Hari Iyer's generals examiners: Akshay Venkatesh (chair, AV), Shouwu Zhang (SZ), Alexandru Ionescu (AI). special topics: algebraic geometry and representation theory of compact Lie groups May 12 2025 at 1 pm in SZ's office. lasted 2.5-3 hours. AV: who's chair oh its me! I was asked what topic to start with, i said i have no preference, AV insisted that i pick, so i started with analysis (questions by AI unless otherwise specified) SZ joked: he studied analytic number theory at harvard (me: "a little", gesturing by crushing my thumb and index finger together) oh look he's being so modest today!! COMPLEX ANALYSIS: write cauchy integral formula, use to evaluate $\int_0^\infty sin(z^2)dz$. i struggled to pick the contour correctly but turns out to be the line $Re(z) = Im(z)$. given a function on the strip $-1 < Im(z) < 1$ bounded by $1 + |z|^2$, can you bound its derivative by the same? made it easier by asking to bound $f'(0)$ by a constant.. apply cauchy integral formula for derivative. AI remarked "oh dear i made the question too easy" what can you say about zeroes of holomorphic functions (isolated), can you construct functions with prescribed zeroes? e.g. vanishing at integers ($e^{2\pi inz}-1$, intended was $sin(2\pi z)$), i wrote weierstrass product and fiddled with correction by an exponential in each factor. state riemann conformal mapping theorem. AV: how would you explain the riemann mapping theorem to a high schooler? so i drew a blob and described that each point in the blob goes to a disk, in such a way that angles between lines are preserved. they were amused by my explanation (i drew dots and lines) fix the simply connected to be the disk, what can you say about the biholomorphisms? fixed a point via mobius transformation and used schwarz lemma to show they are rotations composed with mobius transformations. SZ asked about automorphisms of a related object in number theory... turns out he was referring to the action of $SL_2(\mathbb{R})$ on the upper half plane -- which is conformally equivalent to the disk REAL ANALYSIS: what is $L^p$, defined. do you know the hardy-littelwood maximal function? lebesgue differentiation? i had not studied these and AI seemed unhappy about this. asked about distributions. i had not studied carefully but i guessed they are continuous functionals on the continuous functions (wrong), i though about $L^p$ etc but then he had me write an example of a nontrivial distribution so i wrote $\delta$, which is $\delta(f) = f(0)$... he asked me to differentiate it so i wrote (?) $\delta' = x \delta$ which was wrong but fixed it to $\delta'(f) = f'(0)$... so i guessed the domain should be smooth functions (it is actually schwartz class), then i defined schwartz functions. then there were quesitons about how to define distributions corresponds to integrating against $1/x$, $1/|x|$, $1/x^2$ which i struggled with (the problem is at $x = 0$, so you use derivatives to modify your functional at zero to make the distribution well-defined)... i also didn't know the topology of the schwartz space, which would've helped, though AI directed me to figure it out on the way. AV then asked if i wanted a break, water etc but i insisted on doing algebra before. ALGEBRA: SZ: when can you solve $e^A = B$ for $B$ a real matrix. i struggled quite a bit with this... AV asked me to first define exponential (done.) then i considered jordan form and jordan chevalley decomposition to exponentiate a jordan block efficiently so yes you can take logs of jordan blocks with positive diagonal. but this only works over algebraically closed in general... over R i wrote rational canonical form, which yields quadratic irreducible factors,which were difficult to take logs in the same way. i got stuck so AV suggested writing down a matrix that's not diagonalizable (i chose rotation by theta). i looked at it and still struggled to take the log, but in hindsight i should've written e to the i theta which makes log quite straightforward. at this point SZ asked me to compute the subalgebras of two by two matrices, i was initially confused but they are the same as the rational canonical form decomposition. AV and SZ guided me to examine the irreducible case which is always the complex numbers, which the exponential map sends to $\mathbb{C}^\times$, and from there it was clear to see how to take log (though the answer really was the same as writing rotation as e to the i theta and using that you can square root -1 in matrices). in the middle of this, AV asked about diagonalizing symmetric real matrices. i gave the proof by sending a vector in the unit ball $v \to |Av|$ for A the matrix (compact goes to bounded pick a sequence converging to the max on the ball, AV was happy), there was some discussion between SZ, AV about $v \to |Av|$ versus $v \to (Av, v)$ but both work in the real symmetric case. AV: what does the matrix do to the unit ball? i worked it out explicilty in coordinates to be sure, but it of course stretches the ball in every eigen-direction, giving an ellipsoid. SZ: compute conjugacy classes for GL2(Fp). so i wrote out the rational canonical form again, there are three possibilities and counted them for total $p^2 - 1$. what are its reps? AV stopped at this point and said to save for rep theory, though we didn't return to it. AV: explicitly write a cubic extension of $F_3$. i reasoned that i need a cubic irreducible, it suffices that there's no root in F3 (look at splitting type), so by fermat's little theorem $x^3 - x + 1$ works. what's the norm? i guessed $1$ by looking at the last term, but i was asked to define norm of a field extension... i wrote down for galois but struggled to remember for non-galois... i wrote an element of $L/K$ in a basis for $L/K$ and they were happy. i was asked to explicitly compute $Nm(x)$ in this cubic extn, using matrices it is $-1$ as you suspect anyway... one could've multiplied conjugates from the beginning. SZ: for fun, what's galois group of this extn, i wrote Z mod 3, SZ points out this is F3 and asks me for the action explicitly. i struggled to see this, but he walked me through writing the equation for two roots, subtracting them and seeing that the difference must lie in F3, so the action is adding the element of F3. we stopped there for a break. got water. SZ and AV returned and were chatting about course/semester schedules at the university vs the IAS (which didn't really have a term schedule). SZ told some stories about reading and taking notes with Faltings during grad school... AV : which topic do you want to start with? algebraic geometry ALGEBRAIC GEOMETRY: SZ: what is a curve. smooth projective gometrically connected 1-dim scheme/bar{k} alg closed. AV: what's your definition of smooth? i defined regular, where dimension of ring is dimension of its tangent/cotangent space, i tried to say something about jacobian but one definition was enough. SZ: what are genus zero curves? i said P1 by riemann roch, degree one divisor. SZ: since we did it for complex analysis, what is Aut(P1)? i wrote PGL2... why? i wrote an explicit map and they asked why are these all of them? i said they're all the linear automorphisms (no! you just wrote an algebraic definition of curve), so i though to remove a point from both P1's (reminded by SZ to fix infinity) yielding an aut of A1.. which i said are given by translations because of degree considerations on k[t]. this is correct but they were looking for an argument using pic (O(1) pulls back to itself so every coordinate is in fact given by a hyperplane section, as in mobius transformation), which generalizes to dimension n, SZ remarked that removing a point will not generalize and aut of affine n-space is a hard problem in general. genus one? elliptic curves, went through the weierstrass form argument... what are line bundles on it? split off degree to get degree zero pic, which is the elliptic curve itself. then i define abel jacobi map $E \to Pic^0(E)$ and was asked to show its bijective. i struggled with some details but eventually we talked through the proof. why's the group law algebraic? i drew a curve and they were happy, but i unsuccesfully drew a line on the curve (eliciting laughs), and wrote $P + Q + R = 0$ when they lie on a line... i.e. they're the zeroes of the pullback of a hyperplane section under the degree three embedding in $P^2$. so you get an algebraic group law (upon comparing with some linear section whose divisor is $3(O)$ (SZ: so you have to take the ratio!) SZ and AV: what breaks in genus 2? i took a while but looked at the surjectivity argument and it required picking a section for degree one divisor, which isn't always possible in genus 2.. need to use degree two divisor, so the map has to involve adding two points $P + Q$ and subtracting (what? i guessed that you fix a point, SZ laughed and said to use the canonical.. AV remarked that $P - Q$ should also work)... later SZ said he wanted to walk me through constructing the jacobian for genus 2 but didn't have time. at this point, AV: pick your favorite curve, not genus 1 or 0 and compute its genus. i asked if it could be non-singular he said sure. so i wrote a genus 2 hyperelliptic curve, in two explicit coordinate charts and applied riemann hurwitz. REPRESENTATION THEORY OF LIE GROUPS: AV wanted to move on to rep theory for time. asked me what i know? i said su2, so3, su3 etc. AV: ok, what are reps of su2.. i write them as homogeneous polynomials and as symmetric powers of std... i prove these are all by passing to complexified lie algebra, and playing with e,f,h operators and showing highest weight is an integer for these reps. AV: ok, can you tensor sym2 with itself, so i write two rows of three dots for weights and put them together with dots and circles (to indicate multiplicity), ok split off the factors. sym2 otimes sym2 = sym4 + sym2 + triv AV: can you write down a bilinear form to sym4? i described the silly one which takes two pairs of 2 vectors and gives a symmetrized four vector... AV is there another way to write this? using homogeneous polynomials? how to get quartic from two quadratics? i said multiply but i was unnecessarily concerned about the coordinates on both sym2 factors being different (AV: they are both maps V star otimes V star to k, quadratic forms). AV: can you write a bilinear form to triv? i guessed to use the constant term (why doesn't that work?).. he prompted me to think in so(3) reps instead. i said sym2(std) are harmonics, degree 1 in three variables, i.e. a 3-d vector... so i guessed to take dot product, and he said this is correct. AV: why can you decompose reps? i show complete reducibility by splitting off the orthogonal part, and defining the G-invariant inner product (by integrating the translates of an arbitrary hermitian product) SZ: for fun, what are reps of SL2(R)? (ummm.. its noncompact) write its lie algebra! so i wrote sl2(r) complexified to sl2(c) and people were happy, but i said i was concerned that you can't just lift lie algebra reps when its not simply connected... SZ: so you have a problem! what is pi_1 of SL2(R)? (i didn't know this was algebraic topology exam now...).. i mumbled about upper half plane from the earlier part of the exam SZ : why you mention upper half plane? i said at least there's an action! ok SZ suggests looking at $g \to gi$, i said it has stabilizer so(2).. i struggled to write the fibration sequence (i hadn't formally studied AT) but eventually did and concluded pi_1 is Z. aside: i'm still not sure how this lets you compute irreps... but in hindsight the point was the unitary trick, and picking the correct compact form (so(2) will NOT work, its not zariski dense) in fact you must complexify and identify reps at the lie group level beforee passing to lie algebra -- where simply connectedness of su(2) helps -- though in this particular case the order doesn't matter, since we know enough irreps to make $Rep(G) \to Rep(g)$ bijective. AV: alright lets stop here! they sent me out for a few minutes --which felt like an eternity-- and one by one came out and "congratulations!" and shook my hand. SZ stuck around in his office and we chatted for a bit about the exam, and what to do next. ___________________________________________________________________________________________________________ all in all, this was a rough exam, and i believe the way i've written the transcript doesn't do justice to all the doubt, incorrect statements, guesswork, and pauses and guidance and hints that were generously provided by my examiners along the way! the questions were very fair, though i found it very hard to think and write at the board with three people reacting to every move (which could be helpful or confusing or both!). i was consistently stressed, sweating, dehydrated, at times at a loss for words (dropped my chalk a couple times). overall the examiners were quite sensitive to all this and did their best to make it a pleasant and comfortable experience.