sequence is nearly uniformly convergent. As Littlewood himself wrote, “it is natural to ask if the ‘nearly’ is near enough, and for a problem that is actually soluble it generally is.” In this talk we will discuss these three “nearly” and try to construct counter-examples when certain conditions are violated.#RAFS Session 2Time：Sep. 18, 2020, 15:00--17:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Yang, Rui (Class of 2022)Title：Revisiting introductory real analysis (Part II)Abstract：This is the second half of a 4-hour presentation which serves as a quick overview of a standard real analysis course taught here at NWPU. #RAFS Session 1Time：Sep. 11, 2020, 15:00--17:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Li, Yulong (Class of 2022)Title：Revisiting introductory real analysis (Part I)Abstract：This is the first half of a 4-hour presentation which serves as a quick overview of a standard real analysis course taught here at NWPU. Archive for Past Seminars -- 2019-2020 Mathematical Analysis Tuesday Seminars#MATS Final Session 10Time：Jan. 10, 2020, 15:00--17:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker 1：Mr. Xiong, Jing (Class of 2020)Title 1：A study on Bayesian statistics and frequentist statisticsAbstract 1：In the field of statistical inference, 19th century science was broadly Bayesian in its methodology, while frequentism dominated 20th century scientific practice. The aim of this talk is to compare these two schools by first presenting a basic overview of Bayesian and frequentist statistics as well as the related background of the 270-year debate between them. Then we construct examples from common scenarios including point estimation, interval estimation and hypothesis testing and use specific questions and data to draw a comparison of the corresponding results.Speaker 2：Mr. Yao, Jiahao (Class of 2021)Title 2：Kakeya problem in finite fieldsAbstract 2：The Japanese mathematician Soichi Kakeya posed his famous question in 1917, which could be expressed as follows: a line segment with length 1 moves continuously by rotation and translation on the plane, turns 180 degrees and returns to its original position, then what is the minimum area that has been swapt? Since the work of Besicovitch, Kakeya sets have been found to be related to many other mathematical fields, including harmonic analysis and geometric measure theory. In this talk we are going to present Zeev Dvir's stunningly simple proof in 2008 which gives a full affirmative answer to the Kakeya conjecture in finite fields.#MATS Session 9Time：Jan. 3, 2020, 9:00--11:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Yang, Rui  (Class of 2022)Title：Differentiation of functions Abstract：We are to discuss Chapter 10 of Tao's Analysis I. This includes basic definitions, local maxima/minima and derivatives, monotone functions and derivatives, inverse functions and derivatives, and the well-known L'Hopital's rule.#MATS Session 8Time：Dec. 20, 2019, 9:00--11:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Li, Yulong (Class of 2022)Title：Continuous functions on R (Part II) Abstract：We continue to discuss Chapter 9 of Tao's Analysis I. This includes the intermediate value theorem, monotonic functions, uniform continuity, and limits at infinity.#MATS Session 7Time：Dec. 13, 2019, 9:00--11:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Li, Yulong (Class of 2022)Title：Continuous functions on R (Part I)Abstract：We are to discuss Chapter 9 of Tao's Analysis I. This includes the intermediate value theorem, monotonic functions, uniform continuity, and limits at infinity.***Fall Break***#MATS Session 6 (Guest Talk)Time：Oct. 29, 2019, 13:30--15:30Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Miss Li, Zishang (Class of 2021)Title：From Riemann integral to Henstock integralAbstract：Despite being the primary integration technique taught at undergraduate level, the theory of Riemann integral turns out to be not fully satisfactory. To overcome the drawbacks, in this talk we will define and investigate properties of the Henstock integral, which is a generalization of Riemann's in the sense that in the construction of the filter base with respect to Riemann sum, we change the restriction for parameter of the partition from being less than a constant to some gauge function. This talk has no prerequisite for Lebesgue measure theory. #MATS Session 5Time：Oct. 22, 2019, 15:00--17:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Yao, Jiahao (Class of 2021)Title：Limit along a filter baseAbstract：Our interest stems from the definition of Riemann integral, which is well known to be defined as the limit, if it exits, of the Riemann sum. However this limit seems to be neither a standard sequence limit nor function limit. In this talk we will define and investigate the limit along a filter base, which was introduced in the 1930s by the famous group of (mainly) French mathematicians under the collective pseudonym Nicolas Bourbaki, and which offers a very useful type of convergence to help fully understand the limit of Riemann sum. Properties in parallel with those of the standard limit will also be discussed.#MATS Session 4Time：Oct. 15, 2019, 15:00--17:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Yang, Rui (Class of 2022)Title：An incomplete list of completeness axioms of R (Part II)Abstract：We first present the well-known Theorems of Cauchy-Cantor, Borel-Lebesgue, and Bolzano–Weierstrass, all of which are classical equivalent completeness axioms of the real numbers. Another several propositions regarding existence of suprema/infima, connectedness, as well as convergence of sequences will also serve as members of this very much incomplete list. We will show the proof and briefly bring the role of Archimedean Property to the audience's attention, which is less discussed in the curriculum of mathematics department.#MATS Session 3Time：Oct. 8, 2019, 13:30--15:30Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Xiong, Jing (Class of 2020)Title：Everywhere continuous nowhere differentiable functionsAbstract：Most of the mathematical world had believed that a continuous function could only fail to be differentiable at some collection of isolated points, until Weierstrass published his shocking article in 1872. In this talk, we are mainly concerned with these pathological functions, i.e. everywhere continuous but nowhere differentiable ones. We will discuss three kinds of functions along with the proof of their nowhere differentiability, including the classical Weierstrass function and those presented by Takagi (1903) & van der Waerden (1930), and Swift (1961). #MATS Session 2Time：Sep. 17, 2019, 15:00--17:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Yang, Rui (Class of 2022)Title：An incomplete list of completeness axioms of RAbstract：We first present the well-known Theorems of Cauchy-Cantor, Borel-Lebesgue, and Bolzano–Weierstrass, all of which are classical equivalent completeness axioms of the real numbers. Another several propositions regarding existence of suprema/infima, connectedness, as well as convergence of sequences will also serve as members of this very much incomplete list. We will show the proof and briefly bring the role of Archimedean Property to the audience's attention, which is less discussed in the curriculum of mathematics department.#MATS Session 1Time：Sep. 3, 2019, 15:00--17:00Location：447 Qi Xiang Building, Chang'an CampusSpeaker：Mr. Li, Yulong (Class of 2022)Title：Rolle's Theorem and zeros of generalized polynomialsAbstract：We recollect the classical Rolle's Theorem first and derive Descartes' rule of signs with its generalizations. Bounds of number of zeros of Dirichlet polynomials will be obtained in terms of sign changes in the sequence of coefficients, as well as, in a finer degree, the sequence of their partial sums. My field of interest consists of fractal geometry and dynamic systems.You are very welcome to contact me by shuangshen@nwpu.edu.cn for research, teaching, or even just coffee. 个人相册

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