Jekyll2023-02-10T01:03:44-05:00https://www.math.columbia.edu/~axu/feed.xmlColumbia MathematicsAlex Xu's Columbia websiteAlex Xuaxu[at]math.columbia.eduApplications and Computations of Floer Homology Learning Seminar2023-01-16T00:00:00-05:002023-01-16T00:00:00-05:00https://www.math.columbia.edu/~axu/seminars/floer-homology-seminar<ul>
<li>Time : Thursdays 12-1pm</li>
<li>Location : Mathematics 417</li>
<li>Organizers: Alex Xu</li>
</ul>
<p>Monopole Floer and Heegard Floer homology are essential tools in low dimensional topology and have been crucial to many breakthroughs in the last 20 years. However, this machinery is rather complex and in practice many computations will use formal properties, such as surgery exact sequences, homology pairing formulas, or formal dimension computations. The goal of the seminar is not to prove these results, but rather to use these formal properties as a black box to do concrete computations and applications of Floer homology.</p>
<p>The format is that every week someone presents a paper/preprint on some explicit computation or an application of Floer homology. Following is a tentative list of topics; of course participants are free to present anything on applications of Floer homology (broadly understood).</p>
<p>Tentative topics:</p>
<ul>
<li>Computations of Seiberg-Witten invariants for various 4-manifolds (e.g. elliptic fibrations)</li>
<li>Floer homology detects the Thurston norm</li>
<li>Monopole Floer homology detects the unknot</li>
<li>Computations of the Floer homology of branched double covers</li>
</ul>
<h3 id="seminar-schedule">Seminar Schedule</h3>
<table>
<thead>
<tr>
<th>Week/Date</th>
<th>Speaker</th>
<th>Title</th>
<th>Abstract</th>
</tr>
</thead>
<tbody>
<tr>
<td>2/16</td>
<td> </td>
<td> </td>
<td><a href="https://arxiv.org">ArXiv Link</a></td>
</tr>
<tr>
<td>2/23</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>3/2</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>3/9</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>3/16</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>3/23</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>3/30</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>4/6</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>4/13</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>4/20</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>4/27</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>Alex Xuaxu[at]math.columbia.eduTime : Thursdays 12-1pm Location : Mathematics 417 Organizers: Alex XuFall 2022 Scalar Curvature and Topology Student Learning Seminar2022-09-05T01:00:00-04:002022-09-05T01:00:00-04:00https://www.math.columbia.edu/~axu/seminars/scalar-curvature-seminar-post<ul>
<li>Time : Monday 6-8 PM</li>
<li>Location : Room 507</li>
<li>Organizers: Aaron Chow, Shuang Liang, Jingbo Wan, Alex Xu</li>
</ul>
<p>Scalar curvature is the least understood of all the various notions of curvature: bounded below scalar curvature can tell us something about the topology of the manifold, but a classification of such manifolds remains elusive. And many of techniques used to study them come from many different areas of geometric analysis, such as stable minimal hypersurfaces, spin geometry, harmonic maps, and geometric flows, and in many cases it is unclear whether these techniques are related on some fundamental level or if this is just a strange coincidence. The goal of this seminar is to survey some of these techniques as well as some applications and recent results, using some <a href="https://arxiv.org/abs/1908.10612">notes</a> by Gromov as a starting point and branching out based off of participant interest.</p>
<h3 id="seminar-schedule">Seminar Schedule</h3>
<table>
<thead>
<tr>
<th>Week/Date</th>
<th>Speaker</th>
<th>Title</th>
<th>Abstract</th>
</tr>
</thead>
<tbody>
<tr>
<td>Sept 19</td>
<td>Jingbo Wan</td>
<td>Stern’s Bochner formula on compact three-manifolds</td>
<td>We give the basic ingredients for Stern’s Bochner formula and apply this formula to harmonic maps \(M^3 \rightarrow S^1\) where \(M\) is a compact three-manifold with or without boundary. This gives a beautiful inequality relating the average Euler characteristics of harmonic map’s level sets and the scalar curvature of \(M\). References: <a href="https://arxiv.org/abs/1908.09754">arxiv.org/abs/1908.09754</a>, <a href="https://arxiv.org/abs/1911.06803">arxiv.org/abs/1911.06803</a></td>
</tr>
<tr>
<td>Sept 26</td>
<td>Shuang Liang</td>
<td>Kähler-Ricci flow and estimates</td>
<td>We will cover some basic analytic results concerning the evolution of various curvature tensors along the Kähler-Ricci flow. References: <a href="https://arxiv.org/abs/1508.04823">arxiv.org/abs/1508.04823</a>, <a href="https://arxiv.org/abs/1212.3653">arxiv.org/abs/1212.3653</a></td>
</tr>
<tr>
<td>Oct 3</td>
<td>Shuang Liang</td>
<td>Convergence of Kähler-Ricci flow and examples</td>
<td>We aim to prove the convergence of the Kähler-Ricci flow when the first Chern class is negative or zero. If time allows, we will consider the case when the canonical bundle is nef and big and talk about the formulation of singularities. References: Same as last week</td>
</tr>
<tr>
<td>Oct 10</td>
<td>Nikita Klemyatin</td>
<td>The Kähler-Ricci flow on Fano manifolds</td>
<td>Let \((X, \omega)\) be a compact Kähler manifold with \(c_1(X) > 0\). Such manifolds are called Fano manifolds. In this case, there are several obstructions to the existence of the Kähler-Einstein metrics and the convergence and other results are much more involving than in the case of nonpositive $c_1(X)$. We will discuss the basics about Fano manifolds, together with examples of manifolds that do not admit KE metrics. After that, we will study the normalized Kähler-Ricci flow on Fano manifolds, such as the long time existence and preservation of positivity of the bisectional curvature along the NKRF. Reference: <a href="https://link.springer.com/book/10.1007/978-3-319-00819-6">https://link.springer.com/book/10.1007/978-3-319-00819-6</a></td>
</tr>
<tr>
<td>Oct 17</td>
<td>Nikita Klemyatin</td>
<td>Perelman’s uniform estimates and convergence</td>
<td>This talk is the continuation of the previous one. We will prove the uniform diameter and scalar curvature estimates along the normalized Kähler-Ricci flow. If time permits, we will discuss some applications of this result. Reference: <a href="https://www.cambridge.org/core/journals/journal-of-the-institute-of-mathematics-of-jussieu/article/bounding-scalar-curvature-and-diameter-along-the-kahler-ricci-flow-after-perelman/1793B91D3C3C403219D28058A8C7EBBB">Bounding Scalar Curvature and Diameter along the Kähler Ricci Flow</a></td>
</tr>
<tr>
<td>Oct 24</td>
<td>Aaron Chow</td>
<td>Dirac type operators on complete Riemannian manifolds</td>
<td>This talk will be a preparation to Gromov-Lawson’s proof on the nonexistence of PSC metrics on enlargeable manifolds. We will begin with a crash course on Dirac type operators and derive the Weitzenbock formula. After that we will study analytic properties of Dirac type operators on complete Riemannian manifolds. Reference: <a href="https://link.springer.com/article/10.1007/BF02953774">link.springer.com/article/10.1007/BF02953774</a></td>
</tr>
<tr>
<td>Oct 31</td>
<td>Aaron Chow</td>
<td>Nonexistence of PSC metrics on enlargeble manifolds</td>
<td>We will start by studying the relative index of Dirac type operators on complete Riemannian manifolds. The goal of this talk is to go through Gromov-Lawson’s proof on the nonexistence of PSC metrics on enlargeable manifolds. Reference: <a href="https://link.springer.com/article/10.1007/BF02953774">link.springer.com/article/10.1007/BF02953774</a></td>
</tr>
<tr>
<td>Nov 7</td>
<td> </td>
<td>No seminar due to university holiday</td>
<td> </td>
</tr>
<tr>
<td>Nov 14</td>
<td>Aaron Chow</td>
<td>Nonexistence of PSC metrics on enlargeble manifolds 2</td>
<td>We continue studying the relative index of Dirac type operators on complete Riemannian manifolds and go through Gromov-Lawson’s proof on the nonexistence of PSC metrics on enlargeable manifolds. Reference: <a href="https://link.springer.com/article/10.1007/BF02953774">link.springer.com/article/10.1007/BF02953774</a></td>
</tr>
<tr>
<td>Nov 21</td>
<td>Alex Xu</td>
<td>Introduction to Seiberg Witten theory</td>
<td>This talk will be a crash course on the various aspects of Seiberg-Witten theory most applicable to geometric analysis. We first setup the Seiberg-Witten equations on a closed orientable 4-manifold \(X\) and discuss how existence of irreducible solutions leads to apriori estimates on the total scalar curvature of \(X\). We follow up by defining the Seiberg-Witten invariant and some gluing formulas. If time permits, we discuss the situation on Kähler surfaces where the equations are much more concrete and exhibit some computations of the Seiberg-Witten invariant. References: <a href="https://press.princeton.edu/books/paperback/9780691025971/the-seiberg-witten-equations-and-applications-to-the-topology-of">John Morgan. The Seiberg-Witten equations and applications to the topology of smooth four-manifolds</a></td>
</tr>
<tr>
<td>Nov 28</td>
<td>Alex Xu</td>
<td>Total scalar curvature and Seiberg Witten theory</td>
<td>In the 90’s Witten noted that solutions to the Seiberg Witten equations gave apriori estimates on the total scalar curvature \(\int_X s^2\). LeBrun later used this observation to prove uniqueness of the Einstein metric on compact quotients of \(\mathbb{CH}^2\) and nonexistence of Einstein metrics on a certain class of manifolds. Applying these estimates to exotic pairs of 4-manifolds leads to the striking observation that \(inf \int_X s^2\) depends on the smooth structure of \(X\). The goal of this talk is to survey these results and the techniques used to derive them. References: <a href="https://arxiv.org/abs/dg-ga/9411005">arxiv.org/abs/dg-ga/9411005</a>, <a href="https://arxiv.org/abs/dg-ga/9511015">arxiv.org/abs/dg-ga/9511015</a>, <a href="https://arxiv.org/abs/math/0003068">arxiv.org/abs/math/0003068</a></td>
</tr>
<tr>
<td>Dec 5</td>
<td>Jingbo Wan</td>
<td>A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions</td>
<td>Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically ﬂat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. Geometrically and very roughly, this is analogous to the topological restriction of a stable minimal surface in positively curved 4-manifold. In this reading seminar talk, the speaker is going to talk about Galloway and Schoen’s generalization of Hawking’s theorem to any dimensional Spacetime satisfying the dominant energy condition, asserting that outer apparent horizon is Yamabe positive, except some very special cases. Reference: <a href="https://link.springer.com/article/10.1007/s00220-006-0019-z">A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions</a></td>
</tr>
<tr>
<td>Dec 12</td>
<td>Jingbo Wan</td>
<td>Maximum Principles and applications</td>
<td>In this reading seminar talk, the speaker will present Hamilton’s Maximum Principle for Ricci flow and discuss applications for convergence of Ricci flow under certain positive curvature conditions. Then, as a comparison, the speaker will present a version of Bony’s strict maximum principle for degenerate elliptic equations and discuss its application on rigidity results (where we change the previous positive curvature conditions to non-negative curvature conditions). If time permits, the speaker will present and discuss some results where a version of Hamilton’s Maximum Principle or Bony’s strict maximum principle was applied. Reference: Simon Brendle, “Ricci Flow and the Sphere Theorem”, Chapter 5 & 9.</td>
</tr>
</tbody>
</table>Alex Xuaxu[at]math.columbia.eduTime : Monday 6-8 PM Location : Room 507 Organizers: Aaron Chow, Shuang Liang, Jingbo Wan, Alex XuSeminar Template Post2020-12-01T00:00:00-05:002020-12-01T00:00:00-05:00https://www.math.columbia.edu/~axu/seminars/seminar-template-post<p>This is a template seminar page for future use.</p>
<ul>
<li>Time : Sometime every week</li>
<li>Location : A mysterious classroom</li>
<li>Organizers: Some people</li>
</ul>
<h3 id="seminar-schedule">Seminar Schedule</h3>
<table>
<thead>
<tr>
<th>Week/Date</th>
<th>Speaker</th>
<th>Title</th>
<th>Abstract</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>Someone with a long name (And their current institution)</td>
<td>An uncharacteristically long title with a lot of words in it. And then some</td>
<td><a href="https://arxiv.org">ArXiv Link</a> Lorem ipsum dolor sit amet, \(f: M \rightarrow N\) consectetur adipiscing elit, sed do eiusmod tempor \(Df: TM \rightarrow TM\) incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud \(\nabla_\boldsymbol{x} J(\boldsymbol{x})\) exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.</td>
</tr>
<tr>
<td>2</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>3</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>4</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>5</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>6</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>7</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>8</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>9</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>10</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>11</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>12</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>Alex Xuaxu[at]math.columbia.eduThis is a template seminar page for future use.Teaching Template Post2020-12-01T00:00:00-05:002020-12-01T00:00:00-05:00https://www.math.columbia.edu/~axu/teaching/teaching-template-post<p>This is a template syllabus for future use in a Calculus 1 course. This entire page is fictional and was written so I can be lazy in the future.</p>
<h2 id="basic-info">Basic Info:</h2>
<p><strong>Lecture Times</strong>: Tue/Thu 11:40-12:55<br />
<strong>Location</strong>: TBA<br />
<strong>Instructor</strong>: Alex Xu<br />
<strong>Email</strong>: axu [at] math.columbia.edu ; please include the course number in the subject header<br />
<strong>Office Hours</strong>: Wed 11:40-12:55 or by appointment at (Columbia has not assigned first year grads offices yet)<br />
<strong>Teaching Assistants</strong>: TBA</p>
<h2 id="course-description">Course description</h2>
<p>In this class we will cover</p>
<ul>
<li>Limits of sequences and series, and continuity of functions</li>
<li>Derivatives of functions and their implications</li>
<li>Integration of functions and the fundamental theorem of calculus</li>
</ul>
<p>Ultimately, the goal is to gain an intuition for these abstract mathematical concepts and be able to apply them in contexts outside of a math class.</p>
<h2 id="textbook">Textbook</h2>
<p>The standard reference is <em>Calculus: Early Transcendentals</em>, 8th edition, by John Stewart. This will be used as a reference for examples and exercises.</p>
<p>An interesting supplementary textbook will be <em>The Calculus: A Genetic Approach</em>, by Otto Toeplitz. A PDF is freely available online <a href="http://prima.lnu.edu.ua/faculty/mechmat/Departments/MFAUKR/attachments/toeplitz.pdf">here</a>. This book will be used to help you learn the concepts more concretely</p>
<h2 id="homework">Homework</h2>
<p>Weekly homework sets will be posted online (for a total of 11) on weeks that do not have midterms. The lowest 3 grades will be dropped, for a total of 8 assignments that count towards your grade. But you should still do all of the homework for your own sake. Math classes generally build each lecture on top of the previous one so it will only hurt you if you decide to abuse this policy and skip homework.</p>
<p>Homework is due in class by the end of the first class the following week (Mon/Tues). Alternatively you can also drop them off before class in the drop box on the 4th floor of the Math building. Late homeworks will not be accepted without a note from a doctor or dean documenting a medical or family emergency. Grades will be posted on CourseWorks. You are encouraged to discuss the homework with other students but you must write your solutions individually, in your own words.</p>
<p>In addition, the material to be covered will be posted online in advance and I highly recommend that you at least skim the textbook so you can get more out of lecture.</p>
<h2 id="exams">Exams</h2>
<p>There will be 2 midterms and a final exam. Since later concepts build upon previous concepts, these exams will necessarily be cumulative. Practice problems will be released the week before. If you require special accommodations due to religious or disability reasons, please try to reach out at least 2 weeks beforehand to we can find an arrangement that works.</p>
<h2 id="schedule">Schedule</h2>
<p>Approximate topics covered each class. The material for the week will be approximately taken from Stewart; extra reading from other sources is optional but helpful for intuition and historical contexts.</p>
<table>
<thead>
<tr>
<th>Class # (Date)</th>
<th>Topics</th>
<th>Notes</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>Review of special functions</td>
<td>Stewart Ch 1</td>
</tr>
<tr>
<td>2</td>
<td>Inverse and piecewise functions</td>
<td> </td>
</tr>
<tr>
<td>3</td>
<td>Sequences and their limits</td>
<td>Stewart Ch 2.1-2.2, Toeplitz Ch 6-10</td>
</tr>
<tr>
<td>4</td>
<td>Continuity and limits of functions</td>
<td> </td>
</tr>
<tr>
<td>5</td>
<td>Limit laws, squeeze theorem</td>
<td>Stewart Ch 2.3-2.8</td>
</tr>
<tr>
<td>6</td>
<td>The derivative, motivation and definition</td>
<td> </td>
</tr>
<tr>
<td>7</td>
<td>The derivative of special functions</td>
<td>Stewart Ch 3.1-3.3,3.6</td>
</tr>
<tr>
<td>8</td>
<td>Product and quotient rules</td>
<td> </td>
</tr>
<tr>
<td>9</td>
<td>The chain rule and implicit differentiation</td>
<td>Stewart Ch 3.4-3.5</td>
</tr>
<tr>
<td>10</td>
<td>Related rates and linear approximation</td>
<td>Stewart Ch 3.9-3.10</td>
</tr>
<tr>
<td>11</td>
<td>Midterm 1 review</td>
<td> </td>
</tr>
<tr>
<td>12</td>
<td><strong>Midterm 1</strong></td>
<td> </td>
</tr>
<tr>
<td>13</td>
<td>Extreme value theorem, graph sketching</td>
<td>Stewart Ch 4.1-4.3, Toeplitz Ch 18-21</td>
</tr>
<tr>
<td>14</td>
<td>Mean value theorem</td>
<td> </td>
</tr>
<tr>
<td>15</td>
<td>Indeterminate forms and L’Hospital’s rule</td>
<td>Stewart Ch 4.4-4.6</td>
</tr>
<tr>
<td>16</td>
<td>Second derivatives and convexity</td>
<td> </td>
</tr>
<tr>
<td>17</td>
<td>Optimization problems</td>
<td>Stewart Ch 4.7</td>
</tr>
<tr>
<td>18</td>
<td>Midterm 2 review</td>
<td> </td>
</tr>
<tr>
<td>19</td>
<td><strong>Midterm 2</strong></td>
<td> </td>
</tr>
<tr>
<td>20</td>
<td>Antiderivatives</td>
<td>Stewart Ch 4.9</td>
</tr>
<tr>
<td>21</td>
<td>Area and distance</td>
<td>Stewart Ch 5.1-5.2, Toeplitz Ch 11-16</td>
</tr>
<tr>
<td>22</td>
<td>Riemann integrals and computations</td>
<td> </td>
</tr>
<tr>
<td>23</td>
<td>Fundamental theorem of calculus</td>
<td>Stewart Ch 5.3, Toeplitz Ch 23</td>
</tr>
<tr>
<td>24</td>
<td>Substitution rule and integration by parts</td>
<td>Stewart Ch 5.5, 7.1</td>
</tr>
<tr>
<td>25</td>
<td>Indefinite integrals</td>
<td>Stewart Ch 5.4</td>
</tr>
<tr>
<td>26</td>
<td>Improper integrals</td>
<td>Stweart Ch. 7.2</td>
</tr>
<tr>
<td>27</td>
<td>Review of first half of the course</td>
<td> </td>
</tr>
<tr>
<td>28</td>
<td>Review of second half of the course</td>
<td> </td>
</tr>
</tbody>
</table>
<h2 id="homeworks">Homeworks</h2>
<ul>
<li>Hw 1: (Due by Lecture 3)</li>
<li>HW 2: (Due by Lecture 5)</li>
<li>HW 3: (Due by Lecture 7)</li>
<li>HW 4: (Due by Lecture 9)</li>
<li>HW 5: (Due by Lecture 11)</li>
<li>Hw 6: (Due by Lecture 15)</li>
<li>HW 7: (Due by Lecture 17)</li>
<li>HW 8: (Due by Lecture 21)</li>
<li>HW 9: (Due by Lecture 23)</li>
<li>HW 10: (Due by Lecture 25)</li>
<li>HW 11: (Due by Lecture 27)</li>
</ul>
<h2 id="rubric">Rubric</h2>
<table>
<thead>
<tr>
<th>Category</th>
<th>Weight</th>
</tr>
</thead>
<tbody>
<tr>
<td>Homework</td>
<td>40% (5% per homework)</td>
</tr>
<tr>
<td>Midterm Exams</td>
<td>40% (20% per midterm)</td>
</tr>
<tr>
<td>Final Exam</td>
<td>20%</td>
</tr>
</tbody>
</table>
<p>If your final exam score is higher than the lowest of your midterm exam scores, then the corresponding midterm exam score will be replaced with your final exam score. Grading for this course will be as follows</p>
<table>
<thead>
<tr>
<th>Grade</th>
<th>Score</th>
</tr>
</thead>
<tbody>
<tr>
<td>A</td>
<td>90%-100%</td>
</tr>
<tr>
<td>B</td>
<td>80%- 89%</td>
</tr>
<tr>
<td>C</td>
<td>70%- 79%</td>
</tr>
<tr>
<td>D</td>
<td>60%- 69%</td>
</tr>
<tr>
<td>F</td>
<td>0% - 59%</td>
</tr>
</tbody>
</table>
<h2 id="resources">Resources</h2>
<p>Here are some resources that you might find helpful</p>
<ul>
<li><a href="https://www.math.columbia.edu/general-information/help-rooms/">Columbia Math Help Room</a></li>
</ul>Alex Xuaxu[at]math.columbia.eduThis is a template syllabus for future use in a Calculus 1 course. This entire page is fictional and was written so I can be lazy in the future.