Tailwind mathjax 显示在导航栏上方
Tailwind mathjax is showing above to the navbar
不好意思再问一个问题，我早上也问了一个问题。
这次我遇到了 Mathjax 文本问题。由于我的导航栏是固定的，我的文本在导航栏内，而数学表达式在导航栏上方。从今天早上开始，这让我很烦。
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<p class="mt-4 leading-relaxed text-lg text-justify"> My area of interests include <em>differential geometry</em>, <em>differential topology</em> and <em>algebraic topology</em>. More specifically, I am working on <em class="text-green-600">cut locus of a submanifold</em>. For a given Riemannian manifold $M$ and $N\subset M$ the <em class="text-green-600">cut locus of $N$</em>, $\mathrm{Cu}(N)$, is the collection of points $q\in M$ such that extension of a distance minimal geodesic joining $N$ to $q$ is no longer minimal. Here by the <em class="text-green-600">distance minimal geodesic $\gamma$ joining $N$ to $q$</em> we mean that there exists $p\in N$ such that the length of $\gamma$ from $p$ to $q$ is same as the distance from $N$ to $q$. In my recent <a href="#publications" class="text-blue-500 hover:text-blue-800">paper</a> (joint with Dr Somnath Basu) we have discussed the cut locus of a closed submanifold and described the relation between cut locus, Thom spaces and Morse-Bott functions. </p>
<p class="mt-4 leading-relaxed text-lg text-justify"> Currently, I am working on the cut locus of a quotient manifold and application to the classifying spaces. </p>
<p class="mt-4 leading-relaxed text-lg text-justify:"> </p>
<h2 class="mt-4 mb-4 text-gray-900 text-xl title-font font-medium" id="publications">Publications and Preprints</h2>
<ol class="pl-8 list-decimal">
<li class="text-lg"> <em>A connection between cut locus, Thom spaces and Morse-Bott functions</em> (joint with Dr Somnath Basu) <a href="ml-8 https://arxiv.org/abs/2011.02972" target="_blank" class="text-blue-500 hover:text-blue-800">ArXiv link</a>
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<p class="pl-2 leading-relaxed text-base text-gray-500 text-justify"> <span class="text-gray-900">Abstract:</span> Associated to every closed, embedded submanifold $N$ in a connected Riemannian manifold $M$, there is the distance function $d_N$ which measures the distance of a point in $M$ from $N$. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus $\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flow lines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ is a closed manifold, then we prove that the Thom space of the normal bundle of $N$ is homeomorphic to $M/\mathrm{N}$. We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and a geometric deformation of $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ which is different from the Gram-Schmidt retraction. </p>
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我们可以在上图中看到数学文本位于导航栏上方。我不知道到底是什么问题。任何帮助将不胜感激。
将tailwind class z-10插入到导航栏容器中，使导航栏更高z-Index
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<p class="mt-4 leading-relaxed text-lg text-justify"> My area of interests include <em>differential geometry</em>, <em>differential topology</em> and <em>algebraic topology</em>. More specifically, I am working on <em class="text-green-600">cut locus of a submanifold</em>. For a given Riemannian manifold $M$ and $N\subset M$ the <em class="text-green-600">cut locus of $N$</em>, $\mathrm{Cu}(N)$, is the collection of points $q\in M$ such that extension of a distance minimal geodesic joining $N$ to $q$ is no longer minimal. Here by the <em class="text-green-600">distance minimal geodesic $\gamma$ joining $N$ to $q$</em> we mean that there exists $p\in N$ such that the length of $\gamma$ from $p$ to $q$ is same as the distance from $N$ to $q$. In my recent <a href="#publications" class="text-blue-500 hover:text-blue-800">paper</a> (joint with Dr Somnath Basu) we have discussed the cut locus of a closed submanifold and described the relation between cut locus, Thom spaces and Morse-Bott functions. </p>
<p class="mt-4 leading-relaxed text-lg text-justify"> Currently, I am working on the cut locus of a quotient manifold and application to the classifying spaces. </p>
<p class="mt-4 leading-relaxed text-lg text-justify:"> </p>
<h2 class="mt-4 mb-4 text-gray-900 text-xl title-font font-medium" id="publications">Publications and Preprints</h2>
<ol class="pl-8 list-decimal">
<li class="text-lg"> <em>A connection between cut locus, Thom spaces and Morse-Bott functions</em> (joint with Dr Somnath Basu) <a href="ml-8 https://arxiv.org/abs/2011.02972" target="_blank" class="text-blue-500 hover:text-blue-800">ArXiv link</a>
<div class="mt-2 ml-4 border-l-2 border-indigo-500 hidden md:block xs:hidden">
<p class="pl-2 leading-relaxed text-base text-gray-500 text-justify"> <span class="text-gray-900">Abstract:</span> Associated to every closed, embedded submanifold $N$ in a connected Riemannian manifold $M$, there is the distance function $d_N$ which measures the distance of a point in $M$ from $N$. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus $\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flow lines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ is a closed manifold, then we prove that the Thom space of the normal bundle of $N$ is homeomorphic to $M/\mathrm{N}$. We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and a geometric deformation of $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ which is different from the Gram-Schmidt retraction. </p>
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不好意思再问一个问题，我早上也问了一个问题。
这次我遇到了 Mathjax 文本问题。由于我的导航栏是固定的，我的文本在导航栏内，而数学表达式在导航栏上方。从今天早上开始，这让我很烦。
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<p class="mt-4 leading-relaxed text-lg text-justify"> My area of interests include <em>differential geometry</em>, <em>differential topology</em> and <em>algebraic topology</em>. More specifically, I am working on <em class="text-green-600">cut locus of a submanifold</em>. For a given Riemannian manifold $M$ and $N\subset M$ the <em class="text-green-600">cut locus of $N$</em>, $\mathrm{Cu}(N)$, is the collection of points $q\in M$ such that extension of a distance minimal geodesic joining $N$ to $q$ is no longer minimal. Here by the <em class="text-green-600">distance minimal geodesic $\gamma$ joining $N$ to $q$</em> we mean that there exists $p\in N$ such that the length of $\gamma$ from $p$ to $q$ is same as the distance from $N$ to $q$. In my recent <a href="#publications" class="text-blue-500 hover:text-blue-800">paper</a> (joint with Dr Somnath Basu) we have discussed the cut locus of a closed submanifold and described the relation between cut locus, Thom spaces and Morse-Bott functions. </p>
<p class="mt-4 leading-relaxed text-lg text-justify"> Currently, I am working on the cut locus of a quotient manifold and application to the classifying spaces. </p>
<p class="mt-4 leading-relaxed text-lg text-justify:"> </p>
<h2 class="mt-4 mb-4 text-gray-900 text-xl title-font font-medium" id="publications">Publications and Preprints</h2>
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<li class="text-lg"> <em>A connection between cut locus, Thom spaces and Morse-Bott functions</em> (joint with Dr Somnath Basu) <a href="ml-8 https://arxiv.org/abs/2011.02972" target="_blank" class="text-blue-500 hover:text-blue-800">ArXiv link</a>
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<p class="pl-2 leading-relaxed text-base text-gray-500 text-justify"> <span class="text-gray-900">Abstract:</span> Associated to every closed, embedded submanifold $N$ in a connected Riemannian manifold $M$, there is the distance function $d_N$ which measures the distance of a point in $M$ from $N$. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus $\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flow lines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ is a closed manifold, then we prove that the Thom space of the normal bundle of $N$ is homeomorphic to $M/\mathrm{N}$. We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and a geometric deformation of $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ which is different from the Gram-Schmidt retraction. </p>
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我们可以在上图中看到数学文本位于导航栏上方。我不知道到底是什么问题。任何帮助将不胜感激。
将tailwind class z-10插入到导航栏容器中，使导航栏更高z-Index
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<p class="mt-4 leading-relaxed text-lg text-justify"> My area of interests include <em>differential geometry</em>, <em>differential topology</em> and <em>algebraic topology</em>. More specifically, I am working on <em class="text-green-600">cut locus of a submanifold</em>. For a given Riemannian manifold $M$ and $N\subset M$ the <em class="text-green-600">cut locus of $N$</em>, $\mathrm{Cu}(N)$, is the collection of points $q\in M$ such that extension of a distance minimal geodesic joining $N$ to $q$ is no longer minimal. Here by the <em class="text-green-600">distance minimal geodesic $\gamma$ joining $N$ to $q$</em> we mean that there exists $p\in N$ such that the length of $\gamma$ from $p$ to $q$ is same as the distance from $N$ to $q$. In my recent <a href="#publications" class="text-blue-500 hover:text-blue-800">paper</a> (joint with Dr Somnath Basu) we have discussed the cut locus of a closed submanifold and described the relation between cut locus, Thom spaces and Morse-Bott functions. </p>
<p class="mt-4 leading-relaxed text-lg text-justify"> Currently, I am working on the cut locus of a quotient manifold and application to the classifying spaces. </p>
<p class="mt-4 leading-relaxed text-lg text-justify:"> </p>
<h2 class="mt-4 mb-4 text-gray-900 text-xl title-font font-medium" id="publications">Publications and Preprints</h2>
<ol class="pl-8 list-decimal">
<li class="text-lg"> <em>A connection between cut locus, Thom spaces and Morse-Bott functions</em> (joint with Dr Somnath Basu) <a href="ml-8 https://arxiv.org/abs/2011.02972" target="_blank" class="text-blue-500 hover:text-blue-800">ArXiv link</a>
<div class="mt-2 ml-4 border-l-2 border-indigo-500 hidden md:block xs:hidden">
<p class="pl-2 leading-relaxed text-base text-gray-500 text-justify"> <span class="text-gray-900">Abstract:</span> Associated to every closed, embedded submanifold $N$ in a connected Riemannian manifold $M$, there is the distance function $d_N$ which measures the distance of a point in $M$ from $N$. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus $\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flow lines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ is a closed manifold, then we prove that the Thom space of the normal bundle of $N$ is homeomorphic to $M/\mathrm{N}$. We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and a geometric deformation of $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ which is different from the Gram-Schmidt retraction. </p>
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