-
Notifications
You must be signed in to change notification settings - Fork 0
/
research.html
120 lines (91 loc) · 7.17 KB
/
research.html
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
<!DOCTYPE html>
<html>
<head>
<meta http-equiv="Content-Type" name="viewport" content="text/html, width=device-width, initial-scale=1" charset="utf-8" />
<meta http-equiv="Content-Type" content="text/html" charset="utf-8" />
<title>Tom Mainiero</title>
<link rel="stylesheet" href="styles.css" type="text/css" />
<style>
.picture {padding: 3px; font-size: 60%}
.picture img {vertical-align:middle; margin-bottom: 2px; }
.right { margin: 0.5em 0pt 0.5em 0.8em; float:right; }
.left { margin: 0.5em 0.8em 0.5em 0; float:left; }
</style>
</head>
<body>
<table class="navigation">
<tr>
<td align="center"><h1 class="navname">Tom Mainiero</h1> </td>
<td align="right">
<nav>
<a class="menu" href="index.html"><div class="navit">About</div></a>
<a class="menu" href="research.html"><div class="navit-idem">Research</div></a>
<a class="menu" href="talks.html"><div class="navit">Talks</div></a>
<a class="menu" href="contact.html"><div class="navit">Contact</div></a>
</nav>
</td>
</tr>
<tr> <td colspan="2"><hr class="title"></td> </tr>
</table>
<div class="textbox">
<p>My research interests are broad and constantly expanding, but their largest connected component is a tubular neighborhood of the intersection of geometry, quantum field theory, and quantum information.
<p>A description of some select interests—directed toward a broad audience with some technical background—is below.
<p>A technical summary of my past research directed toward mathematicians can be found <a href="files/statements/mainiero_rstatement_math_2019_v2.pdf">here</a>, and directed toward physicists <a href="files/statements/mainiero_rstatement_physics_2019_v2.pdf">here</a>.
</div>
<h1>Connections Between Geometry/Topology and Information Theory </h1>
<!-- <div class="textbox"> -->
<!-- <table align="center"> -->
<!-- <tr> -->
<!-- <td style="float:right"> <img src="images/cmi.gif" style="max-width:100%" alt=""/> </td> -->
<!-- <td>A bunch of words in here describing stuff. I'm going to keep writing things to see how this looks. Does it look good or bad? I can't tell, here is a bunch of other stuff, why does it not spill over?</td> -->
<!-- </tr> -->
<!-- </table> -->
<!-- <table cellspacing="10" align="center"> -->
<!-- <tr> -->
<!-- <td style="float:right"> <img src="images/cmi.jpg" style="max-width:100%;" alt="Tom Mainiero!"/> </td> -->
<!-- <td>A bunch of words in here describing stuff</td> -->
<!-- </tr> -->
<!-- </table> -->
<!-- </div> -->
<img class="picture right" src="images/cmi.gif" style="max-width:50%" alt=""/>
<div class="textbox">
<p> Amazingly, both classical and quantum information theory have deep connections to geometry and topology.
A small hint of this connection appears when studying <a href="https://en.wikipedia.org/wiki/Mutual_information">mutual information</a>: a numerical quantity that measures how much information is shared between two random variables (for instance, measured in a lab by two separate "experimenters").
Mutual information is calculated as the alternating sum of entropies---another quantity is calculated this way: the <a href="https://en.wikipedia.org/wiki/Euler_characteristic">Euler Characteristic</a> of a topological space!
As it so happens, this seemingly superficial observation has deep underpinnings.
Understanding these underpinnings may help us derive more geometrically motivated measures of shared information, and might even be part of the puzzle of how space-time geometry can emerge from quantum entanglement (an active area of research).
</div>
<h2 align="left">Relevant papers</h2>
<div class="textbox">
<ul>
<li><a href="https://arxiv.org/abs/1901.02011">Homological tools for the Quantum Mechanic</a></li>
<li><a href="files/papers/higher-information-gsi23.pdf">Higher Information from Families of Measures</a></li>
</ul>
</div>
<h1 style="clear:both">BPS Particles and Donaldson-Thomas Invariants</h1>
<img class="picture right" src="images/spectralnetwork.gif" style="max-width:100%" alt=""/>
<div class="textbox">
<p> Often, a firm grasp on the physics most immediately applicable to the real world is too complex to tackle directly.
As a result, physicists sometimes study simpler versions of reality—"toy models"—in order to obtain a better understanding of nature.
Symmetry is a great simplifier in physics and, oftentimes, such toy models typically have much greater symmetry.
This symmetry is often far beyond what exists in the real world, but helps us more easily identify properties that might still exist in nature.
<p>One such family of toy models are "field theories with extended supersymmetry" in four space-time dimensions.
As the name suggests, these theories have quite a bit of symmetry, making them far easier to understand and recognize patterns within.
But questions about physics in these theories also have deep connections to questions in areas of mathematics: particularly algebraic/complex geometry.
<p>For instance one can focus on certain types of particles, called BPS particles (named after Bolgomol'nyi, Prasad, and Sommerfield) within our theory.
These particles are particularly interesting as they are "stable" against small changes to physics itself.
The "count" of certain types of particles in these theories turns out to be closely related to counts of geometric objects of interest to algebraic/complex geometers called "Donaldson-Thomas (DT) Invariants".
<p>Among a myraid of other things, using ideas from physics, one can show that these "DT invariants" are actually encoded in special functions can be directly computed by studying webs or graphs that reside on a two-dimensional surface.
<!-- <p>In work with colloboration with Andrew Neitzke, Gregory Moore, using techniques from physics I computed DT invariants for a simple toy model. -->
<!-- The results were a surprising voliation of a conjecture, and the physics of this toy model bears some relation to the existence of the <a href="https://en.wikipedia.org/wiki/Hagedorn_temperature">Hagedorn temperature</a> that appears in the physics of the real world. -->
<!-- In later work, using machinery from physics, I sketched a suspected argument that the special functions alluded to above are always solutions of some algebraic equations—an observation that has deeper geometric implication, and showed the constraints on asympototics of the counts of BPS particles/DT invariants using this algebraic fact. -->
</div>
<h2 align="left">Relevant papers</h2>
<div class="textbox">
<ul>
<li><a href="https://arxiv.org/abs/1305.5454">Wild Wall Crossing and BPS Giants</a>: with Dmitry Galakhov, <a href="https://www.katalog.uu.se/profile/?id=N15-1714">Pietro Longhi</a>, <a href="http://www.physics.rutgers.edu/~gmoore/">Greg Moore</a>, and <a href="https://gauss.math.yale.edu/~an592/">Andrew Neitzke</a>.</li>
<li><a href="https://arxiv.org/abs/1606.02693">Algebraicity and Asymptotics: An explosion of BPS indices from algebraic generating series</a></li>
</ul>
</div>
</body>
</html>