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 ${T_d(L)}$. ${\blacksquare}$

I’ve imagined many things on my blog – mathematics, chess, cricket, India, the US, good movies.. Culinary recipes is so not a thing I imagined on my blog! I think I owe an apology. Firstly, this post is not a precursor to future gastronomic blog posts. (The label: food starts with the liquid diet I used to have with the wired jaw, and ends with this post on the tawa chicken.) Secondly, I am blogging this because Shashank was desperately looking for  an easy chicken recipe and this is the answer. Without further ado, here we go.

• Thaw chicken and rub it with limejuice, salt, pepper, chilly powder (& turmeric?)
• Marinade for say, two hours
• Bake over aluminium foil (make holes in the chicken with a fork / knife so it cooks uniformly) at 350F for about 20-25 min
• (While baking, ensure that the chicken is enclosed in the foil like an air-tight bag)
• Let it sit unopened for 15 min
• (Don’t throw away the juices! They are going to be your gravy. If you’re a low-fat person who discards these fat juices, this dish isn’t meant for you)
• Fry (three) chopped onions in oil (long time, till it turns golden brown)
• Add two chopped tomatoes to it
• Copious amount of garam masala (गोडा मसाला), enough chilli powder and reasonable amount of salt. Keep mixing till you get semi-solid state
• Open the chicken foil bag and peel off chicken slices vertically. Put it back in the juice
• After frying the onion-tomato mixture, add chicken
• Based on the consistency you want, add the juice
• Cook for about 15 minutes
• Serve hot!

In this previous post, we saw the existence of a common eigenvector, namely $\phi(n) = a_n =$ number of nonzero solutions to $x^2=d$ modulo $n$. This was not a coincidence. Indeed, it was based on the fact that $\{ T_p : p \nmid N := 4|d| \}$ is a family of self-adjoint and commuting operators on the space of complex-valued functions on $G = (\mathbb Z/N \mathbb Z)^*$.

(Here, by self-adjoint, I’m talking about the inner product

$\langle f,g \rangle = \displaystyle\sum_{r\in G} f(r) \overline{g(r)} . \qquad )$

This post is a generalization.

Let $f$ be a holomorphic function on the upper half complex plane. We say $f$ is modular if it satisfies a technical condition called “holomorphic at the cusps” and the following.

$\displaystyle f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z) \quad \forall \gamma = \begin{pmatrix} a&b\\c&d \end{pmatrix} \in \Gamma := SL(2,\mathbb Z).$

Given any $f$ holomorphic on the upper half plane and $\gamma \in \Gamma(1)$, define

$\displaystyle (f|_\gamma)(z) := (cz+d)^{-k} (\text{det} \gamma)^{k/2} f\left(\frac{az+b}{cz+d}\right), \quad \forall \gamma = \begin{pmatrix} a&b\\c&d \end{pmatrix} \in \Gamma(1)$.

It is a fact that for any $\alpha \in \text{GL}(2,\mathbb Q)^+$, there is a double-coset decomposition $\Gamma(1) \alpha \Gamma(1) = \displaystyle\bigcup_{i=1}^l \Gamma(1) \alpha_i$.

Define for such a decomposition,

$f|_{T_\alpha} := \displaystyle\sum_{i=1}^l f|_{\alpha_i}$.

Observe that $(f|_{\alpha})|_{\beta} = f|_{\alpha\beta}$ so that defines a well-defined action of $\Gamma(1)$ on the $f$‘s. There is a vector space called the space of modular forms and a $T_\alpha$-invariant subspace – $S_k(\Gamma(1))$ – the space of cusp forms (similar to $V_N$ in the previous post) and for varying $\alpha$, the operators $T_\alpha$ (called the Hecke operators)

$T_\alpha : S_k(\Gamma(1)) \to S_k(\Gamma(1))$,

$T_\alpha(f) := f|_{T_\alpha}$.

It’s a cool theorem that the Hecke algebra is commutative and the Hecke operators are self-adjoint with respect to an inner product (the Petersson inner product). A standard result in linear algebra tells that these can be diagonalized; there is a common eigenvector, called the Hecke eigenform. When suitably normalized, it’s associated $L-$ function has an Euler product (similar to the $\zeta$ function). This Euler product gives the Ramanujan’s identity -

$\displaystyle\sum_{n=1}^\infty \tau(n) n^{-s} = \prod_p \frac{1}{1-\tau(p) p^{-s} + p^{11-2s}}.$

(Here, $\tau$ is the Ramanujan-$\tau$ function. )

Pretty cool stuff, eh!

——————————————————————————————————————

Tailpiece: References (since I am very vague here) -

• A first course in modular forms – Diamond, Shurman
• Automorphic forms and representations – Daniel Bump

Also, I was interested in the properties the L-function corresponding to the $a_p$‘s in the earlier post. I haven’t seen any book that mentions about these.

Last week, I went to a summer  workshop in Salt Lake City, Utah. As I checked in the university guest house, an elderly gentleman with a white shaggy beard walked to me and asked, “You must be a mathematician.”

Turns out, (as he later introduced me) he was Gregg Zuckerman, a famous mathematician working in representation theory. He knew my advisor Shahidi and was astonished Shahidi didn’t ever mention to me about him.

To his question, I replied in true mathematical spirit, “That depends on your definition of a mathematician!” leaving a chuckle behind.

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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