Last week, I attended the Thematic Program in Discrete Groups in Topology and Algebraic Geometry, an undergraduate summer school that ran for about a week.

In the week leading up to the program, I did some prereading to prepare: Löh’s Geometric Group Theory and Silverman and Tate’s Rational Points on Elliptic Curves. Both were excellent and helped me avoid getting lost during the lectures.

Math

There were three speakers, each of which gave a mini lecture series.

Geometric Group Theory by Daniel Studenmund: The talks were generally good, covering foundational geometric group theory, which was fresh in my mind. While much of the material was familiar and felt a bit slow at times, some of the problems presented were quite memorable.
Why $e^{i\pi\sqrt{163}}$ is almost an integer by Shuddhodan Kadattur Vasudevan: This session was really interesting but the pacing was very challenging, as it oscilatted between being slightly slow to going way too fast, often concluding with hand-wavy explanations. This is a topic that I had already encountered at mcsp23, but this presentation offered far greater depth. We began with a review of Riemann surfaces and then moved to elliptic curves, which was fine, but then once we started talking about moduli spaces and complex multiplication, I got slightly lost.
Rational Tangles by Nick Salter: aimed to tie together the previous two lecture series we'd been following, offering a kind of grand finale. I thought the explanations were clear and the presentation was well-executed. Despite this, I didn't find the subject of rational tangles particularly interesting, and my attention wandered a lot.

Here is a day by day list of what we covered; detailed notes can be found here.

Monday, June 2

AM Session 1: Group Actions: Definition of a group action; properties (orbit, stabilizer, free, transitive); examples ( $S_n$ on $\{1,...,n\}$ , $GL_n(k)$ on $k^n$ , Cayley graph, $SO(n)$ on $S^{n-1}$ , conjugation, fundamental group on universal cover); free actions of groups on $\mathbb{R}$ and $\mathbb{R}\backslash\{0\}$ ; finite group action on a tree has a global fixed point; hyperbolic plane geodesics and isometries; $SL_2(\mathbb{R})$ action verification; mapping of points by $SL_2(\mathbb{R})$ isometries; generalized semicircles; hyperbolic area calculations; hyperbolic triangles are slim.
AM Session 2: Hyperbolic Geometry: Poincaré upper half-plane ( $\mathbb{H}$ ) as a model for hyperbolic geometry; Riemannian metric on $\mathbb{H}$ ; hyperbolic length of a smooth curve; hyperbolic metric and distance function $d_{\mathbb{H}}$ ; geodesics in a metric space and in $\mathbb{H}$ (Euclidean semicircles and vertical rays); Riemannian isometries of $\mathbb{H}$ ; $SL(2,\mathbb{R})$ acting on $\mathbb{H}$ by Möbius transformations as orientation-preserving Riemannian isometries; generators of $SL(2,\mathbb{R})$ (translations, dilations, inversion) and their properties as isometries; transitivity of $SL(2,\mathbb{R})$ action on $\mathbb{H}$ and stabilizer of $i \in \mathbb{H}$ ( $SO(2)$ ).
PM Session 1: Introduction to Riemann Surfaces: Number systems ( $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ ); C-differentiability (holomorphicity/analyticity); power series representation of holomorphic functions; real vs. complex differentiability; real manifolds (circle, sphere, torus); definition of a Riemann surface (charts, transition maps); examples of Riemann surfaces ( $\mathbb{C}$ , open subsets of $\mathbb{C}$ , unit disk, upper half-plane); definition of holomorphic functions on Riemann surfaces; holomorphic maps between Riemann surfaces; Liouville's Theorem application (no non-constant holomorphic map from $\mathbb{C}$ to $\Delta$ ).
PM Session 2: Introduction to Riemann Surfaces II: Riemann sphere ( $S^2$ , $\mathbb{P}_{\mathbb{C}}^1$ ) as a Riemann surface; biholomorphism between $\mathbb{H}$ and $\Delta$ ; tori and lattices ( $\mathbb{C}/\Lambda$ ); definition of a lattice; $\mathbb{C}/\Lambda$ as a Riemann surface; uniqueness of complex structure on $S^1 \times S^1$ ; holomorphic maps between tori are affine linear; biholomorphism of tori via scaling of lattices; parameter space for lattices ( $SL_2(\mathbb{R})/SO(2)$ ).

Tuesday, June 3

AM Session 1: Group Presentations: Group presentation notation ( $G \cong \langle S | R \rangle$ ); $\mathbb{Z}^2$ presentation; definition of a generating set; examples of generating sets for $\mathbb{Z}$ and $\mathbb{Q}$ ; words and reduced words; free groups; universal property of free groups.
AM Session 2: Trees: Kernel of homomorphism from $F_2$ to $\mathbb{Z}^2$ ; normal closure of a subset; relation of kernel to commutator subgroup; definition of a graph and a tree; Cayley graph of a free group is a tree; group acts freely on a tree iff it's a free group.
PM Lecture 1: Riemann Surfaces III: Classification of compact Riemann surfaces by genus; $S^2$ has unique complex structure ( $\mathbb{P}_{\mathbb{C}}^1$ ); complex projective space $\mathbb{P}_{\mathbb{C}}^n$ ; tori complex structures parameterized by homothety classes of lattices; action of $GL_2(\mathbb{R})$ on lattices and upper half-plane; relation between $GL_2(\mathbb{Z})\backslash(\mathbb{H}^+ \sqcup \mathbb{H}^-)$ and $SL_2(\mathbb{Z})\backslash\mathbb{H}^+$ ; compactification of $Y(1)$ to $Y(1)$ ; introduction to projective varieties (homogeneous polynomials, smooth projective curves).
PM Session 2: Elliptic Curves: Compact Riemann surfaces are algebraic curves; $\mathbb{C}/\Lambda$ is algebraic; Weierstrass $\wp$ -function and its differential equation; embedding of $\mathbb{C}/\Lambda$ into $\mathbb{P}_{\mathbb{C}}^2$ ; definition of elliptic curves (smoothness condition via discriminant); j-invariant and isomorphism classes of elliptic curves; modular forms (definition, transformation property, holomorphicity at cusps); $g_2, g_3, \Delta$ as modular forms; modular forms as holomorphic sections of line bundles.

Wednesday, June 4

AM Session 1: Trees: Group presentation notation ( $G \cong \langle S | R \rangle$ ); relation to normal closure; a group is free if and only if it acts freely on a tree; sketch of proof for the theorem.
AM Session 2: Farey Graphs: Definition of the Farey graph; properties of vertices and edges; visualization in the upper half-plane; construction of the tree $T_{far}$ from the Farey graph; action of $SL_2(\mathbb{Z})$ and $PSL_2(\mathbb{Z})$ on $T_{far}$ ; definition of free product of groups ( $G*H$ ); presentation of free product; theorem on group action on a tree and free product decomposition; $PSL_2(\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/3\mathbb{Z}$ .
PM Session 1: Moduli I: Principal congruence subgroup $\Gamma(N)$ ; modular curves $Y(N)=\Gamma(N)\backslash\mathbb{H}$ ; $Y(1)$ is isomorphic to $\mathbb{C}$ via j-invariant; compactification of $Y(N)$ to $X(N)$ ; j-invariant depends on homothety class of lattice; definition of a modular form of weight 2k and level N; relation between functions on lattices and modular forms; $g_2, g_3, \Delta$ as modular forms; vector space of modular forms; modular forms and sections of line bundles.
PM Session 2: Moduli II: Definition of a line bundle; tautological line bundle over $\mathbb{P}_{\mathbb{C}}^{1}$ ; pullback line bundles; line bundles on simply connected spaces are trivial; factor of automorphy; modular forms as holomorphic sections of line bundles; moduli interpretation of modular curves; $Y(1)$ as a coarse moduli space for elliptic curves; universal family of elliptic curves; Hodge bundle.

Thursday, June 5

AM Session 1: Braid Groups I: Braided strands illustration; equivalence of braids via isotopy; braid product and inverse operations form a group ( $B_n$ ); generators $\sigma_i$ for $B_n$ ; formal definition of a braid; braid diagram.
AM Session 2: Braid Groups II: Braid relations: $\sigma_i\sigma_j=\sigma_j\sigma_i$ for $|i-j|\ge2$ and $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ ; Artin braid group definition; isomorphism between Artin braid group and geometric braid group; $B_n$ presentation; $B_3 \cong \langle x,y|x^3y^{-2}\rangle$ ; homomorphism from $B_n$ to $S_n$ ; pure braid group $P_n$ ; ordered and unordered configuration spaces; $B_k \cong \pi_1(C(\mathbb{C},n))$ .
PM Session 1: Complex Multiplication I: Absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ ; abelian extensions of $\mathbb{Q}$ ; cyclotomic fields $\mathbb{Q}(\zeta_n)$ are Galois extensions of $\mathbb{Q}$ ; $Aut(\mathbb{Q}(\zeta_n)/\mathbb{Q})\cong(\mathbb{Z}/n\mathbb{Z})^{\times}$ ; Kronecker-Weber Theorem; Hilbert's twelfth problem; torsion points of elliptic curves $E[n]$ ; endomorphism ring $End(E)$ of an elliptic curve; complex multiplication (CM); example of CM (Gaussian integers $\mathbb{Z}[i]$ ); orders in imaginary quadratic fields; maximal order $\mathcal{O}_K$ .
PM Session 2: Complex Multiplication II: Classification of elliptic curve endomorphism rings; proof sketch relating $\alpha$ to quadratic polynomial; ideal class group $Cl(\mathcal{O}_K)$ ; finiteness of ideal class group; main theorem of complex multiplication (bijection between isomorphism classes of CM elliptic curves and ideal class group); finiteness of CM elliptic curves; j-invariant of a CM elliptic curve is an algebraic integer; $K(j(E))$ is Hilbert class field.

Friday, June 6

AM Session 1: Mapping Class Groups: Braid as isotopy class of paths in configuration space; mapping class group $Mod(\overline{\mathbb{D}},n)$ definition; isomorphism $B_n \cong Mod(\overline{\mathbb{D}},n)$ ; mapping class group of a surface; Dehn twists; relations of Dehn twists ( $T_{f(a)}=f\circ T_a\circ f^{-1}$ ); braid relation $T_\alpha T_\beta T_\alpha = T_\beta T_\alpha T_\beta$ ; geometric intersection number and subgroup structure of $\langle T_\alpha, T_\beta \rangle$ .
AM Session 2: Rational Tangles I: Rational tangles; horizontal Twist (T) and vertical Rotation (R) operations; tangle invariant $\tau$ ; Möbius transformations induced by T and R; Euclidean algorithm for untangling; untangle is isotopic if $\tau(\sigma)=0$ ; bdpq symmetry of a tangle; stabilizer of 0 in $PSL_2(\mathbb{Z})$ ; relation between matrix operations and tangles; kernel of homomorphism $F_2 \rightarrow PSL_2(\mathbb{Z})$ ; trivial action of kernel elements.
PM Session 1: Ramanujan's Constant: Imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D})$ ; ring of algebraic integers $\mathcal{O}_K$ ; elliptic curve endomorphism ring $End(E)$ ; bijection between class group $Cl(\mathcal{O}_K)$ and isomorphism classes of elliptic curves with CM by $\mathcal{O}_K$ ; CM points on modular curve $Y(1)$ ; equidistribution of CM points; explanation of Ramanujan's constant $e^{\pi\sqrt{163}}$ .
PM Session 2: Rational Tangles II: Group actions ( $F_2 \subset \mathcal{T}$ , $PSL_2(\mathbb{Z}) \subset \mathbb{Q}\cup\{\infty\}$ , $B_3 \subset \mathcal{T}$ ); $B_3$ action on tangles factors through $PSL_2(\mathbb{Z})$ ; center $Z(B_3)$ and isomorphism $B_3/Z(B_3)\cong PSL_2(\mathbb{Z})$ ; relation between $R\cdot_{A1}\tau$ and $(\sigma_1\sigma_2\sigma_1)\cdot_{A3}\tau$ ; $B_3$ as fundamental group of space of monic cubic polynomials with distinct roots; homotopy equivalence $\text{Conf}_3(\mathbb{C})\simeq S^3\backslash trefoil$ ; link of a singularity; link of $x^p+y^q=0$ .

Nonacademics

Some memorable activites and places:

Hesburgh Library: This is one of the nicest libraries I’ve ever visited. The lower levels felt like the lobby of a high-end hotel in NYC. It was so easy to relax there. Upstairs, everything shifted into a historic library. It became the kind of quiet, classic library where it’s almost effortless to forget about everything else and just focus.
Saint Joseph and Saint Mary's Lake: We had planned to visit the beach by one of the lakes, but that didn’t quite work out. Still, that small change led us to a trail instead. While walking toward the other lake, we came across two swans and a little island that looked like something out of a storybook. We eventually found a bench with a perfect view to relax on.
Raclin Murphy Museum of Art: My own college has a great art museum, so I didn’t think this one would really stand out. But it was better than I had expected, and they had some really interesting faux paintings. Outside, the sculpture park was slightly ugly but the concept was cool.
Food: The food was much better than I had expected. I won’t forget the tacos at Street Fare, my first time trying Dave’s Hot Chicken, the cozy dinner at Parisi’s Ristorante Italiano, and that one pizza spot on Angela Boulevard.
Swings: The swings were tall and smooth and just really fun.
Howard Park: Howard Park really surprised me. It's probably the nicest park I've seen in a long time. I was expecting a simple place, but it turned out to be huge, with so much to do. It had way more features than I imagined. If there was one thing from this trip I would do again, it would be this one.

The area around Notre Dame has a quiet calm to it that’s hard to describe. It’s not busy like the city, but it’s not empty either. It sits in this gentle middle ground, where time seems to slow down just enough to let you breathe.

You can tell the neighborhood surrounding the school isn’t the most well-off, but that gives it a kind of warmth. There’s something very genuine about the people here. They seem at peace, like they’ve learned how to enjoy life in the small, everyday ways. Perhaps this is the power of religion.

Walking around, it's easy to notice that things aren't perfectly polished. The grass grows a little too tall in places. Some of the buildings have seen better days. But somehow, all of that gives the place a quiet charm. It feels lived-in, like it has stories to tell. There’s wildlife everywhere: exotic birds displaying every color of the rainbow, elegant swans floating in the ponds, and grassy patches where ten rabbits play.

During this program, I kept reminding myself to enjoy it, because one day I would miss it. And I tried, I really did. But now that it’s ending, I can’t help but feel I didn’t hold on tightly enough. The math was excellent, as it always is. But that’s never been the reason I come to things like this. I could learn the same material in solitude. The magic is in youth, in new places, in memories made.

CMND Discrete Groups in Topology and Algebraic Geometry

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Math

Nonacademics