I’ve been in Grenoble for two days now and am loving it. Grenoble is surrounded by mountains (the Vercors, the Chartreuse and the Belledonne (wow, I could recall all of them!)) on all sides with two rivers — Le Drac (the lion) and Le Isere (the serpent) — going through it. Indeed, it is the capital of the Alps.

Today I went on a hike in the Vercors mountain range, climbing the peak Le Moucherotte (1910m). I started in the morning by taking a bus from the Grenoble station to Saint-Nizier (110m) and started my hike from there.

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Signboard at the start

It was a perfect sunny day as I looked up to La Moucherotte.

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La Moucherotte

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The path went through scenic woods but it was a brutal, merciless ascent without resting spots.

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Crazy mountain biker!

Tired as I was climbing, a dog ran past me followed by it’s owners. “Bon jour”, I replied my ego crushed. I saw quite a few people on the way – people and their dogs and bicycles – doing the trek. Here is a mountain biker.

Finally I reached the summit after the promised two and a half hours, much to my surprise.

Beer time!

Beer time!

The circular table is called an orientation table and it helps to find which mountain range you can see in every direction.

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Snow!

There was some snow on the way, to say the least.

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View during descent

Waiting for the bus, chilling with some Italian food.

Waiting for the bus, chilling with some Italian food.

I am on a trip to CIRM, Luminy which is a research institute in the town of Marseille in France, for a conference in mathematics. CIRM seems to be a wonderful place for Mathematics, away from the hustle and bustle of the city. The campus is on a mountainous terrain which meets the Mediterranean sea at stone’s throw. The result is a beautiful trail leading to what the French call a “calanque“.

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Since this has been my first visit to France, I couldn’t help but observe a few quirks about the place. At the train station, there was a mobile charger which you needed to pedal, to charge your phone; can’t imagine this in the US! I traveled from Paris to Marseille, which is a distance of 800 km in 4 hours in the famous TGV train. Bullet trains are also unusual in the US.

CIRM is a purely research institute and in fact, is purely for conferences. Every week there is a conference in some area of Mathematics and researchers around the world in specific areas gather here to attend it. The lunch and dinner at the institute are elaborate and methodical and indicate to me that the French take their food seriously. Every meal is a five-course meal : entrée –> main dish –> bread + cheese (baguette et la fromage) –> dessert –> café. One of the days, we were served the famous Bouillabaise of Marseille, which is a popular local fish dish. It was served with white wine, customized to pique the mathematicians’ taste buds (read the description).

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The Mathematician’s wine

Today I went to the local supermarket to buy fruits and other groceries. It was amusing to see the number of aisles reserved for cheeses and vin (wine). Language has been a problem because only youngsters seem to know English. But politeness and the willingness to learn their culture and language go a long way in making good conversations.

Umm… what else? Well, cars park on sidewalks, cats roam freely. Roads seem narrower and cigarettes are longer. But my sample-space of a part of a small town may be too small to generalize. Tomorrow I hope to conquer Mont Puget which might offer an excellent view of the Mediterranean. More later, au revoir!

In the coffee hour discussion today, Nick gave me an interesting explicit example of an exceptional isomorphism of Lie groups.

G = \mathbb{SL}(4, \mathbb C) \to \mathbb{SO}(6, \mathbb C)

is an isomorphism via the \bigwedge^2 map. Let me elaborate. The group G acts naturally on V = \mathbb C^4. If \{e_1, e_2, e_3, e_4\} is a basis of V, then a basis of \bigwedge^2 V could be taken as \{ e_1 \wedge e_2, e_1 \wedge e_3, \cdots, e_3 \wedge e_4 \}. Thus, \mathbb{GL}(\bigwedge^2 V) is \mathbb {GL}(6, \mathbb C) and our G-action gives a monomorphism of groups:

\bigwedge^2 : G=\mathbb{SL}(4,\mathbb C) \to \mathbb{GL}(\bigwedge^2 V) = \mathbb {GL}(6,\mathbb C).

The image must actually be inside \mathbb{SL}(6,\mathbb C).

The quadratic form

(\; , \; ) : \bigwedge^2 V \times \bigwedge^2 V \to \mathbb C,

(e_i \wedge e_j , e_k \wedge e_l ) \mapsto e_i \wedge e_j \wedge e_k \wedge e_l

is symmetric (two permutations) and preserves the G– action, i.e.,

(g . \left(e_i \wedge e_j\right) , \left( g . e_k \wedge e_l \right) ) = (e_i \wedge e_j, e_k \wedge e_l) .

Hence the image of G in \mathbb{GL}(6, \mathbb C) \simeq \mathbb{GL}(\bigwedge^2 V) must also preserve this symmetric bilinear form. Thus, \bigwedge^2(G) \subseteq \mathbb{SO}(\bigwedge^2 V). By dimension consideration, they must be equal.

I ran the risk of opening a pandora’s box if I posted this on Facebook (with random ‘friends’ starting discussions on my timeline) so I decided to post my views here. This post is for me to walk down the memory lane by reading it in the future rather than start a fight here. Plus, it’s been a notoriously long time since I posted anything here.

Modi was unanimously elected the Prime Minister in May 2014 when BJP won the elections with an overwhelming majority. With over an year of the government being in power, issues are being raised about the growing intolerance. There are two things I would like to point out.

(1) Everything Modi and the BJP does is being scrutinized. You either love Modi or hate him; there isn’t a middle ground. Why didn’t people focus on the Congress and it’s wrongdoings? Incidents like the Dadri lynching are outrageous and bring shame to my country but they haven’t cropped up only after the BJP rose to power, they only became more visible. The problem with Indian secularism is that it works only in favour of minorities. The fact that media is no longer unbiased in it’s reporting and goes only for sensationalism doesn’t help either.

(2) As citizens with the power of (just) “one” vote, if good governance is our objective, we should vote for the party with ‘lesser evil’. That automatically gives the onus on politicians to perform. If BJP doesn’t live up to it’s standards, vote it out! If it does improve my country, re-elect Modi. With a highly active and involved ministry, I think (hope) Modi’s cabinet gets another chance. After all, don’t we all want the Uniform Civil Code re-implemented, as was promised in the BJP manifesto?

I was wondering the rationale behind naming parabolic subgroups of linear algebraic groups. The answer, interestingly, comes from the action of SL(2,\mathbb R) on the upper half plane. (I came up with this little discovery on my own.)

The orbit of the point i under the action of the standard parabolic subgroup of SL(2,\mathbb R) is a parabola.

The upper half-plane is an object that comes up in many parts of mathematics, hyperbolic geometry, complex analysis and number theory to name a few. The group SL(2,\mathbb R) acts on it by fractional linear transformations:

\begin{pmatrix} a & b \\ c & d \end{pmatrix} .\ z \mapsto \displaystyle \left( \frac{az+b}{cz+d} \right) .

A parabolic subgroup is it’s subgroup P such that quotienting by P gives a compact variety. Upto conjugation, the only parabolic subgroup of SL(2,\mathbb R) is \begin{pmatrix} * & * \\ 0 & * \end{pmatrix}. It’s action on the point i is given by:

\begin{pmatrix} x & y \\ 0 & x^{-1} \end{pmatrix} . i = \displaystyle \left( \frac{xi+y}{0i+x^{-1}} \right) = y + i x^2

whose locus is a parabola.

I wonder why textbooks in algebraic groups don’t mention this!

EDIT: (21 April, 2015) The above calculation is WRONG. I don’t know the answer to “parabolic”.

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Now I’ll think twice before going to Greyhouse. Here is why.

I went to the coffee-house today evening for a cup of GreyCap and Cogdell’s book on Automorphic L-functions. Greyhouse is a wonderful place to do some Mathematics and at times, just gaze around and see what other people are doing. My Chess buddies often meet here for some rapid games. Today the place was less crowded since the school is closed for summer.

A lady approaches me and asks me my age. I have seen her earlier around the campus and even in Greyhouse. I remember last summer, she once interrupted our Chess discussion and made an awkward conversation. Kerem pointed out she’s crazy but “not guilty until proven” is my principle.

Not wanting to create a fuss, I told her my approximate age and she left me alone. After a while, she came to me and asked if I could share the table with her. Honouring the etiquette of the coffee-house, I welcomed her. Now she asks if I could be her friend. I say yes. But soon things start getting weird.

“Would you like to be my friend with benefits?”

“No.” (What the fcuk!!)

“Do you know what that means?”

“I think I know.”

“Hugging, kissing, … ”

“Yes I know” (This was getting creepy)

“Do you have a girlfriend?”

“No”

“Why?”

“Because I love some things (pointing at my book) more than others.”

She laughed aloud enough for everyone around to hear. I could feel the pain hidden in that laughter. I felt pity toward her.

She tore off a page from her pocket-diary and wrote her phone number on it. When I accepted it, she asked mine. I refused to share my cell number with a creepy lady I just met. She asked her note back and I gladly obliged. With some more awkward comments, she left. I shared a moment with the Indian girl sitting at the adjacent table; I could tell she overheard our conversation.

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The conversation had an impact on me I hadn’t imagined. It was not just any other crazy experience but somewhat traumatizing one. I felt shaken, mildly violated although all she did was have an extremely weird conversation. I wonder what it would be to be a girl in India. Eve-teasing, stalking, groping are real problems because of roadside Romeos – jokers who don’t know how to woo a girl but know a lot of other creepy mischief. The askew sex-ratio also doesn’t help. My (female) friend used to tell me the little ways she used to retaliate against these miscreants. They should be brought to book. Hopefully “Ab ki bar Modi sarkar” or AAP improve things.

पूजा: अाईला सांगू नकोस!

I came across a simple statement in finite group theory that I’m almost upset no one told me earlier. The source is Serre’s book ‘Linear Representations of Finite Groups’. Serre used this statement below to define / prove Brauer’s theorem on induced representations of finite groups, using which one proves the meromorphicity of Artin-L-functions. Here it goes.

G is a finite group and p is a fixed prime. An element x of G is called punipotent if x has order a power of p and pregular if it’s order is prime to p.

Cool result:  Every element x in G can be uniquely written as

x = x_u x_r;

where

  • x_u is p-unipotent, x_r is p-regular,
  • x_u and x_r commute and
  • they are both powers of x.

The proof is really easy. Just replace G by the (finite) cyclic group generated by x!

I call it the Jordan decomposition because we have a similar decomposition for endomorphisms (among other things).

Let V be a finite dimensional vector space over an algebraically closed field of characteristic zero (just in case!). Each x \in \text{End }(V) can be uniquely written as

x = x_s x_u;

where

  • x_s is semisimple (diagonalizable), x_u is nilpotent,
  • they both commute and
  • they are polynomials in x without a constant term.

Pretty cool, huh!

This post is just a collection of basic results I have compiled for referring to in desperate times Nothing too deep except Haar-von Neumann’s theorem.

A measure \mu on a locally compact (Hausdorff, always) group G is left-invariant if

\boxed{ \int_G f(s^{-1}x) \text{d} \mu(x) = \int_G f(x) \text{d} \mu(x). }

The most important theorem in this topic is the existence and uniqueness of the Haar measure (proven by Haar and von Neumann, respectively).

Theorem 1: On every locally compact group G there is a unique (up to a positive constant of proportionality) left-invariant positive measure \mu \neq 0.

Proposition 2: G, \mu as usual. For G to have a finite measure, it is necessary and sufficient that G be compact!

Proposition 3: There is a continuous group-homomorphism (called right modulus), \triangle_r : G \to \mathbb R^*_+ such that

f(xs^{-1}) \text{d} \mu(x) = \triangle_r(s) \int f(x) \text{d} \mu(x).

Proposition 4: Let \mu (respectively, \nu) be a left-invariant Haar measure on locally compact groups H (resp., K). Then the product integral on G = H \times K is left-invariant and

\boxed{\triangle^G_r(s,t) = \triangle^H_r(s) . \triangle^K_r(t).}

Corollary 5: G is unimodular (i.e., \triangle_r =1) precisely when H and K are so.

The most important example (for me) is the general linear group over number fields or p-adics and its subgroups. All semisimple (and more generally reductive) groups are unimodular. Compact groups are unimodular. Abelian groups are trivially so. However the measure on Borel (and parabolic) subgroups (i.e., upper triangular matrices) is not unimodular. The above proposition 4 allows one to transfer the Levi decomposition on the groups to their measures.

About me

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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