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van Kampen’s theorem via covering spaces
If is a connected topological manifold, and is a point in , the (topological) fundamental group of at is traditionally defined as the space of equivalence classes of loops starting and ending at , wit… more →
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van Kampen's theorem via covering spaces
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wrote 6 months ago: If is a connected topological manifold, and is a point in , the (topological) fundamental group of a … more →
Cayley graphs and the algebra of groups
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wrote 1 year ago: This is a sequel to my previous blog post “Cayley graphs and the geometry of groups“. In … more →
The Hilbert-Smith conjecture
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wrote 1 year ago: The classical formulation of Hilbert’s fifth problem asks whether topological groups that have … more →
The Guth-Katz bound on the Erdős distance problem
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wrote 2 years ago: Combinatorial incidence geometry is the study of the possible combinatorial configurations between g … more →
Mazur's swindle
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wrote 3 years ago: Let be a natural number. A basic operation in the topology of oriented, connected, compact, -dimensi … more →
At the AustMS conference
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wrote 3 years ago: This week I was in my home town of Adelaide, Australia, for the 2009 annual meeting of the Australia … more →
Recent progress on the Kakeya conjecture
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wrote 4 years ago: Below the fold is a version of my talk “Recent progress on the Kakeya conjecture” that I … more →
Cohomology for dynamical systems
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wrote 4 years ago: A dynamical system is a space X, together with an action of some group . [In practice, one often pl … more →
The Kakeya conjecture and the Ham Sandwich theorem
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wrote 4 years ago: One of my favourite family of conjectures (and one that has preoccupied a significant fraction of my … more →
285G, Lecture 19: The structure of Ricci flow at the singular time, surgery, and the Poincaré conjecture
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wrote 4 years ago: In the previous lecture, we studied high curvature regions of Ricci flows on some time interval , an … more →
Outside in
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wrote 5 years ago: Given that this blog is currently being devoted to a rather intensive study of flows on manifolds, I … more →
上同调论讲义习题（林金坤）Chapter 2
wrote 5 years ago: Exercise 1:Able群G叫做可除的，如果对任意 和每个正整数, 存在 使得 .证明：若B可除，则. Proof.只需要证明，给定A的自由分解 ，则 也是正合列即可。而由本章定理，这只要证明 … more →
上同调论讲义习题（林金坤）Chapter 2
wrote 5 years ago: Exercise 1:Able群G叫做可除的，如果对任意 和每个正整数, 存在 使得 .证明：若B可除，则. Proof.只需要证明，给定A的自由分解 ，则 也是正合列即可。而由本章定理，这只要证明 … more →
上同调论讲义（林金坤）Chapter 1
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wrote 5 years ago: 学数学不做题目，如入宝山而空返。——华罗庚 我们的很多博士生已经不做题了。这大概是目光短浅的一个体现。他们觉得做题耗费时间，可是时间长了，差距就慢慢的造成了。这学期我选学林金坤的上同调论的课程，说心里 … more →
285G, Lecture 5: Finite time extinction of the third homotopy group, I
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wrote 5 years ago: In the previous lecture, we saw that Ricci flow with surgery ensures that the second homotopy group … more →
285G, Lecture 4: Finite time extinction of the second homotopy group
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wrote 5 years ago: Returning (perhaps anticlimactically) to the subject of the Poincaré conjecture, recall from Lecture … more →
285G, Lecture 2: The Ricci flow approach to the Poincaré conjecture
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wrote 5 years ago: In order to motivate the lengthy and detailed analysis of Ricci flow that will occupy the rest of th … more →
Equivariant DeRham Cohomology
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wrote 5 years ago: My lectures on equivariant cohomology are spinning a bit out of control. The questions and live … more →
The Cohomology of Quotients
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wrote 5 years ago: We’ve organized a mostly informal Topics in Noncommutative Algebra seminar this semester, … more →
