mypage/info.html at main · you-qi2121/mypage · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN">
<html>
<head>
  <title>Information</title>
  <meta name="description" content="You Qi's research, teaching and service." >
  <link type="text/css" rel="stylesheet" href="style.css">
</head>

<body>

<!-- Main content -->

<p><img src="researchpic.png">
<h1>You Qi - 2024 Research and Promotion Material </h1>

<p><a href="./index.html">Back to my homepage</a>

<hr><b>Research Description:</b>
<br>My main interest in mathematics is on higher representation theory, algebraic geometry, and their applications to low-dimensional topology. Currently, I'm working on categorification of small <a href="http://en.wikipedia.org/wiki/Quantum_group"><i>quantum groups</i></a>-a special kind of finite dimensional Hopf algebra, and trying to construct a four-dimensional <a href="http://en.wikipedia.org/wiki/Topological_quantum_field_theory"><i>topological quantum field theory</i></a> out of them. This is the original motivation of categorification initiated by Louis Crane and Igor Frenkel. The picture on the top-right is a categorical interpretation of one of the so called <i>quantum Serre relations</i> for the small quantum group sl(3).
I'm also looking into symmetries of various link homology theories, as well as exploring the algebro-geometric meaning behind the small quantum groups and their applications in topological field theories.<br>


<ul>
<li>My <a href="candidatestatement.pdf">candidate statement.</a><br>
<li>My <a href="qiCV.pdf">CV</a> and <a href="researchstatement.pdf">research and teaching statement.</a><br>
</li>
<li>My profiles on on <a href = "https://scholar.google.com/citations?user=_RAJnYgAAAAJ&hl=en">Google Scholar</a> and <a href="https://www.researchgate.net/profile/You-Qi">ResearchGate </a>.</li>
<li>My <a href = "https://scholar.google.com/citations?user=_RAJnYgAAAAJ&hl=en">Published Work</a> (in zip).</li>
</ul>
<br>

<hr><b>Published and accepted works (before joining UVA):</b><br>
[1] Hopfological Algebra, Compositio Mathematica, Volume 150, Issue 01, 2014, pp. 1-45.  <a href="publication/hopfo.pdf">journal version</a>, <a href="http://arxiv.org/abs/1205.1814">arXiv:1205.1814</a>.<br>
[2] An approach to categorification of some small quantum groups, with <a href="http://www.math.columbia.edu/~khovanov/">Mikhail Khovanov</a>, Quantum Topology, Volume 6, Issue 2, 2015, pp. 185-311. <a href="publication/approach.pdf">journal version</a>, <a href="http://arxiv.org/abs/1208.0616">arXiv version</a>.<br>
[3] An approach to categorification of some small quantum groups II, with <a href="http://pages.uoregon.edu/belias/">Ben Elias</a>, Advances in Mathematics, Volume 288, 2016, pp. 81-151. <a href="publication/approach2.pdf">journal version</a>, <a href="http://arxiv.org/abs/1302.5478">arXiv version</a>.<br>
[4] A categorification of the Burau representation at a prime root of unity, with <a href="https://sites.google.com/site/joshuasussan/home">Josh Sussan</a>, Selecta Mathematica, Volume 22, Issue 3, 2016, pp. 1157-1193, <a href="publication/burau.pdf">journal version</a>, <a href="http://arxiv.org/abs/1312.7692">arXiv version</a>.<br>
[5] A categorification of quantum sl(2) at prime roots of unity, with <a href="http://pages.uoregon.edu/belias/">Ben Elias</a>, Advances in Mathematics, Volume 299, 2016, pp. 863-930, <a href="publication/sltwo.pdf">journal version</a>, <a href="http://arxiv.org/abs/1503.05114">arXiv version</a>. <br>
[6] The differential graded odd nilHecke algebra, joint with <a href="http://pages.uoregon.edu/ellis/">Alexander P. Ellis</a>, Communications in Mathematical Physics, Volume 344, Issue 1, 2016, pp. 275-331. <a href="publication/oddsltwo.pdf">journal version</a>, <a href="http://arxiv.org/abs/1504.01712">arXiv version</a>.<br>
[7] Categorification at prime roots of unity and hopfological finiteness, with <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a>, AMS Contemporary Mathematics, Volume 683, pp. 261-286, 2017. <a href="publication/finite.pdf">journal version</a>, <a href="http://arxiv.org/abs/1509.00438">arXiv version</a>.<br>
[8] The center of small quantum groups I: the principal block in type A, with <a href="https://people.epfl.ch/anna.lachowska">Anna Lachowska</a>, IMRN, Volume 2018, Issue 20, pp. 6349-6405, 2018. <a href="publication/center.pdf">journal version</a>, <a href="http://arxiv.org/abs/1604.07380">arXiv version</a>.<br>
[9] A categorification of a quantum Frobenius map, Journal of the Institute of Mathematics of Jussieu, Volume 18, Number 5, pp. 899-939, 2019. <a href="publication/frobenius.pdf">journal version</a>, <a href="http://arxiv.org/abs/1607.02117">arXiv version</a>. <br>
[10] The center of small quantum groups II: singular blocks, with <a href="https://people.epfl.ch/anna.lachowska">Anna Lachowska</a>, Proceedings of the London Mathematical Society, Volume 118, pp. 513-544, 2019. <a href="publication/centerII.pdf">journal version</a>, <a href="https://arxiv.org/abs/1703.02457">arXiv version</a>. <br>
[11] Morphism spaces in stable categories of Frobenius algebras, Communications in Algebra, Volume 47, Number 8, pp. 3239-3249, 2019. <a href="publication/morph.pdf">journal version</a>, <a href="https://arxiv.org/abs/1801.07838v1">arXiv version</a>. <br>

<hr><b>Published and accepted works (after joining UVA):</b><br>
[12] A faithful braid group action on the stable category of tricomplexes, with <a href="http://www.math.columbia.edu/~khovanov/">Mikhail Khovanov</a>, SIGMA 16 (2020), 019, 32 pages. <a href="publication/tricomplex.pdf">journal version</a>, <a href="https://arxiv.org/abs/1911.02503">arXiv version</a>.<br>
[13] Evaluating thin flat surfaces, with <a href="http://www.math.columbia.edu/~khovanov/">Mikhail Khovanov</a> and <a href="https://math.unc.edu/staff/rozansky-lev/">Lev Rozansky</a>,  Communications in Mathematical Physics, Volume 385, pp. 1835-1870, 2021. <a href="publication/ethin.pdf">journal version</a>, <a href="https://arxiv.org/abs/2009.01384">arXiv version</a>.<br>
[14] Remarks on the derived center of small quantum groups, with <a href="https://people.epfl.ch/anna.lachowska">Anna Lachowska</a>, Selecta Mathematica, Volume 27, Article Number 68, 40 pages, 2021. <a href="publication/remarks.pdf">journal version</a>, <a href="https://arxiv.org/pdf/1912.08783.pdf">arXiv version</a>. <br>
[15] A categorification of cyclotomic integers, with <a href="http://sites.math.rutgers.edu/~rul2/">Robert Laugwitz</a>, Quantum Topolology, Volume 13, Issue 3, 2022, pp. 539-577. <a href="publication/cyclo.pdf">journal version</a>, <a href="https://arxiv.org/abs/1804.01478">arXiv:1804.01478</a>.<br>
[16] A braid group action on a p-DG homotopy category, with <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a> and <a href="http://www.math.nagoya-u.ac.jp/~m03039e/">Yasuyoshi Yonezawa</a>, Journal of Algebra, Volume 598, 15 May 2022, Pages 470-517. <a href="publication/braidact.pdf">journal version</a>, <a href="https://arxiv.org/abs/2012.15181">arXiv version</a>.<br>
[17] On some p-differential graded link homologies, with <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a>, Forum of Mathematics, Pi, Volume 10, E26. pp. 1-58, 2022. <a href="publication/pdglink.pdf">journal version</a>, <a href="https://arxiv.org/abs/2009.06498">arXiv version</a>.<br>
[18] On some p-differential graded link homologies II, with <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a>, Algebraic & Geometric Topology Volume 23, Issue 7, pp. 3357-3394, 2023. <a href="publication/pdglink2.pdf">journal version</a>, <a href="http://arxiv.org/abs/2108.10722">arXiv version</a>.<br>
[19] Actions of sl(2) on algebras appearing in categorification, with <a href="http://pages.uoregon.edu/belias/">Ben Elias</a>, Quantum Topology, Volume 14, Number 4, pp. 733-806, 2023. <a href="publication/actionsl2.pdf">journal version</a>, <a href="https://arxiv.org/abs/2103.00048">arXiv version</a>.<br>
[20] A Rickard equivalence for hopfological homotopy categories, Journal of Pure and Applied Algebra, Volume 227, Issue 5, Article Number 107252, 2023. <a href="publication/rickard.pdf">journal version</a>, <a href="https://arxiv.org/abs/2204.14220">arXiv version</a>.<br>
[21] Categorifying Hecke algebras at prime roots of unity, part I, with <a href="http://pages.uoregon.edu/belias/">Ben Elias</a>, Transactions of the AMS, Volume 376, pp. 7691-7742, 2023. <a href="publication/hecke.pdf">journal version</a>, <a href="https://arxiv.org/abs/2005.03128v1">arXiv version</a>.<br>
[22] p-DG cyclotomic nilHecke algebras, with <a href="http://www.math.columbia.edu/~khovanov/">Mikhail Khovanov</a> and <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a>, Memoirs of the American Mathematical Society, Volume 293, Number 1462, 2024. <a href="publication/memo1091.pdf">journal version</a>, <a href="https://arxiv.org/abs/1711.07159">arXiv:1711.07159</a>.<br>
[23] p-DG cyclotomic nilHecke algebras II, joint with <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a>, Memoirs of the American Mathematical Society, Volume 293, Number 1463, 2024. <a href="publication/memo1092.pdf">journal version</a>, <a href="https://arxiv.org/abs/1811.04372">arXiv version</a>.<br>
[24] Symmetries of gl(N)-foams, with <a href="https://lrobert.perso.math.cnrs.fr/">Louis-Hadrien Robert</a>, <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a> and <a href="http://wagner.perso.math.cnrs.fr/index.html/">Emmanuel Wagner</a>, Quantum Topology, <a href="https://ems.press/journals/qt/articles/14297777">DOI:10.4171/QT/215</a>, published online on May 3, 2024.  <a href="publication/foamsym.pdf">journal version</a>, <a href="https://arxiv.org/abs/2212.10106">
arXiv version</a>.<br>
[25] A braid group action on an A-infinity category for zigzag algebras, joint with <a href="https://sites.google.com/site/cooperbenj/">Benjamin Cooper</a>, <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a>, accepted by AMS Contemporary Mathematics, 2024. <a href="https://arxiv.org/abs/2305.02824">arXiv:2305.02824</a>.<br>

<hr><b>Book Chapters:</b><br>
[1] <a href="https://link.springer.com/chapter/10.1007/978-3-030-48826-0_21">Connections to Link Invariants</a>, joint with Johannes Flake. In <a href="https://www.springer.com/gp/book/9783030488253">Introduction to Soergel bimodules</a>, edited by Ben Elias, Shotaro Makisumi, Ulrich Thiel and Geordie Williamson. RSME Springer Series, Volume 5, pp. 421-440, Springer, Cham. 2020. <a href="publication/connections.pdf">published version</a>.<br>

<hr><b>Preprints:</b><br>
[1] A categorification of the colored Jones polynomial at a root of unity, joint with <a href="https://lrobert.perso.math.cnrs.fr/">Louis-Hadrien Robert</a>, <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a> and <a href="http://wagner.perso.math.cnrs.fr/index.html/">Emmanuel Wagner</a>, 2021, <a href="https://arxiv.org/abs/2111.13195">arXiv:2111.13195</a>.<br>
[2] Symmetries of equivariant Khovanov-Rozansky homology, joint with <a href="https://lrobert.perso.math.cnrs.fr/">Louis-Hadrien Robert</a>, <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a> and <a href="http://wagner.perso.math.cnrs.fr/index.html/">Emmanuel Wagner</a>, 2023, <a href="https://arxiv.org/abs/2306.10729">arXiv:2306.10729</a>.<br>

<hr><b>Unpublished notes:</b><br>
[1] On the axioms of module algebras over Hopf algebras, 2022, available <a href="reduction.pdf">here</a>.<br>

<hr><b>Work in progress:</b><br>
[1] A categorification of the colored Jones polynomial at a root of unity, part II, joint with <a href="https://lrobert.perso.math.cnrs.fr/">Louis-Hadrien Robert</a>, <a href="https://sites.google.com/site/joshuasussan/home">Joshua Sussan</a> and <a href="http://wagner.perso.math.cnrs.fr/index.html/">Emmanuel Wagner</a>,  in preparation.<br>

[2] Categorifying Hecke algebras at prime roots of unity, part II, joint with <a href="http://pages.uoregon.edu/belias/">Ben Elias</a>, in preparation.<br>

[3] Hopfological algebra and algebraic K-theory, in preparation.<br>


<hr><b> MPhil and PhD Theses:</b><br>
<ul>
<li>This is a printer-friendly version of my <a href="mphilthesis.pdf">MPhil Thesis</a>, written under the guidance of <a href="http://www.math.ust.hk/~mameng/">Professor Guowu Meng</a> at Hong Kong University of Science and Technology. DOI: 10.14711/thesis-b1023323.</li>
<li> This is an updated version of my <a href="thesis.pdf">PhD Thesis</a>, supervised by <a href="http://www.math.columbia.edu/~khovanov/">Professor Mikhail Khovanov</a> at Columbia University. DOI:10.7916/D8G73M28.</li>
</ul>

<hr><b>Awards and Grants:</b><br>
[1] Carl B. Boyer Memorial Fellowship, Columbia University, 2012-2013.<br>
[2] NSF Research Grant "Categorication at Roots of Unity," DMS-1763328, PI, 2017-2021.<br>
[3] NSF Conference Grant "Categorical Methods in Representation Theory and Quantum Topology" DMS-2204700, PI, 2022-2023.<br>
[4] NSF Conference Grant "Southeastern Lie Theory Workshop Series" DMS-2303977, coPI, 2023-2025.<br>
[5] Collaboration Grants for Mathematicians, Simons Foundation, PI, 2022-2028.<br>
[6] NSF Collaborative Research Grant "Small quantum groups, their categorifications and topological applications", DMS-2401376, PI, 2024-2027.<br>

<hr><b>MPhil and PhD Students:</b><br>
<ul>
<li> David Winters 2021-2023. Currently a PhD student in Mathematics at Georgia Tech</li>
</ul>


<address>Updated 08/27/2024<br>
  by You Qi.</address>

<!-- Start of StatCounter Code for Default Guide -->
<script type="text/javascript">
var sc_project=11163445;
var sc_invisible=0;
var sc_security="e617d327";
var scJsHost = (("https:" == document.location.protocol) ?
"https://secure." : "http://www.");
document.write("<sc"+"ript type='text/javascript' src='" +
scJsHost+
"statcounter.com/counter/counter.js'></"+"script>");
</script>
<noscript><div class="statcounter"><a title="web analytics"
href="http://statcounter.com/web-analytics/"
target="_blank"><img class="statcounter"
src="//c.statcounter.com/11163445/0/e617d327/0/" alt="web
analytics"></a></div></noscript>
<!-- End of StatCounter Code for Default Guide -->

</body>
</html>