Vol. 1, No. 1 Spring 2019

UT Austin UndergraduateMath Journal

A Student Publication of UT Austin

Table of Contents

1 From the Editors 2

2 Clifford Algebras 3

Orthogonal Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Hyperplanes and the Cartan-Dieudonne Theorem . . . . . . . . . . . . . . . . . 4

The Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 An Interview with Prof. Freed 10

4 Building and Solving Differential Equations Using Electronic Circuits 17

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Building Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Operational Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 A Word from the Students 23

6 Something to finish on. . . 25

SPECIAL FEATURE

1

From the Editors

Thank you so much for taking the time to check out the UT Austin Undergraduate

Math Journal! As you are probably aware, this is the first issue of the journal. There

have been many learning experiences along the way – ironing out LATEX issues, editing

articles, conducting and transcribing longform interviews, etc. There are of course many

things we can (and will) improve on in future volumes. As such, we welcome any and all

constructive feedback you might have. Whether it’s comments on content or aesthetics,

please send your thoughts to utmathjournal@gmail.com. We hope you enjoy reading

this journal as much as we enjoyed putting it together. Ciao!

STUDENT ARTICLE

2

Clifford Algebras

Jeffrey Jiang, ’19

Jeffjiang@utexas.edu

Orthogonal Groups

An important group in mathematics is the orthogonal group On, which consists of

linear transformations T : Rn → Rn such that 〈Tv, Tw〉 = 〈v, w〉 for all v, w ∈ Rn (such

transformations are said to preserve the inner product 〈·, ·〉). This inner product induces

a norm | · | on Rn given by |v| = 〈v, v〉1/2, which in turn introduces a notion of length in

Rn. This also gives us a notion of the angle θ between two vectors v, w ∈ Rn via

θ = arccos

(|〈v, w〉||v||w|

).

Since On preserves the inner product, it also preserves these notions of angle and length.

The induced norm makes Rn into a metric space with distance function d : Rn ×Rn → Rgiven by d(v, w) = |v − w|. It follows that On is the group of linear isometries of the

metric space (Rn, d).

The orthogonal group plays a large role in physics, since many of the linear transfor-

mations we want to observe in the natural world preserve our perceptions of angle and

length. For intuition, try thinking about what kind of shape these groups can have. In

one dimension, we don’t have too much to work with since any linear map preserving the

absolute value must be multiplication by ±1. Therefore, we find that O1 = {± id}.Things are a little more interesting in two dimensions. Note first of all that any A ∈ O2

preserving length and angle must necessarily preserve the unit circle S1 ⊂ R2. Any such

transformation is then (almost) uniquely determined by the angle by which it rotates

4

a single vector, as well as whether it flips the orientation of R2. Heuristically, what we

mean by orientation is a choice of clockwise or counterclockwise. A rotation by some

angle θ preserves orientation, but a reflection, say across the y-axis, flips the orientation.

From our description, we now see that O2 should look like two disjoint circles: one circle

for all the rotations, and another separate circle of rotations following a reflection. We

call the orientation preserving component SO2, which you can think of as the group of

rotations. The components of O2 form a group, which we denote π0(O2). This follows

from the fact that the composition of two orientation reversing transformations preserves

orientation, which gives a group isomorphism π0(O2) ∼= Z/2Z.

Going back a bit, why did we say that the transformation is almost uniquely determined

by the angle? The reason is that a rotation by angle θ is the same as a rotation by 2π− θ.This means that SO2 might not be parameterized by a circle as we might have thought!

In fact, though, SO2 is isomorphic to a circle (why?) – a fact that does not generalize

to higher dimensions. Indeed, the same observation as above in the three-dimensional

case gives us that SO3 is isomorphic not to S2 but rather to RP3! It’s a good mental

exercise to figure out what happens here, and why the two- and three-dimensional cases

are different. You might want the think of RP3 as the unit sphere S3 ⊂ R4 with antipodal

points identified via v ∼ −v. Can you see why this is the right picture? If not, don’t

worry: after some work we will have a more rigorous explanation.

Summarizing our results from tinkering around with orthogonal groups in low dimen-

sions:

1. The composition of an even number of orientation reversing maps is orientation

preserving, i.e. π0(On) ∼= Z/2Z.

2. Using spheres to describe orthogonal transformations has redundancies, which can

be attributed to antipodal points. Somehow, v and −v encode the same data for

an orthogonal transformation.

Hyperplanes and the Cartan-Dieudonne Theorem

Definition. Let V be a real vector space of dimension n. A hyperplane in V is an

(n− 1)-dimensional subspace P ⊂ V .

If V has an inner product then any hyperplane P is determined by the line given by

its orthogonal complement P⊥. In addition, since we have a notion of length, a line is

determined by the unique vector v ∈ P with norm 1 (called the unit normal). If V

doesn’t have an inner product then there is in general no way to make these distinctions.

But wait, something is wrong with what we just said! There isn’t a unique vector with

norm 1 in P : −v is just as good. This is one of the key points we observed about the

orthogonal group, which suggests that something deeper is going on here. We can capture

this deeper relation by taking a hyperplane P ⊂ Rn and defining a map RP : Rn → Rn

5

that reflects Rn about P (e.g., reflection about a line in R2 or reflection about a plane in

R3). If we let v be one of the two unit normal vectors then RP is given by the formula

RP (w) = w − 2〈w, v〉v.

How do we interpret the formula? A hyperplane reflection should not change the

components of a vector that lie in P , instead flipping only the component orthogonal to

P . This is accomplished exactly by subtracting 2〈w, v〉v from w.

We’re now ready for a key piece of the puzzle.

Theorem (Cartan-Dieudonne). Any orthogonal transformation A ∈ On can be written

as a composition of ≤ n hyperplane reflections

Proof. We prove a slightly more general statement. Let V be a finite dimensional vector

space equipped with an inner product 〈·, ·〉, and let O(V ) denote the group of linear maps

V → V that preserve the inner product. We claim that any element of O(V ) can be

written as a composition of ≤ dimV hyperplane reflections. The proof is by induction

on dimV .

The 1-dimensional case is easy, since the only hyperplane in a 1-dimensional vector

space V is the zero vector, so the only elements of O(V ) are exactly idV and − idV (we

say that idV is the composition of 0 hyperplane reflections).

Now, let V be an n-dimensional vector space with inner product 〈·, ·〉. Fix A ∈ O(V )

and a nonzero vector v ∈ V . We want to find a hyperplane reflection R : V → V such

that RAv = v. To do this, let R be the hyperplane reflection about the hyperplane

bisecting v and Av. Explicitly, R is given by the formula

Rw = w − 2〈Av − v, v〉

〈Av − v,Av − v〉v.

Then, we have that RA is an orthogonal transformation that fixes v since

Rv = v − 2〈Av − v, v〉〈Av − v,Av − v〉

v

= v − 2〈Av, v〉 − 2〈v, v〉〈Av,Av〉 − 2〈Av, v〉+ 〈v, v〉

v

= v − 2〈Av, v〉 − 2〈v, v〉2〈Av, v〉 − 2〈v, v〉

v

= v − (−1)(Av − v)

= Av.

Since RA ∈ O(V ), RA therefore fixes the orthogonal complement v⊥. This is a hyperplane

in V with inner product obtained by restricting the inner product on V . The restriction

of RA to v⊥ is an orthogonal transformation of v⊥ and so is the composition of ≤ (n− 1)

hyperplane reflections in O(v⊥) by the inductive hypothesis. Since RA fixes v, we can

6

extend each of these hyperplane reflections to a hyperplane reflection on all of V by

first extending each hyperplane to a hyperplane in V (just take the span with v) and

then taking the corresponding reflection in V . Thus, RA is a composition of ≤ n − 1

hyperplane reflections in V . Since R2 = idV , composing with R gives that R2A = A is a

composition of ≤ n hyperplane reflections. This completes the induction. �

The stage is now set for the construction of the Clifford algebra.

The Clifford Algebra

Definition. Equip Rn with the standard inner product 〈·, ·〉. The Clifford algebra for

Rn is a unital associative R-algebra Cliff(n) generated by Rn, subject to the relations

1. v2 = −1 if |v| = 1,

2. vw = −wv if 〈v, w〉 = 0.

By unital, we mean that we have thrown in a new element e that acts as the multi-

plicative unit. Therefore, we have that for any λ ∈ R and v ∈ Cliff(n), we must have

λe · v = λv, so by a slight abuse of notation, we follow the standard convention in which

the multiplicative unit is denoted 1. When we say that Cliff(n) is generated by Rn, we

mean that every element of Cliff(n) can be written as a finite formal sum of products

of vectors in Rn (which then reduces by the relations specified above). Therefore, the

standard basis {ei} of Rn is a generating set for Cliff(n), yielding the basis

{ei1 . . . eik : 0 ≤ k ≤ n, 1 ≤ i1, . . . ik ≤ n}.

Thus, a basis for Cliff(n) is just the set of all products of basis vectors with increasing

indices, along with 1. For example, a basis for Cliff(3) as a vector space is given by

{1, e1, e2, e3, e1e2, e1e3, e2e3, e1e2e3}.

This characterization allows us to see that the dimension of Cliff(n) as a real vector space

is 2n. We also note that Cliff(n) contains a subspace

Span(e1, . . . , en) ∼= Rn.

What is the motivation for this definition? A unit vector v ∈ Rn ⊂ Cliff(n) should

represent a hyperplane reflection about the plane perpendicular to v. In particular, since

hyperplane reflections square to the identity, we want the same to be true here (but with

an added sign).1 Another item of motivation is that hyperplane reflections about planes

1Our choice of sign is simply convention: the choice v2 = 1 works just as good. One benefit of our choice

is that some formulas will look cleaner.

7

determined by orthogonal vectors commute. Try messing around with such reflections in

the two- and three-dimensional settings: pay attention to signs!

We now derive a helpful formula.

Lemma. Let v, w ∈ Rn ⊂ Cliff(n). Then, vw + wv = −2〈v, w〉.

Proof. Let v =∑

i viei and w =

∑j w

jej . We compute

vw + wv =∑i

∑j

(viwjeiej + viwjejei)

=∑i

∑j

viwj(eiej + ejei)

Since e2i = −1 and 〈ei, ej〉 is 0 for i 6= j and 1 for i = j, this sum collapses to∑i

viwi(e2i + e2i ) = −2∑i

viwi = −2〈v, w〉. �

Another useful observation about Cliff(n) is that it comes with a linear map T from

Cliff(n) to itself such that T 2 = id (called an involution). This map is uniquely determined

by reversing the order of products of basis vectors a la

T (ei1 . . . eik) = eik . . . ei1

and then extending linearly to the rest of Cliff(n). For an arbitrary g ∈ Cliff(n), we use

the notation gT = T (g). The similarity to the notation for matrix transposition is not a

coincidence!

Consider now the multiplicative group Cliff(n)× of invertible elements of Cliff(n). Note

that we have to be a little careful working with this group since it is nonabelian. There

is a nice subgroup G ⊂ Cliff(n)× generated by the unit vectors of Rn. Identifying Rn as

a subspace of Cliff(n), we claim that there is a (somewhat) natural left action of G on

Rn given by

g · w = gwgT

for g ∈ G and w ∈ Rn. Of course, at the outset we have no reason to even believe that

gwgT is an element of Rn. To verify that we do in fact have a well-defined action of G, it

suffices to check on the generating set of unit vectors. Let v ∈ Rn with |v| = 1. Writing

v =∑

i viei, we have

vT = T (v) = T

(∑i

viei

)=∑i

viT (ei) =∑i

viei = v.

8

Using the lemma, we compute

v · w = vwv

= (−2〈v, w〉 − wv)v

= −2〈v, w〉v − wv2

= w − 2〈v, w〉v ∈ Rn,

which is exactly hyperplane reflection about the hyperplane perpendicular to v! Thus,

not only do we have a group action on our hands, we also know that the generating

set for G acts exactly as the generating set for On. This gives us a natural group

homomorphism ϕ : G → On that sends g ∈ G to the linear transformation w 7→ g · w.

This map is surjective by the Cartan-Dieudonne Theorem and so On ∼= G/ kerϕ by the

First Isomorphism Theorem.

A little work gives that kerϕ = {±1} and so ϕ is a 2-to-1 map. Indeed,

(−v)w(−v) = vwv,

which is an expression of the fact that hyperplane reflection about v⊥ and −v⊥ are

exactly the same. Our larger group G allows us to distinguish between v and −v. In

addition, since any given g ∈ G is a product of an even number of unit vectors, the map

determined by g is orientation preserving since it is a composition of an even number of

hyperplane reflections.

What we just discovered is the group G = Pin(n) that double covers On, as well as

its subgroup Spin(n) generated by even products that double covers SOn. Thus, in one

fell swoop, we’ve addressed two of our earlier observations with the orthogonal group

and encoded them in a new mathematical object – the Clifford algebra. Because of the

way that multiplication in Cliff(n) seems to encode the geometry of Rn, some call the

Clifford algebra Clifford’s geometric algebra.

It’s a useful exercise to characterize these Clifford algebras in more familiar terms. For

example, Cliff(1) is isomorphic to the R-algebra C of complex numbers and Cliff(2) is

isomorphic to the R-algebra H of quaternions, both of which have an involution given

by conjugation q 7→ q. If you’re familiar with computer graphics, you might recall that

rotations are often more compactly represented as quaternions, where the action of a

quaternion q ∈ H on R3 is given by v 7→ qvq. This formula should look awfully familiar:

it tells us that Spin(3) is isomorphic the multiplicative group H× of unit quaternions. The

2-to-1 map ϕ onto SO3 is just the quotient by antipodal points, giving an isomorphism

SO3∼= RP3.

Although the geometric insights involving Pin and Spin are nice, there’s a very beautiful

theory concerning the algebras themselves. If we replace the inner product 〈·, ·〉 with

an arbitrary symmetric bilinear form b : Rn × Rn → R then we can repeat the same

construction as above to obtain Cliff(Rn, b). Every nondegenerate symmetic bilinear form

is uniquely determined by its signature – the number of 1’s and −1’s on the diagonal of

9

its matrix after diagonalizing – so we get an infinite family of algebras Cliff(p, q) (where

p denotes the number of 1’s and q the number of −1’s). Amazingly, this collection of

Clifford algebras is closed under taking tensor products. Via something called Bott

periodicity, it turns out that Cliff(0, n) and Cliff(n, 0) for 0 ≤ n ≤ 8 provide us with

enough information to reconstruct all Clifford algebras by taking tensor products.

With that, I leave you with the abstract definition of a Clifford algebra, which is the

usual context in which one first sees it. Can you see why this abstract definition agrees

with our own definition?

Definition. Given a vector space V and a symmetric bilinear form b : V × V → R, the

Clifford algebra is the data of a unital associative algebra Cliff(V, b) along with a linear

map ι : V → Cliff(V, b) such that, for any linear map ϕ : V → A of V into another unital

associative algebra A satisfying

ϕ(v)2 = b(v, v),

ϕ factors through Cliff(V, b) to give a unique map ϕ : Cliff(V, b) → A such that the

diagram

V

Cliff(V, b) A

ιϕ

∃! ϕ

commutes.

INTERVIEW

3

An Interview with Prof. Freed

Zachary Gardner, ’20

zacharygardner137@utexas.edu

Prof. Dan Freed is a Sid W. Richardson

Foundation Regents Chair in Mathematics

and Professor of Mathematics at the Univer-

sity of Texas at Austin. Prof. Freed’s work

centers around global issues in geometry and

global analysis.

When did your interest in math be-

gin?

I think at a pretty early age. My father

used to go bowling on Sundays and I would

go and try to bowl, though not very suc-

cessfully – I was probably four or five at the

time. Eventually I would wander over and

watch the men play. Pretty soon I was keep-

ing score. I suppose that was a sign that

I had some interest in numbers. That con-

tinued in a strong way through elementary,

middle, and high school. Anyway, as far as

I can remember I’ve always been interested

in math.

When it came time to go to college,

did you know if you wanted to major

in math?

I don’t recall making a conscious decision

– I think it was something I just knew. I

loved math and still love it. I just kept pur-

suing math, through college and into grad

school. At some point, maybe halfway into

my first year of grad school, I fell into a

small depression (if you can call it that)

11

when I realized that I had backed into a

mathematics career without quite choosing

it. I wasn’t depressed in the sense that I felt

I had made a bad choice – it was more like

an awareness of the situation overwhelmed

me for a little bit.

How long did you feel that way?

Not long, maybe a month or two. It didn’t

stop me from functioning or anything like

that. It was like, “Oh, this is where I’m

going. This is my future.” I don’t think

I had ever quite thought about planning a

career in that sense. I just kept following

my interests and that led me where it did.

Do you think your experience was

typical for the time?

Well I can’t say anything certain. Among

those who decided to continue on with their

studies, I think they felt similarly. Some

people were the children of mathematicians

and so it was ingrained in their family. For

others, I can’t really say. For me, being

a mathematician was something I couldn’t

not do.

Did your graduate experience unfold

in the same way? That is, did you

just sort of naturally stumble into

your thesis topic and other things like

that?

Well, even in high school I knew differen-

tial geometry was something I was really in-

terested in. I don’t know why exactly. Even

now, it’s not like I do differential geometry

in a form that most people would recognize,

meaning that I don’t really write papers

focused on the questions or tools of differ-

ential geometry. Probably the best way to

put it is to say differential geometry is my

first mathematical home. I was very fortu-

nate in college to have a group of graduate

students arrive during my junior year who

were all interested in differential geometry.

They had their own learning seminar going

through Spivak’s books on differential ge-

ometry. One of the students needed to learn

Lie algebras and so the group organized a

Sunday morning seminar on Lie algebras

following these beautiful notes written by

Hans Samelson. My job, as the undergrad-

uate, was to provide bagels! Anyway, this

group of graduate students really took me

in. It was a terrific experience and I’m very

thankful to them.

As for my PhD thesis, at some point my

advisor suggested three problems and one

of them developed into my thesis. At that

point loop groups were important infinite

dimensional Lie groups and Lie algebras

were being developed – and a Kahler met-

ric had been introduced on the based loop

group of a compact Lie group. Singer, my

advisor, asked me to compute the curvature

of that metric. I did so – laboriously at

first filling pages of a notebook, but in the

final version just a few paragraphs – and

that computation suggested other problems

and so I was on my way. I strongly believe

that computations are a great way to start

off, and a thesis advisor can do much worse

than suggest a well-chosen computation.

I know from talking to others, grad

students especially, that one of the

things a lot of people have ended up

greatly enjoying and depending on

at times is the social environment of

their institution. It’s a place for peo-

ple to lean on others, ask questions,

and be dumb from time to time.

Well, I think you have to be willing to

be dumb always. The moment you’re not

willing to be dumb is the moment you stop

12

learning and being able to move forward.

If you’re doing research then you’re always

dumb in a sense. But yes, as far as social

environment goes, I think the social environ-

ment is important at all levels and especially

in grad school. When people are picking

out grad schools, they often focus on who

is at each school and what specific research

they are doing. And if you’re advanced

enough then that’s very appropriate. But

you shouldn’t lose sight of the fact that the

grad school years are great years of your life.

If you’re not happy with the environment,

both in terms of the social life in the depart-

ment and physical aspects like geography

and climate, then you’re not going to do

good work. So, choosing a grad school has

to be a decision that takes environment into

account.

To give some personal background, I did

my graduate work at Berkeley. Before that,

I had lived in Chicago and Boston. Berkeley

in the early 80s was a little bit different –

not like Berkeley in the 60s but still a place

with its own quirkiness. In terms of the

math department, it was an amazing time

to be there, especially if you were doing

geometry. There were lots of classes, semi-

nars, and faculty doing interesting geometry

research. I was also playing lots of music

(orchestral trombone), and the Bay Area

offered many wonderful opportunities I was

able to take advantage of.

While I was at Berkeley, MSRI (Mathe-

matical Sciences Research Institute) opened.

I quickly learned that a lot of the people

visiting had left their families behind on

the east coast and come out to MSRI for

a semester. The professors in the math de-

partment were very busy – they often had

closed doors, didn’t want to be in their of-

fices, or were hassled with administrative

tasks and other things of that nature. The

people at the institute were a bit lonely, so

they were happy to talk math with me for

hours. I got a great education out of that!

Berkeley had strong programs in areas that

were of direct interest to me, which was

a great aspect of my graduate experience.

MSRI came along and accelerated the pace

of everything.

I know you were involved in the

founding of PCMI (Park City Math

Institute). Did your experience with

MSRI have any role to play?

I don’t think so, though I’ve never re-

ally thought about that. MSRI and PCMI

are actually very different from each other

– MSRI is primarily a research institution

that runs year-round, while PCMI is much

less focused on research and runs for three

weeks in the summer. I conceived of what

is now called PCMI in terms of what people

call vertical integration, which is the idea

that you should have researchers, graduate

students, undergraduates, and others all in-

teracting in a low-pressure environment. In

a way, the motivation behind PCMI was to

give back. I’ve always had amazing teach-

ers and opportunities, as well as a support-

ive and stimulating environment. PCMI

gives younger people something like that

with a chance to interact with older peo-

ple in a more relaxed setting away from

everyone’s home institution. The PCMI

philosophy also goes the other way around.

From the very beginning, you’re instilling

within young people this idea of mentoring

the next generation by showing them what

mentorship looks like.

Who else was involved in the found-

ing of PCMI?

13

I worked closely with Karen Uhlenbeck,

who was a colleague of mine at UT Austin

for many years and a great friend. Then

there were some mathematicians from the

University of Utah, Jim Carlson and Herb

Clemens. John Polking, the head of the

NSF math division at the time, also played

a very big part. Many others joined in –

building institutions which last is a commu-

nity effort. After a few years, it became

clear that the seed money for PCMI was

going to run out. So, Phil Griffiths, the

director of the IAS (Institute for Advanced

Study) at the time, stepped in. Phil under-

stood that adopting PCMI would be good

for the IAS and so the Institute basically

took over the administrative side of things

and now PCMI is a program of the IAS.

Of course, PCMI has continued to get huge

funding and support from the NSF.

Switching gears a little bit, you’ve

said on your website and elsewhere

that you’ve collaborated with physi-

cists and that your work has overlap

with physics. Could say more about

that?

Well, geometry has a long history of en-

gagement with physics: trigonometry was

introduced to understand astronomical ob-

servations; Gauss came up with the Gauss-

Bonnet Theorem while he was director at

an astronomical observatory; and Newton

developed calculus in conjunction with his

physics theories. There are many more mod-

ern examples too, and our current period is

one of vigorous interaction. Of course, the

degree of engagement has varied over time.

Often, new ideas come into mathematics

from physics and elsewhere, resulting in a

period of internal mathematical work as

these ideas are absorbed. This involves the

development of formalism as a framework

for ideas, as well as theorems and useful

structures. Afterward, these mathematical

fruits are applied back to the physics and

elsewhere.

Deviating from your question a bit,

there’s this question of whether math is

a science or an art. This is of course a

false dichotomy, one of many such (pure vs.

applied, theory-builder vs. problem-solver,

etc.), as math is both a science and an art

in both its input and output. By input, I

mean motivation and inspiration. Math cer-

tainly draws from science (i.e., the physical

world), but it also grows from an artistic

perspective in that we may do math either

because it’s beautiful or the things we’re

working on are of internal importance to

mathematics. By output, I mean the actual

mathematics that is created. Math is an

achievement of the human spirit just like ar-

chitecture, painting, sculpture, music, and

other things typically considered to be arts.

At the same time, math applies to the real

world through either the theoretical aspects

of science or the development of technology,

economy, etc.

Getting back to your question, there’s the

example of the 20th century development

of symplectic geometry as a framework for

classical mechanics, and specifically, classi-

cal field theory, which is something I studied

as an undergraduate. When I got to grad

school, my advisor, Is Singer, was already

teaching courses on topics in theoretical

physics. His advisor, Irving Segal, was very

much involved with physics and the math-

ematics of quantum field theory. Is was

an early champion of the idea that certain

problems in physics could be a huge boon

to mathematics and, conversely, mathemat-

14

ics could have something to say about the

physics. He had already written important

papers solving problems in quantum field

theory and, at the same time, developing

ideas in geometry. With these ideas in mind,

Is ran a weekly seminar at Berkeley with the

aim of learning about supermanifolds and

quantum field theory. Is would teach for

the first two hours, then there would be a

seminar for the next two hours, often with

a physicist as speaker. Discussion would

continue over dinner for another two hours.

Altogether that’s six consecutive hours in

one day! There were many great visitors,

and getting to know them at those dinners

was an important part of my graduate ex-

perience.

Nowadays, interaction between math and

physics is much more typical, with con-

ferences, papers, and even entire research

institutes organized around collaboration.

Currently, quantum field theory and string

theory are a big focus here and elsewhere.

Another thing that’s gaining in popularity

is the interaction between topology and con-

densed matter physics. For me, physics has

always been a topic of interest and a source

of ideas. It’s tremendous fun to collabo-

rate with all sorts of different people with

different backgrounds, even despite the dif-

ficulty involved in learning new languages

and frameworks for certain concepts.

I’ve heard some mathematicians say

that the process of formalizing ideas

in a hot new area of math can some-

times kill a bit of the magic. Do you

have anything to say about that?

Well, I’m an unapologetic mathematician

and so I do believe firmly in the value of set-

ting up a strong mathematical framework.

However, such a framework should be not

be a prison in which we trap ourselves. The

process of developing new math starts by

freely exploring without constraints. Basi-

cally, if you never allow yourself to move

beyond existing frameworks, then you won’t

be able to say something truly new worth

codifying. At the same time, I think new

definitions arise only from engaging with

and solving specific problems. Those defi-

nitions lead to theorems, the theorems to

solutions to other problems, and so the cy-

cle continues. You see that theory-building

and problem-solving go hand-in-hand, and

the magic emerges from the combination.

There are many fundamental and beauti-

ful definitions in math. Take the notion of

a group, for example, which came in part

out of algebraic work of Galois and Gauss

as well as the desire to have an object that

encodes geometric symmetries. Many of the

important properties that groups have come

from the choice of an elegant, sparse defini-

tion. Another beautiful notion is that of the

real numbers. We have a definition that is

also a theorem, which says that the field of

real numbers is the unique complete ordered

field. It’s amazing that just three words –

complete ordered field – can encompass so

much of the power that real numbers give

us. This is especially because, if you were to

go back in time, you likely wouldn’t guess

that such a simple definition would be the

end result of the analysis that preceded it.

So in this case it’s the definition providing

magic rather than taking it away.

Formalization can also provide magic

by connecting seemingly disparate ideas.

For example, there are many different ap-

proaches to QFT (quantum field theory).

There’s a physicist, Nati Seiberg, who likes

to say that there are too many coincidences

15

for us to think that we know the right start-

ing point for QFT. Basically, the coinci-

dences are a sign that we haven’t really un-

derstood the theory. Now, one can hope

that there will be some framework that

helps explain the coincidences, and in a

certain sense there is one already that has

been put forward by Graeme Segal and oth-

ers. Their axiom system is sparse, in the

same way that saying the real numbers are

a complete ordered field is a sparse charac-

terization. But that axiom system is only

a beginning, and has only been fully de-

veloped for theories that are special in the

world of all quantum field theories. All of

that aside, there are limitations to defini-

tions. The definition of a manifold doesn’t

tell you how to construct a manifold, and it

doesn’t give you examples. The definition

simply conveys what a manifold is and, his-

torically speaking, what a manifold ought to

be. The characterization we have of QFTs

gives us some inroad, but there’s still a long

way to go.

Are you actively involved in that

quest for better understanding

QFTs?

In some broad sense, sure. At any given

time, there’s a selection of problems that

I’m working on, poking at, and seeing if I

can make a little contribution here or there.

Dennis Sullivan once told me that he likes

to sit down a mathematician and ask them

to tell the story that weaves together all the

papers they’ve written in order. If I was to

sit down and do that then QFTs would be

part of the story.

Have you had any long periods of

time in which you were stuck on a

problem and maybe felt discouraged?

Sure, all of the time! I like to say that if

you’re not confused then you’re not work-

ing. Most endeavors aren’t quite like that,

but math research is. Fortunately, we have

the luxury in mathematics of being very

nimble, meaning that we can change direc-

tion on a dime because we don’t have to

buy expensive equipment or manage large

teams as researchers in laboratory sciences

do. We also don’t have to plan ahead nearly

as much – there are astronomers now who

are planning telescopes that won’t be online

for fifteen years.

As an aside, when I was in my last year

of graduate school, Ed Witten came to MIT

and lectured on a new formula for global

anomalies. I understood something about

it, and where it fit into some global analysis.

When I talked to my advisor, Is, about it he

told me to drop everything and work on this.

That made my thesis submission late! Also,

just prior to that, I had submitted a post-

doc application to the NSF. A year later,

I wasn’t working on any of the problems I

had proposed in my application since I was

following up the new ideas about anomalies.

Naturally, I was a little bit concerned and so

asked my elders about it. They told me not

to worry, which was a lesson to me about

the nimbleness of mathematics research and

how we should embrace that.

Coming back to the question, there are

basic techniques you can apply to get un-

stuck. Polya’s book How to Solve It has

good pointers that apply just as well to

research as any other kind of mathemat-

ical pursuit – break problems down into

smaller problems, look for related problems,

change the problem, play with the problem,

etc. The goal in general is not to spin your

wheels. This is one of the many benefits of

16

having great colleagues and collaborators.

I’ve found that I can easily start spinning

my wheels when I’m working by myself. You

get stuck on one idea, one thought, one di-

rection. You’re too stupid – you can’t do

it. So you ask someone else. And some-

times, just a few minutes of conversation

can clear things up. Of course, working

with collaborators is also lots of fun. You

need to spend many long hours working by

yourself to produce mathematics, but the

social aspect shouldn’t be understated.

Some strategies for unsticking yourself in-

clude working on different things in parallel,

focusing on administrative work, and simply

deciding to pick up a book and learn some-

thing completely new for a change. Another

important strategy is to focus on teaching.

As I said earlier, mentoring the next gen-

eration is very important. And one of the

many benefits of teaching is that it provides

you with a routine and the psychological

satisfaction of knowing that you’ve accom-

plished something no matter how research

is going. There are challenges there, too, so

everyone has to find their own balance.

To close, what do you enjoy most

about being a professor?

There are many things I enjoy about be-

ing a professor. It is a charmed life, with

a tremendous amount of freedom to pursue

research I’m interested in and to teach in

a flexible way. There are enjoyable inter-

actions with colleagues (both here at UT

and elsewhere), grad students, and younger

students looking for mentorship. There re-

ally are so many positives. The situation

is largely analogous to that of the artist or

novelist who has a position at a university

teaching and such but also produces their

own creative work.

STUDENT ARTICLE

4

Building and Solving Differential

Equations Using Electronic Circuits

Vic Frederick, ’20

Vic79@me.com

Introduction

Ordinary Differential Equations (ODEs) create a relationship between a function of

one variable and its derivatives. Circuits can be used to build and solve these ODEs.

Such circuits have historically been called analog computers. Primarily used to calculate

projectile motion during WWII, these analog computers now find their home in control

systems. This article will go into detail on how circuits capture differential behavior, how

to turn a differential equation into a circuit, and how to solve differential equations using

circuits.

Building Blocks

Kirchhoff’s circuit laws provide the first building block for modeling differential equa-

tions using circuits. These two laws relate current and voltage to the graph-like structure

of a circuit, with current being a quality of edges and voltage a quality of vertices.

Kirchhoff’s Voltage Law says voltage sums to zero in a cycle (i.e., an ordered collection of

vertices beginning and ending with the same vertex), while Kirchhoff’s Current Law says

the weighted sum of current relative to a vertex is zero (i.e., the sum of incoming current

equals the sum of outgoing current). In engineering parlance, vertices are the components

of a circuit and nodes are the places where two or more components meet. Kirchhoff’s

18

circuit laws therefore allow us to construct differential equations by inspecting cycles and

nodes.

Figure 1: Linear Circuit Components

The next step is characterizing the electric components shown in Figure 1. Current

through a component can be related to the voltage across it. For a resistor with constant

resistance R, the relationship between voltage and current is described by Ohm’s Law:

V = RI (1)

Current through a component can also be related to the change in voltage across it.

For a capacitor with constant capacitance C, the relationship is described by

I = CdVCdt

(2)

Here, VC is the voltage across the capacitor. Sources, the remaining components, either

add voltage across a node or specify a current.

Building Differential Equations

Consider the following series circuit, so called because it consists of a single cycle.

Figure 2: RC Circuit

19

Kirchhoff’s Voltage Law and Ohm’s Law together give

0 = VR + VC − V1 = RI + VC − V1 (3)

The current is constant throughout the series circuit because each node is the junction

of only two vertices. Using Equation (2) to write I in terms of VC and solving for V1gives the first-order linear non-homogeneous differential equation

V1 = RCdVCdt

+ VC (4)

We can perform a similar procedure to find a second-order linear non-homogeneous

equation characterizing the circuit shown below.

Figure 3: Second Order RC Circuit

Kirchhoff’s Current Law and Equation (2) together give

I1 = IC1 + IC2 = C1dVC1

dt+ C2

dVC2

dt(5)

Kirchhoff’s Voltage Law and Ohm’s Law together give

VC1 = VR + VC2 = RC2dVC2

dt+ VC2 (6)

Substituting this into Equation (5) and collecting like terms gives

I1 = C1d

dt

[RC2

dVC2

dt+ VC2

]+ C2

dVC2

dt= RC1C2

d2VC2

dt2+ (C1 + C2)

dVC2

dt(7)

We now have two examples of recreating differential equations using circuits, and thus

two cases in which we can approximate solutions by taking measurements. Namely, we

can use an oscilloscope to read the relevant capacitor voltages and thereby measure

solutions to Equation (4) and Equation (7). Unfortunately, this approach has two key

issues.

1. Linear electric circuits will not have outputs that rise above the input – i.e., the

waveform will be chopped off above the maximum of the non-homogeneous side of

the equation.

20

2. Such circuits are often difficult to synthesize even if they are easy to analyze.

Fortunately, there is an electronic component known as an Operational Amplifier that

allows us to address these issues.

Operational Amplifiers

Consider the following schematic.

Figure 4: Operational Amplifier with Supply Lines

Ideally, Operational Amplifiers (Op-Amps for short) keep the same voltage at nodes

A and B shown in Figure 4. This is done using feedback from Vout. V+ and V− are the

power supply lines, which limit Vout via

V− ≤ Vout ≤ V+ (8)

Staying within these limits is important for minimizing distortion. For the sake of

simplicity, we will largely ignore supply lines when building equations. One big advantage

of Op-Amps is that they allow us to neatly package the available operations of sums,

coefficients, and derivatives – the building blocks of linear differential equations. This is

illustrated by using Kirchhoff’s Current Law at node A of the figures below (note that

node A is not explicitly marked but can be identified by comparison with Figure 4).

Figure 5: Coefficient Circuit

Vout = −R1

R2Vin (9)

21

Figure 6: Summing Circuit

Vout = −R1

R2Vina −

R1

R3Vinb (10)

Figure 7: Differentiating Circuit

Vout = −RC d

dtVin (11)

Selecting appropriate resistances and capacitances therefore allows us to control any

coefficients that appear. To simplify the process of building equations, we will treat these

circuits like black boxes and adopt the notational shorthand shown in the following figure.

Note that each operational block changes the sign of its input, which is why such circuits

are called Inverting Amplifiers.

Figure 8: Circuit Shorthand

The symbol A in the Coefficient Circuit represents a choice of coefficient, which should

be clear in context. Because we are dealing with an Inverting Amplifier, the output is

−A times the input. Now, let’s put these tools to use. Consider the following differential

equation.

Vin = Bd

dtVout + Vout (12)

22

Here, Vin is the input voltage, Vout is the desired output voltage, and B is a constant.

A first attempt at representing this equation is as follows:

Figure 9: ODE Block Representation 1

When reading these sorts of diagrams, note that all inputs enter the left of a block

and all outputs exit the right of a block. The result of such an analysis is

Vout = −Vin +d

dtVout (13)

A slight modification to account for signs and the coefficient on the derivative yields

the desired equation.

Figure 10: ODE Block Representation 2

To close, try writing out the equation associated to the following diagram. You will

see just how much Op-Amps simplify the design process

Figure 11: Differential Equation Solver Implemented with Operational Amplifiers

SPECIAL FEATURE

5

A Word from the Students

David Green is a third-year undergrad math major at UT interested in representation

theory and the theory of tensor categories. Here, David speaks on his REU experience.

I attended the Summer 2018 REU at Texas A&M University, working under Professor

Eric Rowell in the Mathematics Topological Computation group. Since I had no prior

research experience, the change in both pace and content was, to say the least, somewhat

jarring. Learning primarily from papers instead of textbooks and solving a problem over

a couple of months instead of somewhere between a couple of hours and a day presented

me with some newfound challenges. I had to brainstorm new strategies for categorizing

information which didn’t neatly build off of things I already knew. Not making progress

on a problem for a week (or sometimes much longer. . . ), though relatively typical within

research mathematics, was psychologically taxing for me. In my own experience, I’m used

to solving problems I’m given almost immediately. For the REU, despite having only a

single problem and 8 weeks in the program to work on it, I still have some calculations

to do before I can really say I’m done. Even though I was learning a lot and working

hard most of the time, I always felt like I hadn’t done anything – an experience which

was difficult for my self-image. At the end of the day, though, I got through it and have

a better idea of what doing “real” math is like. I feel more confident about applying to

graduate schools for their PhD programs, which I count as a success for the REU.

As to the actual content of my research, I helped with a classification program for rank

6 modular tensor categories. I succeeded in showing that in the case where such objects

have a certain type of Galois group, all possible instances are in fact already known up

to a certain equivalence (subject to the calculations actually checking out).

24

Tom Gannon is a third-year math grad student at UT interested in algebraic geom-

etry, representation theory, and number theory. Here, Tom reflects on how his perceptions

of math have changed since his undergrad years.

One thing that has changed drastically for me is my concept of what math actually

“is.” As an undergrad, I assumed that I would master one area (say algebra) then master

another area (say algebraic geometry), continuing until everything would all come together

and then I would finally be able to produce research. I realized sometime during my

second or third year of grad school that math doesn’t really work like that. Quite often,

theories are built upon results that people haven’t mastered yet. The most “cutting edge”

math research would only become accessible after more than 7 years if you started with

the foundations, filling in all the little holes one at a time. Instead, you learn to build a

working knowledge of everything that serves as a guide.

To give an example, for the first time in my life I am participating in a learning

seminar whose explicit goal is not to achieve complete mastery of the theory of “infinity

categories,” whatever that means. Instead, the goal is to get a broad sense of the outline

of the theory. Before, I would have scoffed at this idea, but now I see that such an

approach is sufficient and even beneficial for doing research.

Another thing that has changed for me is my perspective on the role of intelligence in

math. Perhaps due to things going well for me in high school math-wise, I spent a lot of

time as an undergrad thinking that I was “naturally” good at math and any struggle

was a sign that I wasn’t good enough. I now look at things in totally the opposite way.

Math is a skill that can be developed with practice – what really matter are your passion

and drive. If I could give one piece of advice for soon-to-be grad students, it would

be that you’re not as “far away” from the field as you might think. “Natural talent”

is almost worthless for doing research-level mathematics, which usually involves many

medium-depth insights into a problem rather than one big magical insight that could

only come from your “intelligence.”

Challenge Problem

Here is a version of Fermat’s Little Theorem for matrices. Let n be a positive

integer, I the n× n identity matrix, A an n× n integer matrix, and p a positive

prime. Let tr(A) denote the trace of A, given by the sum of the diagonal entries

of A. Show that

tr(Ap) ≡ tr(A) (mod p).

Think you have a proof? Send your answer to utmathjournal@gmail.com.

SPECIAL FEATURE

6

Something to finish on. . .

Some of us here at UT had the pleasure last fall of taking Prof. Gordon for M382C

Algebraic Topology. A jovial man with a quriky and ever-present sense of humor, Prof.

Gordon produced for us a number of memorable quotes and expressions. We record a

few of them below.

♣ Sorry that I’m always chuckling to myself. I guess I’m just easily amused.

♣ I used to give problems that were wrong on purpose. . . because that’s how life is.

♣ A chain complex is. . . well, exactly what it is.

♣ Once you’re confident it’s right, it all works out!

♣ You never need to prove anything, you just stare at it and say “yes, yes, yes!”

♣ This is all an illusion, we haven’t proven anything!

♣ I’m going to chicken out on the full proof of excision. Besides, you guys know how

to read.

♣ At least it’s true, regardless of whether or not it’s useful [regarding reduced

homology].

♣ You all know how comparing different groups goes in an introductory algebra course.

You have four groups and the question is which groups are isomorphic. The answer

is: yes, yes, no, and yes.

♣ What are we trying to do here? I’ve already forgotten [stated at the end of a rather

short proof].

Thanks for reading!