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Cliff Harvey
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Cliff Harvey

Homotopy Type Theory  - 
 
Conor McBride has been doing some insightful blogging lately, with an entertaining flair to boot. Here he discusses the problem of getting a computationally useful notion of univalence, considering Observational Type Theory as a prototype.

Thorsten Altenkirch, Wouter Swierstra and I built an extensionality boat. We might imagine that one day there will be a fabulous univalence ship: the extensionality boat is just one of its lifeboats. But nobody’s built the ship yet, so don’t be too dismissive of our wee boat. You might learn something about building ships by thinking about that boat. I come from Belfast: we built the Titanic and then some prick sailed it into an iceberg because they made a valid deduction from a false hypothesis.

Along the same lines, his previous couple posts about OTT are also worth a look. For reference I'll link the OTT paper as well:

https://pigworker.wordpress.com/2015/01/06/observational-type-theory-the-motivation/
https://pigworker.wordpress.com/2015/01/08/observational-type-theory-delivery/
http://strictlypositive.org/ott.pdf
"Why do you hate homotopy type theory?" is question I am sometimes asked, but I never answer it, because the question has an inaccurate presupposition. I am not happy when people forget that functi...
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Mathematics  - 
 
 
Diving for Calabi-Yau manifolds. Initially, i had written the following paragraphs in reverse order, more like being washed up ashore, saved from the deep. But i figured you might prefer to peacefully approach K3 surfaces, the secret topic of this post. Assuming you know what polynomials are, we'll start nice and slow:

What are Rational functions? You get those by dividing one polynomial by another N(x)/D(x). Their degree is handily defined to be the maximum power appearing in any of the two polynomials. Since the denominator mustn't be zero, we'll get funny "singularities" wherever the D(x) polynomial has a zero. Just as polynomials themselves, rational functions are quite an easy subject, so don't be shy. 

Some facts: They're representative for meromorphic functions. Just like holomorphic functions but singularities are allowed to appear at a finite set of points. You can "Taylor" them to your required precision, too.

https://en.wikipedia.org/wiki/Rational_function


The Elliptic integral appeared when people set out to calculate arc lengths of ellipses. It's an integral like this:

 x
∫ R(t, √P(t)) dt
 c  

That's a path integral, and you have to pick a starting point c for your path.

The result of the integration cannot, in general, be stated using elementary functions. But when we add the three Legendre canonical forms, also known as the elliptic integrals of 1st, 2nd, and 3rd kind, we can write the solutions as integral over rationals combined with these forms. The reduction formula does this.

https://en.wikipedia.org/wiki/Elliptic_integral

I just hate those notations, but luckily, there's also the Carlson symmetric form:
https://en.wikipedia.org/wiki/Carlson_symmetric_form


The Algebraic variety is the object of a projective geometric theory, in contrast to the linear algebraic groups, giving the better known affine theory. That's to be expected, because there's a division happening in rational functions, and you get projective spaces by "dividing" lines out, so you can look at the remainder.

One can define algebraic varieties as "the set of solutions of a system of polynomial equations". They're just like a regular manifolds, but may have singular points.

https://en.wikipedia.org/wiki/Algebraic_variety


A special case is the Abelian variety, defined as: A projective algebraic variety that is also an algebraic group. Such a group is given by regular functions, which are just another name for polynomials in this context. It means we call a curve Abelian variety when one can find a set of polynomial functions that generate a group.

Historically, the abelian variety played the role of generalizing the elliptic integral from involving polynomials of 3rd and 4th degree to ones involving 5th or higher degree. Originally studied over the complex numbers, over which they turned out to be exactly the complex tori that can be embedded in a complex projective space. Solomon Lefschetz worked that out, and also was the first to use the term Abelian variety.

Example, an elliptic curve is an Abelian variety of dimension 1.

https://en.wikipedia.org/wiki/Abelian_variety


Abelian varieties appear naturally as Jacobian variety of algebraic curves. We define: J(C) of a nonsingular algebraic curve C of genus g is the moduli space of degree 0 line bundles. You fill your surface with nonoverlapping lines, forget that a line has many points, and look at how lines are related.

As a group, the Jacobian of a curve is isomorphic to [...] divisors of rational functions. Our lines correspond to multiples of rational functions, just like any fraction can be given as a multiple of itself:

a/b = (x·a)/(x·b)

Note that plotting f(x) = x·a gives a line going through the origin.

https://en.wikipedia.org/wiki/Jacobian_variety


Abelian varieties also appear as Albanese variety of a curve C, a very natural generalization of the Jacobian to higher dimensions. The construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms (our "lines") generalizes straight forward to give the Albanese variety. Btw, when we're working over complex numbers, holomorphic means polynomial, 1-forms are the linear ones.

https://en.wikipedia.org/wiki/Albanese_variety


Enriques-Kodaira Classification of compact complex surfaces into 10 classes. For 9 of them we have a fairly complete understanding, but the remaining "general" type has too much in it. (Okay, our understanding for type VII depends on the global spherical shell conjecture.)

If you look at the diagram now you'll notice that the 9 well-understood classes are at most 1-d portions, lines of dots. The "general" type is the area left, i'm not surprised it's full of stuff.

https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification


One class are the K3 surfaces, named after Ernst Kummer, Erich Kähler and Kunihiko Kodaira. Actually there are four classes of K3 surfaces, but Andre Weil ran out of people to name fourth one. He chose the mountain K2 instead, which, as you can see, didn't really make it into modern language....

They're the complete smooth surfaces with trivial canonical bundle. That's exactly those 2-dimensional Calabi-Yau manifolds that aren't complex tori. The latter are simple Abelian varieties and much less interesting simpler, so they already got their own class.

Another way to define Calabi-Yau surfaces is as complete smooth surfaces with trivial canonical bundle. The canonical bundle is one passing through zero, it's called trivial when it's made of "lines". Maybe a better way to understand them is to state that all of them have Kodaira dimension 0...

https://en.wikipedia.org/wiki/K3_surface


The Kodaira dimension  k is one less than the number of algebraically independent generators. It's the rate of growth k such that P(d)/d^k is bounded. All Abelian varieties have zero Kodaira dimension! While k > 0 corresponds to hyperbolic, k = 0 to flat, sometimes, k is undefined or negative. We then say k = -oo and those cases correspond to surfaces with positive curvature.

So our surfaces are the "flat" ones in the above sense.


Initially, i had written this post in reverse, trying to figure out what this thread by +David Roberts mentioning K3 surfaces, Chern forms, and the magic number 24 was all about:
https://plus.google.com/+DavidRoberts/posts/CoBq5Y5Haqp


More links

Since the Jacobian variety is the connected component of the identity in the Picard group of our curve, i should have told you about Picard varieties and their associated Picard groups as well. I guess i'll have to do that next time, and maybe then we can also talk about how to blow something up:
https://en.wikipedia.org/wiki/Blowing_up

See also the Abelian surface, which are just Abelian varieties in 2d.
https://en.wikipedia.org/wiki/Abelian_surface

Hodge diamonds give another invariant, just like the Kodaira dimension is:
https://en.wikipedia.org/wiki/Homological_mirror_symmetry#Hodge_Diamond

The diagram is by R.e.b.:
https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification#/media/File:Geography_of_surfaces.jpg

The portraits i found here:
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Kodaira.html
https://upload.wikimedia.org/wikipedia/commons/5/52/Federigo_Enriques.jpg

#ScienceSunday - #algebraic #geometry , #elliptic integrals, #jacobian #variety
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There's a nice Scholarpedia article by Yau: http://www.scholarpedia.org/article/Calabi-Yau_manifold
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This is some fantastic reporting by +Quanta Magazine on recent progress on the Monstrous Moonshine conjecture,  definitely one of the most fantastically intriguing and yet bafflingly inexplicable connections in mathematics and physics.

It's a story that relates number theory, string theory, the Monster group, and optimal sphere-packing patterns, among other things. The grandest insight in the story was proved in 1992: that the modular j-function which knows so much about number theory exactly describes bosonic string theory on a particular spacetime – a spacetime which corresponds to a particularly optimal packing of spheres. But there is clearly much more of this relationship to uncover, in particular relating to other sporadic groups. This latest article correlates to new proofs published on these other branches of the Moonshine story.

This incredible understanding will undoubtedly be a bonanza for pure mathematics, but there also seems a decent chance that physics will be impacted as well, at least as it pertains to string theory.
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+John Baez well, I feel background-dependence tends to be underestimated so that "basic ideas" is less unequivocal than it sounds. I also find that Moonshine with Ramanujan's constant and the like, exhibit features making it a fascinating real world candidate analogue to the monolith of "2001 Space Odyssey", together with the "magnetic anomaly" that draws attention to it in the movie.

BTW, the natural integer that Ramanujan's constant approximates so closely, cries for being rewritten
640320^3+744 and 640320 in turns flaunts divisibility by 2001 while its being cubed evokes the idea of a parallelepiped. A good enough allusion to the 2001 monolith, as far as I am concerned:)

I've published my own share of the article while introducing it with a discussion of Ramanujan's constant here:

https://plus.google.com/u/0/+BorisBorcic/posts/KAnpb4ht6pN
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This saga about the BICEP measurement of the would-be primordial B-mode radiation continues to be highly fascinating. I noticed this write up by +Matt Strassler that seems to do a pretty good job of explaining in general terms what’s happened recently as the BICEP and Planck experiments collaborate to compare their data, which he analogizes to two people with differently-colored sunglasses needing to communicate to infer the overall state of the sky.

Even the disappointing development of learning this signal can’t be unambiguously attributed to cosmic inflation has had some definite positive aspects in retrospect. It’s clearly the case that seemingly earth-shattering measurements often turn out to be wrong, for a variety of reasons, and watching it unfold is clearly conducive to developing the right kind of skepticism. It really has demonstrated why science simply works, when the right kind of substantive and careful criticism is brought to bear, as it was in this case. Beside this aspect, the initial surge of excitement around the possible signal compelled me to try to understand inflation in much more detail, especially how it generates the gravitational waves and the B-modes, and to learn about the challenges that confront attempts to make fundamental sense of inflation. So from my purely selfish considerations, this was a definite plus.

Of course it’s not just the positive signals that move science forward anyway, but excluding parameter space also gives tremendously valuable information. The limits from various experiments on the two main parameters characterizing inflation are illustrated at the chart in this post:

http://resonaances.blogspot.se/2015/02/weekend-plot-inflation15.html

It’s mentioned briefly on that blog, and its also been pointed out to me by +Urs Schreiber, that one of the only inflationary models that sits right in the center of the preferred region is the Starobinsky model or R^2 model. This model is so named because instead of depending in the most straightforward way on R – the Ricci curvature scalar of general relativity – it adds a term proportional to R^2 to the fundamental equation (the Einstein-Hilbert action). There are a couple reasons this could be a very fascinating possibility from the standpoint of fundamental physics. For one thing, it can embed into supergravity and which reduces the required size of the initial homogeneous region by a couple orders of magnitude. For more on that I’ll refer you to his post and well-sourced wiki page:

https://plus.google.com/108081058828040288656/posts/T1PimSeZEUH
http://ncatlab.org/nlab/show/Starobinsky+model+of+cosmic+inflation

It’s interesting to note in passing that this possibility, more generally called f(R) gravity, seems to be one of the very few ways that general relativity itself can be modified without instantly running into serious problems of one sort or another. I recall Nima Arkani-Hamed’s slogan “Don’t modify gravity, understand it,” which is particularly intriguing when you learn that R^2 gravity turns out to be physically equivalent to standard general relativity coupled to a new scalar field(!). It is this field that acts as the inflaton.

https://en.wikipedia.org/wiki/F(R)_gravity

It seems that no matter what new experimental signals turn up there will be many fascinating insights to uncover, and even if most of the models we focus on turn out to be wrong, if we get to better understand the space of consistent possibilities along the way, progress will still have been made.

#sciencesunday
© Matt Strassler [February 6, 2015] Unfortunately, though to no one's surprise after seeing the data from the Planck satellite in the last few months, the BICEP2 experiment's claim of a discovery o...
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All experts I had talked to in person always just highlighted that Starobinsky is the best fit, but now there is debate about how much better it is than other fits. Table 6 in http://arxiv.org/abs/1502.02114 has the answer to this, but I am not sure if I know how to read it. I gather we are to look at the third and fourth column for the Bayesian relative likelihood (which is what is amplified in the text on the preceding pages). The notation "ln B_0X" seems not to be introduced in this article, though. Help me: what is the meaning of an entry "-k" in one of these columns? Is it: "the corresponding model is e^-k times as likely as the R^2 model?"
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Mathematics  - 
 
From Veneziano to Riemann: A String Theory Statement of the Riemann Hypothesis
http://arxiv.org/abs/1501.01975
We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function. We also discuss how the integral criterion based on the argument of the Riemann zeta function relates to the Li criterion for the Riemann hypothesis. We provide a new generalization of this integral criterion. Finally, we comment on the physical interpretation of our recasting of the Riemann hypothesis in terms of the Veneziano amplitude.
_____________________

Hilbert and Pólya independently suggested a physical realization of the Riemann hypothesis: if the zeroes of the zeta function in the critical strip are the eigenvalues of (1/2)I + iT and T is a Hermitian operator acting on some Hilbert space, then because the eigenvalues of T are real, the Riemann hypothesis follows. We do not know what the operator T is, nor what Hilbert space H it acts on. Nevertheless, the Hilbert-Pólya conjecture constitutes a tremendous insight into how a mathematics problem can be mapped to a physical system. Many recent works seek to establish such a connection explicitly.

In this note, we establish a physical realization of the Riemann hypothesis in terms of the properties of bosonic strings. In particular, we consider equivalent statements of the Riemann hypothesis written as integrals of the logarithm of the zeta function or the argument of the zeta function evaluated on the critical line. We link these expressions to the Veneziano amplitude describing the scattering of four bosonic open strings with tachyonic masses. This opens a fascinating new connection between string theory and the physics of the Riemann zeros. We also discuss the relation to the Li criterion for the Riemann hypothesis discussed in our previous publication, and we generalize the integral criterion based on the argument of the Riemann zeta function evaluated on the critical line.

In this opening section, we point out the fundamental reason why string theory, through the form of the Veneziano amplitude, should know about the Riemann zeta function, and thence about the Riemann hypothesis...
Abstract: We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function.
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+Urs Schreiber thanks for always doing such a great job compiling these references. I still have a long way to go on grasping much of the information on nLab, but I am consistently impressed with it, and I definitely share the general sense of enthusiasm about the main concepts. Your participation here is always very much appreciated! ;)

There are many more things I've been meaning to post about in this community, including some of your own work, but I wanted to make sure I remembered to state this general sentiment.
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For years I've waited for someone in the movie business to understand this one simple truth: no matter what fantastical stuff you might dream up for your sci-fi space flick, none of it will ever be more amazing or more provoking than what actually happens in real-life black holes. Well my wait is over. Chris Nolan and the people working on Interstellar have recruited Kip Thorne to help depict a black hole as it would actually appear, based on the equations of general relativity.

It looks stunning, and regardless of what the rest of the movie is like, seeing some of these world-class renderings of the black hole horizon and its accretion disk would practically be worth the ticket price on their own (but I've heard good things... fingers crossed ). As Kip says, studying the math of GR and black holes at any level has to leave you with some desire to see this kind of spectacle rendered as accurately as possible. (With the kinds of budgets these productions get away with, its almost a no-brainer to fund a mini research program for such a film.)

Apparently its been a productive collaboration even purely in terms of physics and computer graphics research, but I just have to see this imagery they've produced.
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My headache. Is a 3
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Cliff Harvey

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This is a fantastic development, one I've been hoping for and advocating for for many years. I have very little doubt that if this can get through without being derailed the world will be much safer as a result.

The truth is that there exists a tremendous technological capacity to monitor a nuclear program and obtain strong guarantees of its non-weaponization. Many basic aspects of this verification regime have already been in place with the NPT Safeguards agreement, but from the details that were outlined yesterday, all weak points of this existing regime will be comprehensively addressed, over the entire fuel cycle from the Uranium mines to the enrichment itself. And of course besides the intense verification aspect, the significant technological limitations that have been agreed to will ensure any hypothetical decision to pursue weaponization would take significantly longer than it otherwise would – about a year just to get the requisite enriched uranium.

If there is any legitimate gripe to be had against this agreement, the idea that it will "pave the path" to a nuclear weapon is absolutely not among them.

Parameters for a Joint Comprehensive Plan of Action
https://www.whitehouse.gov/sites/default/files/docs/parametersforajointcomprehenisveplanofaction.pdf
The more you look at the terms of the nuclear framework agreement, the more favorable they look for the United States. That's surprising.
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discovered a thing you may be interested in:
http://Paradox-One.today  Don't get spooked!
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Mathematics  - 
 
 
"Towards the proof of the geometric Langlands conjecture"
by Hebrew University of Jerusalem
36 videos, 2390 views. Last updated on 7 Jun 2014
Workshop: "Towards the proof of the geometric Langlands conjecture"

https://www.youtube.com/playlist?list=PLT-roSWIpp1FPjDpEXizVfqrvvZ0U0DnM

Workshop Website - https://sites.google.com/site/geometriclanglands2014/
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Way beyond me. Maybe I will have the chance to learn new advanced stuff during the summer. 
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An anomaly (or statistical fluctuation?) worthy of keeping an eye on at LHC13.
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Quantum Field Theory  - 
 
Simple Recursion Relations for General Field Theories
http://arxiv.org/abs/1502.05057

This looks like an exciting development in the program of reformulating quantum field theory in full generality:

On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior of an n-point scattering amplitude under an m-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory.

Clifford Cheung, Chia-Hsien Shen, Jaroslav Trnka
Abstract: On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to ...
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Videos: lectures, interviews, panels  - 
 
 
Ed Witten-2+1 Dimensional Gravity Revisited

This is a very important self-contained lecture by Witten regarding the famous correspondence between the symmetries of the  BTZ black hole in 2+1 dimension QG and the Monster symmetry group of the dual CFT (related to the Moonshine conjecture in Mathematics).

This famous correspondence was suggested by Witten around 2007 [1].

[1] Three-Dimensional Gravity Revisited

http://arxiv.org/abs/0706.3359

#Physics #Mathematics #quantumgravity  
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Quantum Field Theory  - 
 
A great summary of some special QFT's descending from the (2,0) theory in 6 dimensions by +Giotis Mth.

(Many more of your posts would be greatly welcome here.)
 
Terminology and offsprings of 6d (2,0) SCFT

One of the main properties of 6d (2,0) SCFT of Lie gauge algebra of type g is that via its compactification on certain d-dimensional manifolds provides a geometrical description of certain lower 6-d dimensional Supersymmetric gauge QFTs of type g. Then you can establish a dictionary between observables of the Supersymmetric gauge QFTs and quantities defined on the d-dimensional manifold.

Depending on the construction and their characteristics the classes and sub classes of these 6-d dimensional theories have specific names. Here are the two important classes S and R:

Class S Theories

A Class S theory is the 4d SCFT theory obtained by compactifying 6d (2,0) SCFT on a 2d Riemann surface C with certain genus and with or without punctures (i.e. topological defects).

In that respect the 4d N=4 SCFT MSYM is a class S theory obtained by compactifying 6d (2,0) SCFT if the 2d  Riemann surface C is a 2-Torus T2.

Note that often people use the term Class S theories only for the N=2 SCFTs theories obtained by compactification (partially twisted) of the 6d (2,0) SCFT on generic Riemann surfaces with punctures (punctures are related to flavor symmetries of the 4d theory).

Basically this class of theories and their dualities were introduced in a seminal paper by Gaiotto [1].
  
Tg Theories

A Tg theory, with g denoting the gauge algebra g, is a class S special theory obtained by compactifying the type g 6d (2,0) on three-punctured sphere; it is usually called Tn theory when the Lie algebra g is of type An−1.

It has g^3 flavor symmetry.

Class R Theories

Class R theories are the 3-dimensional analogue of class S theories obtained by compactifying 6d (2,0) SCFT on special 3-dimensional manifolds M. 

These are 3d N=2 SCFTs obtained via Chern-Simons Matter theories (CSM).

For their construction and the relevant dictionary between the 3d manifold M and the 3d theory refer e.g. to some of the original work in [2] and [3]. 

References

[1] N=2 dualities

http://xxx.lanl.gov/abs/0904.2715

[2] Gauge Theories Labelled by Three-Manifolds

http://arxiv.org/abs/1108.4389

[3] 3-Manifolds and 3d Indices

http://arxiv.org/abs/1112.5179

#Physics   #stringtheory  
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Has anybody been involved in creating class R variants for use with coordination chemistry? I'm looking for research dealing with electron momentum for quantized energy states.
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    Physics
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I love high energy physics, computer science and giant robots
Introduction
Above all, Im passionate about physics. More generally Im interested in furthering understanding of this world at all levels, from particle physics to economics to galactic superclusters and everything in between. I have an artistic side I once cultivated aggressively but has somewhat fallen by the wayside. My inner artist/musician is howling to get let out soon.

As a physics major at WPI, I did a couple major research projects on quantum information theory, specifically on proofs of Bell's theorem involving 4 and 5 qubits. Those projects heavily shaped the way I view physics in general, and its deep foundational issues in particular. Despite the difficulty of the subject, I am convinced there is much more confusion than is necessary, as sloppy reasoning and explanations persist.

At the moment Im mostly spending my time learning quantum field theory and string theory. I am consistently amazed by the unity of physical and mathematical logic, and as I've studied the structure of this logic I've inexorably gravitated to the stringy school of thought. Despite the challenges, it seems to me an essentially indispensable set of puzzle pieces that allow the whole structure to make sense. It appears so deeply enmeshed that extrication just does not seem very likely. But, as a scientist, I of course try to challenge any presumptions of mine aggressively.

Ill forever be a theorist at heart, but I also want to find a rewarding way to use my skills for something more down to earth. I think becoming a better computer scientists may be one of the better ways for this to happen.

And I believe that those of us awake to the challenges facing the human race – especially intellectuals, anyone who analyzes things systematically – has a certain obligation to actively stand for the truth and what is right. As a US citizen Im especially focused on the issue of campaign finance as fundamental to all other political problems in the US. 

"There are a thousand hacking at the branches of evil to one who is striking at the root."

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