Math Poetry
If deeper into this theory, you'd like to look
scour no further than my favorite book
with excellent authors Dummit and Foote
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Grad School (Yowzah)
Grad School (Yowzah)
Written Saturday, May 17, 2025
after dying during a research presentations at the AWM Research Symposium at UWMadison
Worrying about grad school,and trapped in my mind’s swirling whirlpool,I can’t seem to fall asleep,wondering if this legacy might run deep.
Did Hilbert overthink and worry?Did he too rush through life in a hurry?I wonder if any of the greats did.Or were they born with it—this innate gift of Euclid?
They all earned immortality through mathematics,changing the world with algebra and combinatorics.They make it look so effortless,Noether, Leibniz, and their infinite cleverness.
Praising the ranks religiously,strips them of their humanity.The whole field of math hallows and hurtswhen you keep a perspective so perverse.
From up here on the future’s high pulpit,we forget they too might have wanted to quit.They weren’t gods, but brilliant beings with burdenscutting through patterns to pull back the curtains.
It discredits you,and the hard work you do,to manifest a mathematical lifenot filled with the subject's serious strife.
You deserve better than this dreary delusion,the idealized intellectual illusion.
Did Hilbert overthink and worry?Did he too rush through life in a hurry?I wonder if any of the greats did.Or were they born with it—this innate gift of Euclid?
They all earned immortality through mathematics,changing the world with algebra and combinatorics.They make it look so effortless,Noether, Leibniz, and their infinite cleverness.
Praising the ranks religiously,strips them of their humanity.The whole field of math hallows and hurtswhen you keep a perspective so perverse.
From up here on the future’s high pulpit,we forget they too might have wanted to quit.They weren’t gods, but brilliant beings with burdenscutting through patterns to pull back the curtains.
It discredits you,and the hard work you do,to manifest a mathematical lifenot filled with the subject's serious strife.
You deserve better than this dreary delusion,the idealized intellectual illusion.
You reject the roots of real resiliencewhen you blindly bleach their brilliance.Praise not some pristine, perfect intellect,but the rigor they refused to neglect.
They worked through the night,they worked through rest,they worked for weeks before they slept.
They worked when incorrect,they worked when deject,they worked through feeling utterly inept.
They pushed and puzzled,they proved and revised,they bled through blackboards with burning eyes.
They reworked and researched,and read through the murk,until at last a proof they unearthed.
Perhaps they shared a subtle smirk but not before getting back to work.
And when they did, they didn’t bow.They didn’t beg. Their pace didn’t break.They rose. They roared. They raged awake.
Because greatness isn't given,it’s not graceful or pure—it’s earned in the sweat of those who endure.
Greatness is self made,accessible to you and I, should we too make that trade
So I won’t wait for gifted grace,I’ll chase the flame and take my place.Because I won’t wait to be born brilliant,I’ll blaze and bleed resilient.
They worked through the night,they worked through rest,they worked for weeks before they slept.
They worked when incorrect,they worked when deject,they worked through feeling utterly inept.
They pushed and puzzled,they proved and revised,they bled through blackboards with burning eyes.
They reworked and researched,and read through the murk,until at last a proof they unearthed.
Perhaps they shared a subtle smirk but not before getting back to work.
And when they did, they didn’t bow.They didn’t beg. Their pace didn’t break.They rose. They roared. They raged awake.
Because greatness isn't given,it’s not graceful or pure—it’s earned in the sweat of those who endure.
Greatness is self made,accessible to you and I, should we too make that trade
So I won’t wait for gifted grace,I’ll chase the flame and take my place.Because I won’t wait to be born brilliant,I’ll blaze and bleed resilient.
Below are some of the theorems referenced in The Galois Poem, each taken from Abstract Algebra by Dummit and Foote chapters 7, 8, 13, and 14
The Galois Poem
The Galois Poem
Written February, 2025
We once hoped all equations had solutions explicitunfortunately radicals work only until the quintic.Two elegant proofs, Ruffini’s fixed by Cauchy,were published near the nineteenth century. The Abel Impossibility Theorem makes it clear, that beyond the fourth, there's a distinct frontier.Specifically, no general root formula may appear.
Take a polynomial f with rational coefficients, For a field with its roots, Q is usually deficient. So we adjoin alpha, a primitive element sufficient.
In UFDs like the integers, irreducibles coincide with primes,but a general theory of rings finds this only happens sometimes.r is irreducible if, into itself and units, it can only refine.Then prime ideals and domain quotient rings, irreducibles define.
Proving f irreducible is our next dilemma, but the rational root test, Eisenstein's Criterion and Gauss’s Lemmaguide the way with a method exact. Since Q[x] is a PID, primes stay intact, so Q[x]/(f) forms a field so fine, isomorphic to Q adjoin alpha by design.
We now study morphisms on Q adjoin athat force the rationals in their place, to stay. Let an automorphism tau freely play, on both sides of f = 0 without delay. Since tau fixes rationals and distributes through, We find that tau(a) satisfies f tooSo permute f’s roots, all automorphisms do.
This is group G, Galois cleverly suspects, but f’s separability, he carefully checks. f splits into linear factors, real or complex, and if roots repeat, G’s structure defects. For inseparable f some roots fail to stand alone, making their permutations unclear or unknown.
Field extensions by separable f are called Galois, and G encodes their structure without flaw. This is the groups/fields link Galois saw, Where subgroups match extensions by his law.
The fundamental theorem unfolds in five parts, Mapping subgroups to fixed fields through inverse charts. A lattice bijection that’s inclusion reversed, since as group size grows, fixed fields are dispersed.
Galois extend field F to K with intermediate field E,Then E is the fixed field of H, a subgroup between 1 and G.And H is the galois group of K on E, generally.H3 in H2 means E2’s in E3,and this maps keeps subgroup/subfield indice;
the extension F to E is wonderfullyguaranteed Galois with H’s normalcy,and with G/H, the galois group agrees.These are theorems Dr. Berele taught meWith help from our excellent TA Anjali
If T and Z correspond to D and P,the group generated by T and Z maps preciselyto the lovely field P intersect D.And P composite D has only Galois group T intersect Z
Then our resolution to the original inquirysays in radicals, f’s roots express clearly when its galois group simply obeys solvability.S_5 is unsolvable, we sadly see,but handling reducible f forms a new query.
The chinese remainder theorem in the subject of ringsoccurring partitions product rings into distinct things. By taking Q[x] on each factor separately, our theorem bringsa field made by multiplying after separately quotienting.Then f’s full galois group glistens by subdirect producting.
But now question this strange equation: (x^2 + 3)(x^2 + x + 1), a curious relation. Each part has cyclic Galois group, that much is true, Yet both adjoin root minus three, an unfortunate issue since both share the same field and automorphism too, No direct product arises—just Z2 will do.
We find when X and S are Galois over F, In composites and intersection fields, galois properties manifest. The automorphisms that agree on both fields align, Forming subgroups that each G defines. with product GAL(XS/F), as structure implies.
If even deeper into this theory, you’d like to look,go no further than my favorite book.It has excellent authors Dummit and Foote…
Take a polynomial f with rational coefficients, For a field with its roots, Q is usually deficient. So we adjoin alpha, a primitive element sufficient.
In UFDs like the integers, irreducibles coincide with primes,but a general theory of rings finds this only happens sometimes.r is irreducible if, into itself and units, it can only refine.Then prime ideals and domain quotient rings, irreducibles define.
Proving f irreducible is our next dilemma, but the rational root test, Eisenstein's Criterion and Gauss’s Lemmaguide the way with a method exact. Since Q[x] is a PID, primes stay intact, so Q[x]/(f) forms a field so fine, isomorphic to Q adjoin alpha by design.
We now study morphisms on Q adjoin athat force the rationals in their place, to stay. Let an automorphism tau freely play, on both sides of f = 0 without delay. Since tau fixes rationals and distributes through, We find that tau(a) satisfies f tooSo permute f’s roots, all automorphisms do.
This is group G, Galois cleverly suspects, but f’s separability, he carefully checks. f splits into linear factors, real or complex, and if roots repeat, G’s structure defects. For inseparable f some roots fail to stand alone, making their permutations unclear or unknown.
Field extensions by separable f are called Galois, and G encodes their structure without flaw. This is the groups/fields link Galois saw, Where subgroups match extensions by his law.
The fundamental theorem unfolds in five parts, Mapping subgroups to fixed fields through inverse charts. A lattice bijection that’s inclusion reversed, since as group size grows, fixed fields are dispersed.
Galois extend field F to K with intermediate field E,Then E is the fixed field of H, a subgroup between 1 and G.And H is the galois group of K on E, generally.H3 in H2 means E2’s in E3,and this maps keeps subgroup/subfield indice;
the extension F to E is wonderfullyguaranteed Galois with H’s normalcy,and with G/H, the galois group agrees.These are theorems Dr. Berele taught meWith help from our excellent TA Anjali
If T and Z correspond to D and P,the group generated by T and Z maps preciselyto the lovely field P intersect D.And P composite D has only Galois group T intersect Z
Then our resolution to the original inquirysays in radicals, f’s roots express clearly when its galois group simply obeys solvability.S_5 is unsolvable, we sadly see,but handling reducible f forms a new query.
The chinese remainder theorem in the subject of ringsoccurring partitions product rings into distinct things. By taking Q[x] on each factor separately, our theorem bringsa field made by multiplying after separately quotienting.Then f’s full galois group glistens by subdirect producting.
But now question this strange equation: (x^2 + 3)(x^2 + x + 1), a curious relation. Each part has cyclic Galois group, that much is true, Yet both adjoin root minus three, an unfortunate issue since both share the same field and automorphism too, No direct product arises—just Z2 will do.
We find when X and S are Galois over F, In composites and intersection fields, galois properties manifest. The automorphisms that agree on both fields align, Forming subgroups that each G defines. with product GAL(XS/F), as structure implies.
If even deeper into this theory, you’d like to look,go no further than my favorite book.It has excellent authors Dummit and Foote…
Below is the book I first used to study the Gauss-Bonnet Theorem years before I wrote this poem! It was recomended to me by my excellent measure theory professor, Dr. Burkard, and I was rereading the sixth chapter, The Gauss-Bonnet Theorem when I had the idea for the poem.
An alien on a surface S looks into the ambient R three,Can he find S's Euler characteristic intrinsically, using only his local geometry?Can he compute Chi(S), with guarantee, just by knowing the curvature locally?
That’s when Gauss and Bonnet appear, a theorem held in hand they say is grandThey tell our alien, “Don’t fear, just look here.
An observer who’s to the surface near— should S be a compact Riemannian Manifold—must only well pioneer S’s frontier.Add the integrals of geodesic and Gaussian curvature and twice pi times Chi(S) will appear”
“But wait!” cried the alien. "Measuring the first fundamental form of S, I found rocky edges!The theorem breaks, because of all these sharp ledges!”Bonnet smiles, “There's no need to stress. With just piecewise smoothness of del(S), a generalization exists and yields two pi Chi(S), no less,”
Corner angles and integrals of curvatures, take the whole.Add them together to grasp S’s soul.Math, like Earth, finds beauty in imperfection.”
See the connection? The alien’s journey is each scholar’s plight.We wander with questions, chasing math’s light,following a path dimly lit by a network of minds, across distance and time, joining together, unraveling the sublime.
It may be a daunting task, conquering science.But like our alien, we each stand on the shoulders of giants.
That’s when Gauss and Bonnet appear, a theorem held in hand they say is grandThey tell our alien, “Don’t fear, just look here.
An observer who’s to the surface near— should S be a compact Riemannian Manifold—must only well pioneer S’s frontier.Add the integrals of geodesic and Gaussian curvature and twice pi times Chi(S) will appear”
“But wait!” cried the alien. "Measuring the first fundamental form of S, I found rocky edges!The theorem breaks, because of all these sharp ledges!”Bonnet smiles, “There's no need to stress. With just piecewise smoothness of del(S), a generalization exists and yields two pi Chi(S), no less,”
Corner angles and integrals of curvatures, take the whole.Add them together to grasp S’s soul.Math, like Earth, finds beauty in imperfection.”
See the connection? The alien’s journey is each scholar’s plight.We wander with questions, chasing math’s light,following a path dimly lit by a network of minds, across distance and time, joining together, unraveling the sublime.
It may be a daunting task, conquering science.But like our alien, we each stand on the shoulders of giants.
The book I learned the Class Equation from during my first group theory course! That's when I wrote the first version of the poem. It was revised studying for my comprehensive algebra exam to include reference to its applications and a tidier ending.
One Classy Equation
One Classy Equation
Written September 2023 and revised July 2024
The class equation's elegance gleams,A theorem of groups with orbiting themes.We count the elements, partition by role,And uncover the structure that governs the whole.
First, consider Z(G), the center we see,elements commute with ease, acting so free.For each element incidentallyConjugate action is precisely the identity.
Beyond the center, conjugacy reigns,where action defines equivalence chains.Each element joins a class so neat, with trivial intersections and union complete.
Elements into equivalence classes, this smears,and the Orbit-Stabilizer theorem appears.Each orbit’s size, a quotient refined,Of ∣G∣ by ∣C_G(x)∣, perfectly aligned.
Thus, ∣G∣, the group we partition,yields this equation, our elegant confession:∣G∣=∣Z(G)∣+∑ ∣C_G (x)∣/∣G∣For p-groups, the equation holds sway,forcing nontrivial centers in a pivotal way.And deeper still, in any finite group’s span,Irreducible representations take their stand.Each conjugacy class has a unique role to play,Counting the forms in which G may display.
Conjugacy classes, equivalence defined,Reveal how symmetries can intertwine.The fixed and the flowing, in balance they play,In algebra’s heart, this equation will stay.
So pause for a moment, let wonder abound,The class equation is a theorem profound.With center and orbits, we trace its design,A harmony pure, both simple and fine.
First, consider Z(G), the center we see,elements commute with ease, acting so free.For each element incidentallyConjugate action is precisely the identity.
Beyond the center, conjugacy reigns,where action defines equivalence chains.Each element joins a class so neat, with trivial intersections and union complete.
Elements into equivalence classes, this smears,and the Orbit-Stabilizer theorem appears.Each orbit’s size, a quotient refined,Of ∣G∣ by ∣C_G(x)∣, perfectly aligned.
Thus, ∣G∣, the group we partition,yields this equation, our elegant confession:∣G∣=∣Z(G)∣+∑ ∣C_G (x)∣/∣G∣For p-groups, the equation holds sway,forcing nontrivial centers in a pivotal way.And deeper still, in any finite group’s span,Irreducible representations take their stand.Each conjugacy class has a unique role to play,Counting the forms in which G may display.
Conjugacy classes, equivalence defined,Reveal how symmetries can intertwine.The fixed and the flowing, in balance they play,In algebra’s heart, this equation will stay.
So pause for a moment, let wonder abound,The class equation is a theorem profound.With center and orbits, we trace its design,A harmony pure, both simple and fine.
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