-
Notifications
You must be signed in to change notification settings - Fork 0
/
index.html
executable file
·422 lines (340 loc) · 26.8 KB
/
index.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
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
<!DOCTYPE HTML>
<html>
<head>
<!-- Google tag (gtag.js) -->
<script async src="https://www.googletagmanager.com/gtag/js?id=G-LJLDWHPL5F"></script>
<script>
window.dataLayer = window.dataLayer || [];
function gtag(){dataLayer.push(arguments);}
gtag('js', new Date());
gtag('config', 'G-LJLDWHPL5F');
</script>
<title>Raj Patil</title>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=no" />
<link rel="stylesheet" href="assets/css/main.css" />
</head>
<body class="is-preload">
<!-- Header -->
<div id="header">
<div class="top">
<!-- Logo -->
<div id="logo">
<span class="image avatar60"><img src="images/photo.jpg" alt="" /></span>
<h1 id="title">Raj Patil</h1>
<!-- <h1 id="title">(राज पाटिल)</h1> -->
<p>Ph.D. student</p>
<p>Albert Einstein Institute</p>
<p>Humboldt University</p>
</div>
<!-- Nav -->
<nav id="nav">
<ul>
<li><a href="#top" id="top-link">About Me</a></li>
<li><a href="#researchgeneral" id="research-link"> <font style="font-size:12px;">(Overview)</font> Research </a></li>
<li><a href="#research" id="research-link"> <font style="font-size:12px;">(Details)</font> Research </a></li>
<li><a href="#looptheloop" id="resources-link"> Loop-the-Loop </a></li>
<li><a href="#pastime" id="resources-link"> Pastime </a></li>
<!-- <li><a href="#test" id="test-link">test</a></li> -->
</ul>
</nav>
<div id="logo">
<!-- <h2><a href=''>Curriculum Vitae</a></h2> -->
<h2> </h2>
<h2>Webpage   <a href="https://www.aei.mpg.de/person/124916/138766" target="_blank">[AEI]</a>   <a href="https://www2.hu-berlin.de/rtg2575/members/doctoral-students/" target="_blank">[Humboldt]</a></h2>
<h2> </h2>
<h2><a href="https://inspirehep.net/authors/1829940" target="_blank">[iNSPIRE]</a></h2>
<h2> </h2>
<h2><a href="https://github.com/raj-physics" target="_blank">[GitHub]</a></h2>
<h2> </h2>
<h2>Email:</h2>
<!-- <h2><a href="mailto:[email protected]">[email protected]</a></h2> -->
<h2>[email protected]</h2>
</div>
</div>
</div>
<!-- Main -->
<div id="main">
<!-- Intro -->
<section id="top" class="one dark cover">
<div class="container">
<header>
Hello. My name is Raj Patil. I am a physicist working on modelling of gravitational waves emitted by compact objects. <br> <br>
Currently, I am a Ph.D. student at Albert Einstein Institute and Humboldt University. I am working here under the guidance of Prof. Dr. Alessandra Buonanno and Dr. Jan Steinhoff on modelling compact objects and their effect on emitted gravitational waves. <br> <br>
Before this, I did my bachelor's and master's in physics from Indian Institute of Science Education and Research Pune. My master's thesis was under the guidance of Dr. Suneeta Vardarajan on studying the effective field approach to the two body problem in gravity.
</header>
</div>
</section>
<!-- Research for non-physicists -->
<section id="researchgeneral" class="two">
<div class="container">
<header>
<h2>Research</h2> <h4>(Overview)</h4>
</header>
<!-- <p> For a general reader, what we do is best explained by the following analogy by Prof. Richard Feynman:
<h5>“…a fun analogy in trying to get some idea of what we’re doing in trying to understand nature, is to imagine that the gods are playing some great game like chess… and you don’t know the rules of the game, but you’re allowed to look at the board, at least from time to time… and from these observations you try to figure out what the rules of the game are, what the rules of the pieces moving are. You might discover after a bit, for example, that when there’s only one bishop around on the board that the bishop maintains its color. Later on you might discover the law for the bishop as it moves on the diagonal, which would explain the law that you understood before – that it maintained its color – and that would be analogous to discovering one law and then later finding a deeper understanding of it. Then things can happen, everything’s going good, and then all of a sudden some strange phenomenon occurs in some corner, so you begin to investigate that – it’s castling, something you didn’t expect. We’re always, by the way, in fundamental physics, always trying to investigate those things in which we don’t understand the conclusions. After we’ve checked them enough, we’re okay.
The thing that doesn’t fit is the thing that’s most interesting, the part that doesn’t go according to what you expected. Also, we could have revolutions in physics: after you’ve noticed that the bishops maintain their color and they go along the diagonal and so on for such a long time and then everybody knows that that’s true, then you suddenly discover one day in some chess game that the bishop doesn’t maintain its color, it changes its color. Only later do you discover a new possibility, that a bishop is captured and that a pawn went all the way down to the queen’s end to produce a new bishop – that can happen but you didn’t know it, and so it’s very analogous to the way our laws are: They sometimes look positive, they keep on working and all of a sudden some little gimmick shows that they’re wrong and then we have to investigate the conditions under which this bishop change of color happened and so forth, and gradually learn the new rule that explains it more deeply.
Unlike the chess game, though, in which the rules become more complicated as you go along, in physics, when you discover new things, it looks more simple. It appears on the whole to be more complicated because we learn about a greater experience – that is, we learn about more particles and new things – and so the laws look complicated again. But if you realize all the time what’s kind of wonderful – that is, if we expand our experience into wilder and wilder regions of experience – every once in a while we have these integrations when everything’s pulled together into a unification, in which it turns out to be simpler than it looked before."
</h5>
</p>
-->
<p>
The observations by Galileo through his telescopes opened new horizons in our understanding of the cosmos. It allowed us to see celestial objects in ways previously unimaginable, revealing moons around Jupiter, the phases of Venus, and much more. Similarly, the first detection of gravitational waves in 2015 has started a new era of astronomical exploration. It allows us to observe cosmic phenomena, such as black hole mergers and neutron star collisions, that were once beyond our reach, fundamentally altering our understanding of the universe.
</p>
<p>
The question I aim to address is the following: When two objects such as black holes and/or neutron stars, undergo a binary coalescence, they emit gravitational waves. We detect these waves using our amazing detectors. What information can we obtain about the properties of the two colliding objects by analyzing the detected gravitational waves? Specifically, what insights can we gain about their mass, spin, equation of state, their surroundings, and so on?
</p>
<p>
What are gravitational waves? <a href="https://www.ligo.caltech.edu/page/what-are-gw" target="_blank">[LIGO]</a>
How are gravitational waves detected? <a href="https://www.gw-openscience.org/s/workshop4/" target="_blank">[GW Open Data Workshop]</a>
</p>
<p>
<img class="image fit" src="./images/60212ligoweb.png" />
<font style="font-size:12px;" >Image credits: <a href="https://www.science.org/content/article/gravitational-waves-einstein-s-ripples-spacetime-spotted-first-time" target="_blank"> [Link] </a> </font>
</p>
<p> One fun excercise is to hear the pleasing melody of graviational waves after converting them into sound waves. Here <a href="https://www.soundsofspacetime.org/sources--sounds-overview.html" target="_blank"> [Soundofspacetime] </a> you can listen to the sound of gravitational waveforms emitted by various binary sources. <br> CAUTION! gravitational waves are not sound waves and this is just a fun excercise.
</p>
<h3>Interesting articles / lectures on gravitational waves:</h4> <br>
<table class="center">
<tr>
<th>Massive Black Holes Shown to Act Like Quantum Particles</th>
<th> <a href="https://www.quantamagazine.org/massive-black-holes-shown-to-act-like-quantum-particles-20220329/" target="_blank"> [Link] </a> </th>
<th> (2022) </th>
</tr>
<tr>
<th>Gravitational-wave physics and astronomy in the 2020s and 2030s</th>
<th> <a href="https://www.nature.com/articles/s42254-021-00303-8" target="_blank"> [Link] </a> </th>
<th> (2021) </th>
</tr>
<tr>
<th>Gravitational waves, Einstein's ripples in spacetime, spotted for first time</th>
<th> <a href="https://www.science.org/content/article/gravitational-waves-einstein-s-ripples-spacetime-spotted-first-time" target="_blank"> [Link] </a> </th>
<th> (2016) </th>
</tr>
<tr>
<th>Gravitational Waves: A New Era of Astronomy Begins <br>
(Panel discussion at World Science Festival) <br>
MODERATOR: Brian Greene, PARTICIPANTS: Barry Barish, Nergis Mavalvala, Frans Pretorius, David Shoemaker, Rai Weiss</th>
<th> <br> <a href="https://www.youtube.com/watch?v=xj6vV3T4ok8" target="_blank"> [Link] </a> </th>
<th> <br> (2016) </th>
</tr>
</table>
<h3>Recent relevant reviews / snowmass papers:</h4> <br>
<table class="center">
<tr>
<th>Gravitational Waves and Scattering Amplitudes</th>
<th> <a href="https://arxiv.org/abs/2204.05194" target="_blank"> [2204.05194] </a> </th>
</tr>
<tr>
<th>Post-Minkowskian expansion from Scattering Amplitudes</th>
<th> <a href="https://arxiv.org/abs/2203.13024" target="_blank"> [2203.13024] </a> </th>
</tr>
<tr>
<th>Classical Gravity from Scattering Amplitudes</th>
<th> <a href="https://arxiv.org/abs/2203.13025" target="_blank"> [2203.13025] </a> </th>
</tr>
</table>
<h4>A list of seminal papers in the field of gravity can be found here <a href="https://pages.jh.edu/eberti2/posts/must-read-paper-list/" target="_blank"> [link] </a></h4> <br>
</div>
</section>
<!-- Research -->
<section id="research" class="two">
<div class="container">
<header>
<h2>Research</h2> <h4>(Details)</h4>
<!-- <a href="https://inspirehep.net/authors/1829940" target="_blank">[iNSPIRE]</a> -->
</header>
<p> Currently, I work on the studying astrophysical compact objects and gravitational waves emitted by them using the techniques of quantum field theories and scattering amplitudes. This problem is interesting because the recent direct detections of gravitational waves by LIGO and Virgo observatories have proved to be a probe for the physics of celestial objects like black holes and neutron stars. Such detections have the potential to unravel the mysteries of cosmic origins, equations of state of compact objects and will prove to be a test to the theory of general relativity. On the other hand, the framework of Quantum field theory has proven to be very successful at predicting physical behavior to high accuracy and precision and is the basic language of modern theoretical physics. Moreover, the techniques of scattering amplitudes provide us with several tools which make the calculation of the physical observables systematic and insightful.
</p>
<div id="floatedL">
<figure>
<img class="image fit" src="./images/scatteringampinGW.jpg" />
<figcaption><font style="font-size:12px;" >Image credits: <a href="https://www.quantamagazine.org/massive-black-holes-shown-to-act-like-quantum-particles-20220329/" target="_blank"> [Link] </a> </font></figcaption>
</figure>
</div>
<p> In my <a href="http://dr.iiserpune.ac.in:8080/jspui/handle/123456789/4705?mode=simple" target="_blank">master's thesis</a>,
<!-- under the guidance of Dr. Suneeta Vardarajan at IISER Pune, -->
I studied the method of Effective Field Theories (EFT) used to model the gravitational waves emitted by various binary sources. Such sources, due to their non-relativistic nature, are analytically studied using the techniques of post-Newtonian (PN) approximation by perturbatively expanding in orders of orbital velocity over the speed of light. I followed the calculation of Einstein-Infeld-Hoffmann (EIH) Lagrangian, which is the 1PN correction to the two body gravitational potential. Then, to consider the effects of the electromagnetic charge of the constituents of the binary system, I newly derived the corrections to this 1PN potential by adding photons in the formalism which can be found here <a href="https://arxiv.org/abs/2009.11107" target="_blank">[arXiv:2009.11107]</a>. To study the effects of the spin of binary constituents on the waveform, I reviewed the calculation of the Spin-Spin, Spin-Orbit, and Spin-squared effective potentials up to 2PN. I also followed the calculation of the Burke-Thorne radiation reaction force at 2.5 PN, using the Schwinger-Keldysh formalism of open quantum field theories.
</p>
<div id="floatedR">
<figure>
<img class="image fit" src="./images/multiloopinGW.jpg" />
<!-- <img class="image fit" src="./images/spin.png" /> -->
<figcaption><font style="font-size:12px;" >Image credits: <a href="https://www.quantamagazine.org/massive-black-holes-shown-to-act-like-quantum-particles-20220329/" target="_blank"> [Link] </a> </font></figcaption>
</figure>
</div>
<p>Currently, for my Ph.D. thesis, I am working
<!-- with Dr. Manoj Mandal, Prof. Dr. Pierpaolo Mastrolia, and Dr. Jan Steinhoff -->
on computing state-of-the-art higher-order effective potentials and fluxes. These potentials and fluxes are important as they are used to compute accurate theoretical waveform templates which are compulsory ingredients for data analysis and reliable physical interpretation of the gravitational wave signals. I am developing a fast and efficient Mathematica routine for the automatic computation of Feynman diagrams and the corresponding integrals using the multi-loop techniques of quantum field theories <a href="https://indico.phys.ethz.ch/event/22/timetable/#60-spinning-compact-objects-us" target="_blank">[Poster at QCDmeetsGravity2022]</a>. Using this I have performed the following analyses:
</p>
<!-- <div id="floatedL"> -->
<!-- <figure> -->
<!-- <img class="image fit" src="./images/spin.gif" /> -->
<!-- <figcaption><font style="font-size:12px;" >Tidal oscillations in the rest frame of NS</font></figcaption> -->
<!-- </figure> -->
<!-- </div> -->
<p>
<ul style="list-style: none;">
<li>Spin-orbit potential at NNNLO (at 4.5PN): <a href="https://arxiv.org/abs/2209.00611" target="_blank"> [arXiv:2209.00611] </a> </li>
<li>Quadratic-in-spin potential at NNNLO (at 5PN): <a href="https://arxiv.org/abs/2210.09176" target="_blank"> [arXiv:2210.09176] </a> </li>
</ul>
Astrophysical black holes and neutron stars usually have a non-zero angular momentum and this "spin" has a huge impact on the emitted gravitational waves. The effects of spin can be characterised as follows. The spin-orbit sector is analogous to the fine structure correction obtained to the hydrogen atom, which describe the interaction between the orbital angular momentum of the binary and the spin of one of its constituents. Whereas, the quadratic in spin are analogous to the hyperfine structure corrections to the hydrogen atom and describe the interaction between each other/self of the spins of the binary constituents. I have computed the effective Hamiltonians at 4.5PN and 5PN, that describe the effect of spin of the constituents on the binary dynamics, thus on the emitted gravitational waves.
</p>
<div id="floatedL">
<figure>
<img class="image fit" src="./images/NSCOMDy.gif" />
<figcaption><font style="font-size:12px;" >Tidal oscillations in NS (blue) due to companion (black). </font></figcaption>
</figure>
</div>
<p>
<ul style="list-style: none;">
<li>Dynamic tidal potential at 2PN: <a href="https://arxiv.org/abs/2304.02030" target="_blank"> [arXiv:2304.02030] </a> </li>
<li>Dynamic and adiabatic tidal potential at 3PN: <a href="https://arxiv.org/abs/2308.01865" target="_blank"> [arXiv:2308.01865] </a> </li>
<li>Adiabatic tidal fluxes at 2PN: <a href="https://arxiv.org/abs/2412.01706" target="_blank"> [arXiv:2412.01706] </a> </li>
</ul>
Neutron stars feature a number of oscillation modes, and to understand them we can picture a basic stellar model that consists of the continuity equation (conservation of mass), Euler’s equation (equation of motion for the fluid elements), Poisson’s equation (that determines the gravitational force from the matter distribution), and the equation-of-state (that describes how pressure and density are related). With these elements combined, we can describe a star in equilibrium, which we can then perturb. The lowest frequency normal mode of perturbed oscillation is known as the fundamental mode (f-mode), which is one of the dominant modes in the context of tidal excitation. I have computed the effective Hamiltonians at 2PN and 3PN, that describe the effect of such tidal f-mode oscillations on the binary dynamics, thus on the emitted gravitational waves. When these oscillation frequency is large as compared to the external driving tidal frequency (which happens in the early inspiral), the tides are locked and move adiabatically with the companion object in the binary. For this case of adiabatic tides, I have computed the effective Hamiltonians at 3PN and also computed the energy flux, the angular momentum flux, the mode amplitudes and the mode phase at 2PN.
</p>
<!--
<p>
Another recently developed method used in the modeling of the spinning compact objects is the scattering amplitudes of massive higher spinning particles. Specifically, spin structure of a Kerr black hole up to the 2^(2s)-multipole is reproduced by considering the scattering of massive spin-s particles which are minimally coupled to gravity. In this direction, I have worked -->
<!-- with Dr. Arnab Rudra, where we developed -->
<!-- on developing a method <a href="https://arxiv.org/abs/2106.05301" target="_blank">[arXiv:2106.05301]</a> to efficiently compute tree-level amplitudes of photon and graviton four-point functions with massive higher spin exchanges. These scattering amplitudes can be then used to compute physical observables for scattering of two Kerr black holes, like momentum transferred or impulse, momentum radiated as radiation, and binding potential with more precision.
</p>
-->
<div id="floatedR">
<figure>
<img class="image fit" src="./images/SEOBNR-PM_waveform.png" />
<figcaption><font style="font-size:12px;" >q is the mass ratio (m1/m2) and chi1 and chi2 are the spins </a> </font></figcaption>
</figure>
</div>
<p> I am also working on developing SEOBNR-PM <a href="https://arxiv.org/abs/2405.19181" target="_blank"> [arXiv:2405.19181] </a> which is a bound-orbit waveform model that uses the post-Minkowskian (PM) results in the effective one body (EOB) formalism. A typical time-domain waveform model uses the Hamiltonian (for conservative effects) and Fluxes (for dissipative effects) to evolve the dynamics of the binary system in the inspiral phase when the orbital radius is slowly decaying due to gravitational wave emission. Then when two compact objects collide on each other, a numerical relativity (NR) and black hole perturbation theory informed templated is used to model the merger and the ringdown. In SEOBNR-PM we use the PM scattering angle (upto 4PM for non-spinning and 5PM for spinning) to construct a PM Hamiltonian, which is a new feature of this model. Here is one example of a waveform generated using the model and its comparison to the corresponding NR waveform.
</p>
</div>
</section>
<!-- Loop-the-Loop -->
<section id="looptheloop" class="three">
<div class="container">
<header>
<h2>Loop-the-Loop</h2>
</header>
<div id="floatedL">
<figure>
<img class="image fit" src="./images/loop-the-loop-poster.png" />
</figure>
</div>
<p> I was part of organising “Loop-the-Loop: Feynman calculus and its applications to gravity and particle physics”, an online workshop that focused on bridge between the fields of particle physics, gravitational wave physics and cosmology, together with cutting edge mathematical techniques and computational methods. The three topics of the workshop were:
<ul style="list-style: none;">
<li>Day1: Applied Mathematics for Feynman Calculus <br>
<font style="font-size:12px;">
Sessions on: Integration-by-parts & Computational Techniques,
Intersection Theory, and
Analytic Structure of Feynman Integrals </font>
</li>
<li>Day2: Scattering Amplitudes in Gravity <br>
<font style="font-size:12px;">
Sessions on: Post-Minkowskian Integrals,
Extreme Mass Ratio Systems, and
Cosmological Correlators
</font>
</li>
<li>Day3: Scattering Amplitudes in Particle Physics <br>
<font style="font-size:12px;">
Sessions on: Amplitudes & Elliptics,
Differential Equations, and
Infrared Subtraction
</font>
</li>
</ul>
The workshop featured 3 review talks, 27 research talks, and 2 afternoon seminars, bringing together leading experts to share insights and foster collaboration, and attracted over 400 registered participants.
For more details, visit the <a href="https://indico.mitp.uni-mainz.de/event/423/" target="_blank"> [workshop webpage]</a>.
</p>
</div>
</section>
<!-- Pastime -->
<section id="pastime" class="three">
<div class="container">
<header>
<h2>Pastime</h2>
</header>
<p>I enjoy solving simple problems in 0+1 dimensional quantum field theories or a.k.a. Quantum Mechanics (thanks to Dr. Arnab Rudra, who introduced me to this). Here is the <a href="https://sites.google.com/iiserb.ac.in/rudra/teaching/quantum-field-theory-i?authuser=0" target="_blank">[link]</a> to the webpage of QFT 1 course by Arnab where he discusses about this formalism in detail. The challenging assignments on his page are very interesting and I would urge you to have a look at them.</p>
<p>
I enjoy coding while solving physics problems. Here I aim to put all the fun things that I did using Mathematica.<br>
Few of the nice packages of Mathematica are <a href="https://homepage.cem.itesm.mx/jose.luis.gomez/quantum/" target="_blank">[Quantum Notation]</a> for quantum mechanics, <a href="https://feyncalc.github.io/" target="_blank">[FeynCalc]</a> for computing amplitudes, <a href="http://www.xact.es/index.html" target="_blank">[xTensor]</a> for tensor algebra, and <a href="https://www.slac.stanford.edu/~maitreda/Spinors/" target="_blank">[S@M]</a> for massless spinor helicity algebra.<br>
Here are few of the projects that I was involved in,
<table class="center">
<tr>
<th>Toy Waveform Model <br> (Modified from <a href="https://arxiv.org/abs/1810.06160" target="_blank"> [1810.06160] </a>) </th>
<th> <a href="https://github.com/raj-physics/ToyWaveformModel" target="_blank"> [GitHub] </a> </th>
</tr>
<tr>
<th>Harmonic oscillator</th>
<th> <a href="https://github.com/raj-physics/HarmonicOscillator" target="_blank"> [GitHub] </a> </th>
</tr>
<tr>
<th>Toy Standard Model at 1 Loop <br> (Building up from simpler models)</th>
<th> <a href="https://github.com/raj-physics/ToySMat1Loop" target="_blank"> [GitHub] </a> </th>
</tr>
<tr>
<th>Amplitudes using Spinor helicity</th>
<!-- <th> <a href="" target="_blank"> [GitHub] </a> </th> -->
<th> Incomplete </th>
</tr>
</table>
</p>
<!-- <p>I also enjoy reading non-fiction books. The one that I would recommend are Sapiens (Yuval Noah Harari), 21 lessons for the 21st century (Yuval Noah Harari), The case for reason (Narendra Dabholkar), Selfish gene (Richard Dawkins), Outgrowing god (Richard Dawkins), Guns germs and steel (Jared Diamond), Philosophy of science (Samir Okasha), The signal and the noise (Nate Silver).
Some biographies that I would also recommend is by Joseph Polchinski <a href="https://arxiv.org/abs/1708.09093" target="_blank">[arXiv:1708.09093]</a>
</p>
-->
<!--
<p> Treks
</p>
-->
<!-- <p> -->
<!-- I also like to spend time on the internet. Some interesting link are given <a href="./subpages/interesting_links.html" target="_blank"> [here] </a> -->
<!-- </p> -->
</div>
</section>
</div>
<!-- Footer -->
<div id="footer">
<!-- Copyright -->
<ul class="copyright">
<li>
Last updated on 18-12-2024
Design: <a href="http://html5up.net">HTML5 UP</a>
</li>
</ul>
</div>
<!-- Scripts -->
<script src="assets/js/jquery.min.js"></script>
<script src="assets/js/jquery.scrolly.min.js"></script>
<script src="assets/js/jquery.scrollex.min.js"></script>
<script src="assets/js/browser.min.js"></script>
<script src="assets/js/breakpoints.min.js"></script>
<script src="assets/js/util.js"></script>
<script src="assets/js/main.js"></script>
</body>
</html>
<!--
<embed src="aaa.pdf" width="400px" height="400px" />
<a href="#" class="image fit"><img src="FINAL_Header_Gravitational_waves_LIGO_092916.jpg" alt="" /></a>
<div class="row">
<div class="column leftB">
<p>
blablabla
</p>
</div>
<div class="column rightS">
<img class="image fit" src="./images/41586_2019_1129_Fig2_HTML.jpg" />
</div>
</div>
-->