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.

I just finished watching The Silence of The Lambs (1991).  Starring Jodie Foster and Anthony Hopkins, it is probably the first of its kind and has won many Academy Awards. The mystery thriller is rated 8.6/10 on IMDB so you can imagine how amazing it would be.

It might be just me but I find an uncanny resemblance with The Call (2013) starring Halle Berry. I watched it yesterday; the story is about a 911 call. Both movies involve psychopath killers, a kidnapped girl and a female cop. In The Silence …, Jodie Foster is an FBI agent whereas Halle Berry is a 911 operator. Both movies are centered around rescuing the victim. Both movies end with cops being misled to the wrong location whereas the actress locates the right house, barges in alone without any reinforcements, is about to be killed by the perpetrator but finally manages to overpower the bad guy!

Too much for coincidence? !!

————————————————————————

Tailpiece: I just came across the most widely spoken constructed international auxiliary language – Esperanto. http://en.wikipedia.org/wiki/Esperanto

It has been almost a week since I returned to Purdue. The past month in India passed so quickly it just felt like a week. I was glad I could meet my family and so many friends.

I attended a workshop and conference on Galois representations at the TIFR. Situated besides the sea-shore, it always feels great to visit the Tata Institute. I was very surprised Prof. Ghate recognized me although we just met  once, that too three years ago. I met Prof. Rajan (my VSRP mentor) and my Masters advisor Prof. Anandavardhanan. It was nice to finally meet Sandeep Verma, a student of my advisor Dr. Shahidi. I couldn’t meet Anand Sawant since he was making an academic visit to Germany but it was good to talk to Sachin Sharma, my old friend at IMSc doing his post-doc at Tata. Although I couldn’t meet Arghya, I made friends with his friends – Shaunak, Aditya, Vineet and Ashay – and we discussed some Mathematics. There were prominent mathematicians attending the conference including Marie-France Vigneras, Sujatha Ramadorai, Pierre Colmez, Dipendra Prasad and Chandrashekhar Khare.

Conference photo

Back home, it was after a long three-year wait that my parents, sister and I were together. We celebrated by going on a short vacation to Jaisalmer and Jodhpur in Rajasthan. There were sand dunes around (it’s a freaking desert) and we had a camel safari. It was fun!

The vibrant colours of India (cloth shop inside Jodhpur fort)

Rajasthani folk dancer (Yes, they are glasses she’s balancing on)

On the pretext of giving his iPad to his folks, I met Partha’s dad.  Fortunately he didn’t ask me usual questions Indian parents ask! Shashank is one of my best friends and he too came to India to attend Avdhut’s marriage. We went to visit Vaibhav (photo below) and had an interesting discussion on genetics and evolution. As always, it was fun meeting Amar – my roomie at IITB. He went paragliding near Pune a day before I left for the US and seeing these pics, I regret not being able to join. Yes, I also met my Purdue roommate Nikhil, who came all the way from Pune to meet me. Nikhil, Rubin and I had a great time cracking Purdue jokes here in India. Another close friend Ameya married last year and in this India visit, I met Asmita and him. (Everyone seems to be getting married 😦 I’m getting older.)

Vaibhav and Shashank

The star of the show was Avdhut, whose marriage was a pushing reason I visited India. His brothers had arranged a bachelors’ party a week before the wedding and I won’t go into details because what happens in Murud stays in Murud. But yes, I did meet Arun and have curious philosophical discussions with him. Avdhut’s marriage was a good excuse for VNITians to meet up – Raju, Tapa, Swapnil, Milind, SKS, Khan and the great Maythegod! Bless the newly wed couple!

Avdhut weds Aasawari

My awesome dream-visit was brought down to earth after landing at Chicago. It was -27 degree Celcius cold!

Tailpiece:

* Just yesterday, I got a Macbook Pro! I promised myself I’d write this post on my new Mac.

* I was stuck up in Chicago since the interstate I-65 was dysfunctional due to heavy snowfall; shuttles were canceled and the train was overbooked. Kyle rescued me by giving a ride since he too was driving from Chicago back to WL. Hail Facebook!

Kyle gave me a ride to Purdue

* Steve Spallone is visiting Purdue next week. He is an American mathematician working in India and I’m quite the opposite. We discovered each others’ blogs accidently. Curiously, he too works in number theory and in fact is Shahidi’s collaborator. I’m looking forward to his talk!

The land of organized chaos

Let $G = GL_2(F)$ for some field $F$ and $B$ be the subgroup of upper triangular matrices in $G$. As usual, let $\mathbb P^1 = \mathbb P^1(F)$ denote the projective plane over $F$.

As $F$-varieties, $G/B$ is isomorphic to $\mathbb P^1$ because $B$ is the stabilizer of the usual (Mobius) action of $G$ on $\mathbb P^1$. The resulting action of $G/B$ on $\mathbb P^1$ is 1-transitive and fixed-point free. Indeed, counting points when $F$ is a finite field with $q$ elements agrees with the facts that

$\# G(\mathbb F_q) = (q^2-1)(q^2-q)$

and

$\# \mathbb P^1(\mathbb F_q) = q+1$.

This generalizes in that parabolic subgroups (closed subgroup containing a Borel subgroup, i.e., containing the upper triangular matrices upto conjugation, for the general linear group) of an algebraic group give rise to projective varieties. They are precisely the stabilizers of a flag and conversely, the set of flags can be given the structure of an $F$-space and in particular a topology wherein they are compact spaces (being projective spaces). They are called flag varieties.

It is but with a heavy heart that I take to pen down my feelings. Sachin batted in what is very likely his last international innings today. No more Sachin. And to make matters worse, Anand lost the fifth game to Carlsen after drawing the last four games.

They measure life by the moments that take your breath away. I can reminisce my life so far by the Sachin moments I can recount. He was always there all my life. I distinctly remember enjoying what would be known as the Desert Storm, watching on my our old TV, in seventh grade. Then there was Sachin’s 98 in the India-Pakistan match in the 2003 world cup, two days before my Maths exam in HSC. And how can I forget the first one-day double century ever by Sachin; frenzied IIT-B hostel-junta was all jumping on the mess tables. And there was the World Cup India won, in 2011, when Virat Kohli carried Sachin on his shoulders. Those Sachin moments I associate with the good times I had studying Mathematics at IMSc. With Sachin retired, there won’t be any more of these moments. Dhoni puts it succinctly – “With Sachin Paaji, a part of me will be gone too.”

I respected, loved, adored, worshipped Sachin but frankly, I never wanted to be a cricketer myself. But at least for a little while as a kid, I wanted to be a grandmaster. Like Sachin, I grew up seeing Vishwanathan Anand. Chess was amazing and Vishy kept winning those rapid and blindfold games and I just idolized him. He became the World Champion in 2007 and has retained the title until now. After four consecutive draws, finally blood is drawn he loses to Magnus Carlsen. With seven more matches to go, I am badly hoping Anand resurges like a Phoenix in the remaining matches and clinches the title. It doesn’t matter if Carlsen snatches the title back from him next year. It might sound silly but, for me it’s like a battle between Sherlock Holmes and Professor Moriarty; it would be the crowning achievement of Holmes’ career if he could defeat Moriarty. Not that I hate Magnus Carlsen, but it’s just that it would pain me to witness Anand fall.

I believe it’s not just the fact that they have been the best players in their respective sports that puts Anand and Sachin on the same pedestal. Of course, both are Indians and I have grown up worshiping them both. But the real similarity between them is that both have their feet firmly on the ground. They both are the epitome of humbleness, a quality too difficult to exaggerate. It comes only with immense patience and respect towards your surroundings. In 23 years of his professional career, I don’t recall a single outspoken statement Sachin made, nor the slightest harsh action on the field. Hit by a speedy bouncer, he would calmly pick himself up and continue to bat in pain. And Anand, well, I’m honoured to see him nay, play a game in an exhibition match with him. It was at the the ICM at Hyderabad in 2010. People asked random, stupid questions to him and he answered them all with aplomb. Take a bow, masters!

#### Why do we need idols?

My sadness at Sachin retiring and Anand losing a game to Carlsen got me thinking; why did I feel sad? Because I wouldn’t be witnessing my role models in action. And why do I need role models? I study Mathematics, neither of them are anyway remotely related to Mathematics. That’s not the reason I idolize them. I revere them because seeing them in action makes me believe I can excel in my field too. I equate myself with them. I want to be a part of their success, and even failure. If Sachin can persevere and perfect that straight drive, I too can push my cognitive limits and understand my Mathematics. If Anand can produce a masterpiece, may be one day I too can.

I hope Anand fights back tomorrow and comes up with something brilliant. He always has.

_________________________________________________

An article I wrote about Sachin long ago — here.

A moment to cherish:

In my recent Riemann surfaces class, Donu started talking about de Rham cohomology and it’s generalization to Hodge theory. I was fascinated by it (in particular, the Hodge decomposition) so started reading some related stuff. I initially intended to write a blog post about it, but soon it grew in size than I had intended, so I’m attaching it to this post as a pdf file. Talking to Nick and Partha and looking at Forster’s Lectures on Riemann Surfaces was also helpful.

A mild introduction to Hodge_theory.

In this post, I will give a counter-example, that the image of a hyperspecial maximal compact subgroup under a surjective map need not (even) be maximal. In particular, this will tell us that a maximal compact subgroup (e.g., $SO(n)$ inside $SL_n(\mathbb R)$) need not be maximal under surjection.

Let $G$ be a connected reductive algebraic group over a local non-archimedian field $F$. A subgroup $K$ of $G(F)$ is called hyperspecial maximal compact subgroup if

• $K$ is a maximal compact subgroup of $G(F)$,
• There is a group scheme $\mathcal G$ such that $\mathcal G(\mathcal O_F)=K$ and $\mathcal G(\mathcal O_F / \varpi \mathcal O_F)$ is a connected reductive group over $\mathbb F_q$, where $\varpi$ is a uniformizer in $\mathcal O_F$ and $q$ is the cardinality of the residue field.

Denote by $\phi$ the canonical surjection

$\phi : SL_2 \to PGL_2 = GL_2 / G_m$

(over some local field $F$, or surjection of group schemes, to be pedantic). Choose a non-unit element $x$ of $\mathcal O_F$. Then

$\begin{pmatrix}x & \\ & x^{-1} \end{pmatrix} \not\in SL_2(\mathcal O_F)$,

but this matrix is in $PGL_2(\mathcal O_F)$ because

$\begin{pmatrix} x & \\ &x^{-1} \end{pmatrix} = \begin{pmatrix} x & \\ & x^{-1} \end{pmatrix} . \begin{pmatrix} x & \\ & x \end{pmatrix} = \begin{pmatrix} x^2 & \\ & 1 \end{pmatrix}$.

Hence the image $\phi(SL_2(O_F) \subsetneq PGL_2(O_F)$ is compact but not maximal compact (so in particular, not hyperspecial).

Remarks:

• A good reference to read about hyperspecial maximal compact subgroups is Tits’ article in Corvallis (Volume 1). Nick and I are planning to start a reading seminar next semester in Spring 14 on the 40-odd page article.
• I found this counter-example somewhere on MathOverflow, while I was preparing the Borel-Tate articles (Corvallis, Volume 2) to speak in our “Arthur seminar”.
• Your first (half) marathon is the most cherished one. Ironically, it’s also your slowest!
• I ran a half-marathon at the Indianapolis Marathon today, 19 October, 2013 in 2:18:56.
• Running a half-marathon = 13.1 miles = 21 kilometers sounds like the craziest idea until you actually run it. After that it seems trivial.
• It was good to have Jacob and Ryan run the 13.1. Qi ran the full marathon; bow to thee, master! Tianyang’s support in cheering us was invaluable.
• A 81-year old lady completed the 13.1 in just over two and a half hours. Highly impressive. Humans are crazy!
• I liked the way the marathon was organized. Fully planned, everything taken care of including parking and restrooms. The concept of time tracking by a tracker attached to the shoe was amazing. I’m impressed with technology.
• There was so much energy in the atmosphere that you just couldn’t get tired, stop. People cheering, runners discussing their past and future marathons, volunteers offering water and energy drinks, loud music.
• All this excitement made me forget the bad weather, it was 5 degrees (Celsius of course), windy and raining.
• A lady had “13.1 on my 30th birthday” printed on her shirt. I wished her a happy birthday! There were some people from Lafayette, we had a “Go Boilers!” exchange.
• There were volunteers encouraging us to keep going. Randomly, I would tell them they’re doing a great job cheering us 🙂
• I was hoping Jacob, who was ahead of me would cross me and I’d wave to him saying, “*How* is it going?”!
• I was in extreme pain after crossing the finish line, but knew I had made history (at a personal level, in the least).
• There’s only one key to completing the run, don’t think about running, don’t count the miles, don’t calculate your pace — Just RUN!

_______________________________________________________________________________________

Tailpiece:

October 2012 : Biked from Bloomington to Purdue for Habitat for Humanity, 125 miles in 2 days

October 2013 : Ran a half-marathon at the Indianapolis marathon.

October 2014 : (Crazy suggestions?)

Gian Carlo Rota’s Indiscrete Thoughts is a must-read for every budding mathematician. He’s highly opinionated and among articles like “Ten things I should have learnt as a graduate student”, one can also find short biographies of biggies like Emil Artin, Stan Ulam and Solomon Lefshetz. Below is a paragraph taken from the book.

His advisor Jack Schwartz gives Rota the task of cleaning up the tome “Linear Operators” by Dunford – Schwartz for errors, solving exercises, correcting semicolons etc. Here is Rota’s description about one of the questions he wasn’t able to solve.

It took me half the summer to finish checking the problems in Chapter Three. There were a few that I had trouble with, and worst of all, I was unable to work out Problem Twenty of Section Nine. One evening Dunford and several other members of the group got together to discuss changes in the exercises. Jack was in New York City. It was a warm summer evening and we sat on the hard wooden chairs of the corner office of Leet Oliver Hall. Pleasant sounds of squawking crickets and frogs along with mosquitoes came through the open gothic windows. After I admitted my failure to work out Problem Twenty, Dunford tried one trick after another on the blackboard in an effort to solve the problem or to find a counterexample. No one remembered where the problem came from, or who had inserted it.

After a few hours, feeling somewhat downcast, we all got up and left. The next morning I met Jack, who patted me on the back and told me, “Don’t worry, I could not do it either.” I did not hear about Problem Twenty of Section Nine for another three years. A first-year graduate student had taken Dunford’s course in linear operators. Dunford had assigned him the problem, the student solved it, and developed an elegant theory around it. His name is Robert Langlands.

In my recent number theory seminar on “Hilbert’s 90 and generalizations” (notes here), Professor Goins asked the following interesting question.

Let ${K}$ be a field and ${d\in K^*}$. Define ${T_d}$ to be the torus

$\displaystyle \left\{ \begin{pmatrix} x & dy \\ y & x \end{pmatrix} : x,y \in L, x^2-dy^2=1\right\}.$

What values of ${d}$ give ${K}$-isomorphic tori?

(The question was perhaps motivated by the observation that over the reals, the sign of ${d}$ determines completely whether ${T_d}$ would be split (i.e., isomorphic to ${\mathbb R^*}$) or anisotropic (i.e., isomorphic to ${S^1}$).

Here are two ways of looking at the answer.

• For ${d,e \in K^*}$, we determine when two matrices ${\displaystyle\begin{pmatrix} x & dy \\ y & x \end{pmatrix}}$ and ${\displaystyle\begin{pmatrix} u & ev \\ v & u \end{pmatrix}}$ are conjugate in ${\text{SL}_2(K)}$. Solving the system

$\displaystyle \begin{pmatrix} x & dy \\ y & x \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} . \begin{pmatrix} u & ev \\ v & u \end{pmatrix} . \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

gives ${\displaystyle \frac{e}{d} = \left(\frac{b}{c}\right)^2, de = \left(\frac{d}{a}\right)^2}$.

Thus ${e \in d.(K^*)^2}$ i.e., the ${T_d}$‘s are classified by ${\displaystyle \frac{K^*}{(K^*)^2}}$. (For ${K=\mathbb R}$, this is isomorphic to ${\{\pm 1\}}$ so the sign of ${d}$ determines ${T_d}$ upto conjugation.) By Kummer theory, ${\displaystyle \frac{K^*}{(K^*)^2} \cong H^1(\text{Gal}(\overline K / K), \mu_2)}$, where ${\mu_2 = \{\pm 1\}}$ are the second roots of unity. Thus there is a correspondence between isomorphism classes of tori ${T_d \; (d \in K^*)}$ and quadratic extensions of ${K}$.

• Another way to look at the same thing is as follows. Fix ${d \in K^*}$. Let ${L}$ be an extension of ${K}$ wherein ${T_d}$ splits. Now ${T_d(L)}$ is a split torus of rank 1. For an algebraic group ${G}$ over an algebraically closed field, we have the exact sequence

$\displaystyle 1 \rightarrow \text{Inn}(G) \rightarrow \text{Aut}(G) \rightarrow \text{Aut}(\Psi_0(G)) \rightarrow 1,$

where ${\Psi_0(G)}$ is the based root datum ${(X,\Delta,X\;\check{}, \Delta\,\check{})}$ associated to ${G}$. (Here, ${X = X^*(G)}$ and ${\Delta}$ is the set of simple roots of ${X}$ corresponding to a choice of a Borel subgroup of ${G}$.) For details, see Corollary 2.14 of Springer’s paper “Reductive Groups” in Corvallis.

In our case, ${G = T_d}$ so ${\Psi_0(G) = ( \mathbb Z, \emptyset, \mathbb Z \;\check{}, \emptyset)}$.

$\displaystyle \text{Aut}(\Psi_0(G)) \cong \text{Aut}(\mathbb Z) \cong \{ \pm 1\}.$

Now ${L/K}$ forms of ${T_d}$ are in bijective correspondence with

$\displaystyle H^1(\text{Gal}(L/K), \text{Aut}(\Psi_0(G))) = H^1(\text{Gal}(L/K), \{\pm 1\}) \cong \text{Hom}_{\mathbb Z}(\text{Gal}(L/K), \{\pm 1\});$

the last isomorphism because the Galois group acts trivially on the split torus ${T_d(L)}$. ${\blacksquare}$

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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### Quotable Quotes

ABHISHEK PARAB
“Do not think; let the equation think for you”

PAUL HALMOS
”You cannot be perfect, but if you won’t try, you won’t be good enough”

ALBERT EINSTEIN
“Don’t worry about your maths problems; I assure you, mine are greater”

THE BEST MATH JOKE
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