Personal website · West Lafayette, Indiana

01 / 06

Alejandro Cano.

I am a fifth-year mathematics student at Purdue University, working with Dr. Andrew Toms and Dr. Di Qi.

Operator algebras
Pure mathematics
Research · Teaching · Code

01 — About me

A little about me.

Background
Colombia → United States
Mathematics & Economics

I was born in Medellin, Colombia, and later immigrated to the United States in 2014. After settling here, I pursued a dual major in Pure Mathematics and Economics at the University of South Florida. Under the mentorship of Dr. Thomas Bieske, I completed my undergraduate studies in 2022.

I am continuing my academic journey in the Mathematics Department at Purdue University. Under the guidance of Dr. Andrew Toms, my research centers on classifying normal operators in suitable classes of C*-algebras using tools such as the Cuntz semigroup.

Alongside pure mathematics, I work with Research Services in Purdue’s College of Agriculture and contribute computational support to the DARPA eX Vi project.

02 — Research

Operator algebras.

Classification · C*-algebras
Normal operators · Cuntz semigroup

Understanding the fine structure that distinguishes well-behaved algebras.

Elliott’s program to classify simple separable nuclear C*-algebras through K-theory and traces is essentially complete under the regularity assumptions of Z-stability and the Universal Coefficient Theorem. My aim is to move beyond the classifiable case and study the obstructions that arise for normal operators acting on separable Hilbert spaces in a unital C*-algebra of stable rank one.

This may require more delicate invariants, such as the Cuntz semigroup, or specialized tools from algebraic topology. I study the fine structures that help distinguish two well-behaved algebras.

In preparation

  • New structure for Villadsen algebras of the first type 2026
  • Rørdam’s question about the embeddings of the Jiang–Su algebra 2026
  • An existence theorem for *-homomorphisms out of the circle algebra 2026
03 — Academics

Study, teaching & community.

Research appointments
Teaching
Selected talks & conferences

Research appointments and teaching

Fall 2021MAS 3105 · Linear Algebra · Teaching AssistantUniversity of South Florida
Spring 2022MGF 3301 · Bridge to Abstract Math · Teaching AssistantUniversity of South Florida
Fall 2022MA 16200 · Calculus II · Recitation InstructorPurdue University
Spring 2023MA 16100 · Calculus I · Recitation InstructorPurdue University
Summer 2023MA 35100 · Linear Algebra · Teaching AssistantPurdue University
Fall 2023MA 15800 · Precalculus: Functions and Trigonometry · InstructorPurdue University
Spring 2024MA 26100 · Calculus III · Recitation InstructorPurdue University
Fall 2024Emergency Teaching AssistantPurdue University
Spring 2025MA 54600 · Functional Analysis · Teaching AssistantPurdue University
Summer 2025Graduate Summer Research Grant RecipientPurdue University
Fall 2025Mathematics Resource Room · Extended Office Hours in Undergraduate MathPurdue University
Spring 2026MA 52800 · Advanced Mathematics for Engineers and Physicists II · Teaching Assistant; Emergency TAPurdue University
Summer 2026Graduate Research Assistant, full timeResearch Services, Purdue College of Agriculture
Fall 2026Graduate Research AssistantResearch Services, Purdue College of Agriculture

Selected coursework

Measure TheoryAbstract AlgebraLinear AlgebraFunctional AnalysisComplex AnalysisC*-Algebras and W*-AlgebrasAlgebraic TopologyIntroduction to Type III FactorsOperator K-TheoryOrdinary Differential Equations & Dynamical Systems (audited)Cuntz Semigroup & ClassificationGeophysical Fluid DynamicsComputational Optimal Transport & Deep Generative ModelsStochastic Calculus (audited)Filtering Complex Fluid SystemsResearch Thesis · Dr. Andrew Toms

Selected talks

Sep 6, 2024
Group C*-Algebras, the Fourier Transform and Locally Compact Groups
Purdue University
Jan 27, 2025
Classification of Injective Factors I
Purdue University
Feb 3, 2025
Classification of Injective Factors II
Purdue University
Feb 2, 2026
Clifford Algebras and Spin Geometry
Purdue University
Feb 9, 2026
The Exterior Algebra, the Clifford Algebra, and a Hint of Representation Theory
Purdue University
Sep 16, 2026
Vector Bundles and Homogeneous C*-Algebras
Purdue University
Sep 23, 2026
Constructing New Vector Bundles from Old
Purdue University
Oct 5, 2026
New Structure for Villadsen Algebras of the First Type · Invited
Brin Center for Mathematics Research, University of Maryland

Conferences attended

2023Groundwork in Operator Algebras Lecture SeriesPurdue University
2023East Coast Operator Algebras SymposiumPurdue University
2024Noncommutativity in the North — MikaelFestChalmers University of Technology and University of Gothenburg
2024Great Plains Operator Theory SymposiumUniversity of Nebraska–Lincoln
2024East Coast Operator Algebras SymposiumNorth Carolina State University
2025Mathematics and Statistics in Industry: Workshop and Panel DiscussionPurdue University
Sep 2025Scientific Machine Learning: Theory, Algorithms, and ApplicationsPurdue University
2025Great Plains Operator Theory SymposiumUniversidad de Puerto Rico, Recinto de Río Piedras
2026Young Mathematicians in C*-Algebras (YMC*A)St. John’s College, Oxford University
04 — Projects & code

Ideas in motion.

Research support
Machine learning
Optimal transport

01DARPA eX Vi↗

eX Virentia (eXVi) explores how existing, unmodified vegetation can act as a sensor for activities of Department of Defense interest by using natural plant responses to man-made chemical exposures.

eX Virentia (eXVi) Phase 0, Department of Defense title slide

The program brings plant biologists together with remote-sensing and modeling experts to study biochemical responses to chemical exposure and their externally observable characteristics. I provide computational and graph-theoretic support to Purdue’s College of Agriculture Research Services team.

Computational supportGraph theoryDARPA
02Rolls-Royce Data Mine↗

This project addresses the need of the aerospace propulsion system Prognostics & Health Management (PHM) team for synthetic data that mimics actual fleet data. The data can be shared with external research entities, including universities, to explore prognostic algorithms while respecting export and International Traffic in Arms Regulations (ITAR) restrictions on actual data.

Rolls-Royce Data Mine project title slide

I spent one semester organizing a student team and co-developing machine-learning models to support the Rolls-Royce PHM team.

Machine learningSynthetic dataPrognostics
03Two-layer neural network for MNIST↗

A basic two-layer neural network classifies handwritten MNIST digits without PyTorch or TensorFlow. It uses NumPy and Pandas, ReLU activations, softmax, backpropagation, and gradient descent. The original page reports accuracy above 91% after fewer than 500 iterations.

Handwritten digit samples for the MNIST classification project

The implementation loads the MNIST CSV, normalizes grayscale values, trains a 784→10→10 network, evaluates on a development split, and plots sample predictions.

PythonNumPyPandasMNIST
04Wasserstein GAN↗

A PyTorch Wasserstein Generative Adversarial Network generates handwritten images from MNIST. The implementation includes a generator, critic, gradient penalty, training loop, saved generated samples, and training-loss curves.

Animated generated handwritten digit visualization from the Wasserstein GAN
PyTorchWGAN-GPMNIST
05Benamou–Brenier transport and FFJORD↗

This project solves a dynamic optimal-transport problem with a Neural ODE. A Gaussian distribution is transported toward a distribution sampled from an MNIST digit using a continuous normalizing flow, kinetic-energy regularization, and trajectory visualizations.

Scatter plot comparing final frames and target samples in the optimal transport project
PyTorchNeural ODEOptimal transportFFJORD
05 — Student seminar

Student Operator Algebras.

Co-organized with
Hao Wan & Chrisil Ouseph
Wednesdays · 12:30

A student seminar to spark interest in operator algebras and build a community across graduate programs and research areas.

We introduce participants to topics in C*-algebra and von Neumann algebra theory, and to results, concepts, and tools that intersect with their own research and coursework. To give a talk, email cano13@purdue.edu.

Topics

  • Operator K-theory
  • Index theory
  • Topological dynamics
  • C*-algebra theory
  • von Neumann algebra theory
  • Noncommutative geometry and topology

References beyond Wikipedia

Selected seminar schedule

Fall 2026

September 2 · Introductions and primers on research · PSYC 3102

Organizers and attendees introduce their areas of research and expertise, showing the range of mathematics that operator algebras connect with.

Speakers · Hao Wan, Alejandro Cano, Chrisil Ouseph, Ethan Kessinger, Arka Das, Michael Poole, Otto Baier, and Jose Manuel Barrientos · Purdue University

September 16 · Vector bundles and homogeneous C*-algebras · SCHM 112

An introduction to vector bundles, Swan’s theorem, and examples.

Speaker · Alejandro Cano · Purdue University

September 23 · Constructing new vector bundles from old · SCHM 112

A continuation of the vector-bundle discussion, with constructions of new bundles and examples.

Speaker · Alejandro Cano · Purdue University

Spring 2026 · Quantum mechanics, Clifford algebras & index theory

January 12 · Quantum Mechanics for the Working Mathematician · ME 1006

Experimental observations that motivated quantum mechanics and the mathematical framework behind the theory.

Speaker · Chrisil Ouseph · Purdue University

January 26 · Quantum Mechanics for the Working Operator Algebraist · ME 1006

Matrix mechanics, wave mechanics, and von Neumann’s unification of the two frameworks.

Speaker · Chrisil Ouseph · Purdue University

February 2 · Clifford Algebras and Spin Geometry · ME 1006

Tensor algebras, real and complex Clifford algebras, orthogonal groups, and the Hopf fibration.

Speaker · Alejandro Cano · Purdue University

February 9 · The Exterior Algebra, the Clifford Algebra, and a Hint of Representation Theory · ME 1006

Relations between exterior and Clifford algebras, examples of bundles and spin coverings, and a look at representation theory.

Speaker · Alejandro Cano · Purdue University

February 23 · Crossed Product C*-Algebras · ME 1006

Construction of the universal C*-algebra associated with a group action, with examples and ways to characterize crossed products.

Speaker · M. Ali Asadi Vasfi · Purdue University

March 2 · Results on General Toeplitz Random Matrices · ME 1006

Convergence of *-moments for Toeplitz matrices with independent Gaussian or free semicircular entries, including joint moments with functions in L∞[0,1].

Speaker · Ajay Kumar Karri · Texas A&M University

March 9 · Finite Group Actions and Rokhlin-Type Properties · ME 1006

Tracial and weak tracial Rokhlin properties for finite group actions, with examples and implications for crossed products.

Speaker · M. Ali Asadi Vasfi · Purdue University

March 23 · Finite Group Actions with the Weak Tracial Rokhlin Property · ME 1006

Examples of finite group actions with the weak tracial Rokhlin property and what it reveals about associated crossed products.

Speaker · M. Ali Asadi Vasfi · Purdue University

March 30 · An Intro to Dirac Operators and Spectral Triples · ME 1006

Dirac operators on spin manifolds, spectral triples in noncommutative geometry, and the connection between analysis and topological index theory.

Speaker · General Ozochiawaeze · Purdue University

April 6 · Spectral Triples: Examples and Index Theory · ME 1006

Examples of spectral triples, including de Rham and spin Dirac operators, and index pairings with KK-theory.

Speaker · General Ozochiawaeze · Purdue University

April 13 · Quasidiagonal C*-Algebras · ME 1006

A continuation of the previous semester’s discussion, revisiting definitions, proofs, and examples.

Speaker · Arka Das · Purdue University

April 20 · More on Quasidiagonal C*-Algebras · ME 1006

Proof ideas around stable finiteness and traces, with questions about AH-algebras and crossed products.

Speaker · Arka Das · Purdue University

Fall 2025 · KK-theory & quasidiagonality

September 10 · Lectures on KK-Theory · PSYC 3187

A guide through Kasparov’s K-theory, a key tool in modern C*-algebra classification.

Speaker · Marius Dadarlat · Purdue University

September 15 · Lectures on KK-Theory · PSYC 3187

Continuing the lecture series on Kasparov’s K-theory and its role in C*-algebra classification.

Speaker · Marius Dadarlat · Purdue University

September 22 · Lectures on KK-Theory · PSYC 3187

Continuing the lecture series on Kasparov’s K-theory and its role in C*-algebra classification.

Speaker · Marius Dadarlat · Purdue University

September 29 · Lectures on KK-Theory · PSYC 3187

Continuing the lecture series on Kasparov’s K-theory and its role in C*-algebra classification.

Speaker · Marius Dadarlat · Purdue University

October 6 · Lectures on KK-Theory · PSYC 3187

Continuing the lecture series on Kasparov’s K-theory and its role in C*-algebra classification.

Speaker · Marius Dadarlat · Purdue University

October 13 · Lectures on KK-Theory · PSYC 3187

Continuing the lecture series on Kasparov’s K-theory and its role in C*-algebra classification.

Speaker · Marius Dadarlat · Purdue University

October 27 · Lectures on KK-Theory · PSYC 3187

Continuing the lecture series on Kasparov’s K-theory and its role in C*-algebra classification.

Speaker · Marius Dadarlat · Purdue University

November 3 · Lectures on KK-Theory · PSYC 3187

Continuing the lecture series on Kasparov’s K-theory and its role in C*-algebra classification.

Speaker · Marius Dadarlat · Purdue University

November 10 · 0th-Order Pseudodifferential Operators and the Baby Index Theorem on the Sphere · PSYC 3187

Toeplitz operators, Fourier modes, the C*-algebra of 0th-order operators, symbol maps, and K-theory.

Speaker · General Oboira Ozochiawaeze · Purdue University

November 17 · Quasidiagonality in C*-Algebras · PSYC 3187

Motivation, examples and non-examples, permanence under algebraic constructions, Voiculescu’s characterization, and a variant of the Blackadar–Kirchberg conjecture.

Speaker · Arka Das · Purdue University

Spring 2025 · Classification and K-theory

January 27 · Classification of Injective Factors (Part I) · SCHM 117

History of the classification program for von Neumann algebras, focusing on Connes, property Gamma, the Araki–Woods classification, and the work of Murray and von Neumann.

Speaker · Alejandro Cano · Purdue University

February 3 · Classification of Injective Factors (Part II) · SCHM 117

Tools used by Takesaki and Connes to give a full description of type III factors.

Speaker · Alejandro Cano · Purdue University

February 10 · Classification of Approximately Finite-Dimensional C*-Algebras (Part I) · SCHM 117

Elliott’s classification theorem, ordered K₀ and K₁ groups, inductive limits, and the intertwining argument.

Speaker · Hao Wan · Purdue University

February 17 · Classification of Approximately Finite-Dimensional C*-Algebras (Part II) · SCHM 117

Inductive limits, Bratteli’s local characterization and diagrams for AF-algebras, and lemmas toward Elliott’s classification theorem.

Speaker · Hao Wan · Purdue University

February 24 · Classification of Approximately Finite-Dimensional C*-Algebras (Part III) · SCHM 117

Elliott’s proof of the classification theorem for AF-algebras.

Speaker · Hao Wan · Purdue University

March 3 · Measure Equivalence and Orbit Equivalence · SCHM 117

Orbit and measure equivalence for countable groups, their relationship to von Neumann algebras, examples, and applications.

Speaker · Ishan Ishan · University of Nebraska–Lincoln

March 17 · The K₁ Group of a C*-Algebra · SCHM 117

Construction and examples of K₁, homotopy invariance, exactness, and its role in classification by K-theoretic invariants.

Speaker · Jose Manuel Barrientos Lopez · Purdue University

March 31 · An AF Embedding · SCHM 117

Examples of embeddings of C*-algebras into AF-algebras and their place in the classification program.

Speaker · Jose Manuel Barrientos Lopez · Purdue University

April 14 · AF Embeddability · SCHM 117

A timeline of results on AF-embeddability, Schafhauser’s theorem, and tools used in the classification of unital C*-algebras.

Speaker · Jose Manuel Barrientos Lopez · Purdue University

April 21 · Classification of C*-Algebras: The Finale · SCHM 117

A synthesis of the classification program, from early results through the celebrated classification theorem.

Speaker · Chrisil Ouseph · Purdue University

April 28 · Classification of C*-Algebras: The Epilogue · SCHM 117

Abstract classification strategies, total K-theory, Toms–Winter regularity, the Cuntz semigroup, and future research directions.

Speaker · Chrisil Ouseph · Purdue University

Fall 2024 · Groups, index theory & noncommutative tori

August 29 · Introductions and Interests · MATH

Seminar introductions and research interests.

Speakers · Alejandro Cano, Hao Wan, and Chrisil Ouseph · Purdue University

September 9 · Direct Integral Decomposition of von Neumann Algebras · SCHM 314

Direct integrals of von Neumann algebras and applications to rigidity questions for group factors with diffuse center.

Speaker · Adriana Fernández Quero · University of Iowa

September 16 · Group C*-Algebras, the Fourier Transform and Locally Compact Groups

Group C*-algebras, Fourier transforms and dual objects, using representation theory, integration, and functional analysis.

Speaker · Alejandro Cano · Purdue University

September 23 · Hilbert Hotels on Hilbert Spaces · SCHM 314

Set theory and operator algebras, including independence results, the Continuum Hypothesis, weak expectations, ultrapowers, and Calkin algebra automorphisms.

Speaker · Chrisil Ouseph · Purdue University

September 30 · Introduction to Finite Free Probability · SCHM 314

Finite free additive and multiplicative convolutions, random matrices, free probability, geometry, representation theory, and combinatorics.

Speaker · Daniel Perales · Texas A&M University

October 21 · The K-Theoretic Atiyah–Singer Index Theorem · SCHM 314

Index theorems and their connection between global analysis and topology, presented in the language of K-theory.

Speaker · General Ozochiawaeze · Purdue University

October 28 · The C*-Algebra Version of the Topological 2-Torus (Part 1) · SCHM 314

Irrational rotation algebras, traces, AF-embeddings, and K-theoretic invariants.

Speaker · Jose Manuel Barrientos Lopez · Purdue University

November 4 · The C*-Algebra Version of the Topological 2-Torus (Part 2) · SCHM 314

A continuation on irrational rotation algebras, their unique trace, AF-embeddings, and classification by K-theory.

Speaker · Jose Manuel Barrientos Lopez · Purdue University

November 11 · Rigidity for von Neumann Algebras Arising from Groups · SCHM 314

Rigidity results for group von Neumann algebras and the deformation/rigidity ideas and group properties behind them.

Speaker · Patrick Debonis · Purdue University

06 — Photography

Places and light.

The full set from the original gallery

Angels Landing in Zion National Park
Angels Landing, Zion National ParkUtah
Silhouettes beneath an aurora near Purdue University
Aurora near Purdue UniversityIndiana
Canal and rail bridge in Copenhagen
CopenhagenDenmark
Boats along a Copenhagen canal
Copenhagen boat tourDenmark
Rooftops and harbor in Copenhagen
ChristianshavnCopenhagen
Midsummer twilight over a house with a yellow-lit window in Göteborg
Midsummer in GöteborgSweden
Autumn leaves in a park
The autumnal seasonIndiana
Öresundståg train in Scandinavia
ÖresundstågScandinavia
Sunlight through a wooded landscape
Sunlight in the woods
A person beside a lake
By the water
Covered walkway on a university campus
Stanford UniversityCalifornia
Pacific coastline along the 17 Mile Drive
17 Mile DrivePacific Grove, California
Rocky Pacific shoreline
Rocky Pacific shore
Waves breaking against the California coast
Waves along Highway 1
Lighthouse beside the Pacific Ocean
Pigeon PointHighway 1, California
Ocean waves along a rocky shore
Ocean edge
Snow-covered forest in Yosemite National Park
Yosemite National ParkCalifornia
Winter view of Yosemite Valley
Yosemite Valley in winter
Yosemite granite cliffs in winter
Granite and snow
Stone towers at Castello di Amorosa
Castello di AmorosaNapa Valley
Sunset above the Pacific Ocean
Point Reyes, CACalifornia
Sunset over a distant coastline
Last light on the coast
Half Dome in Yosemite National Park
Half Dome
Campus buildings framed by a stone archway
Princeton University · 2025New Jersey
A visitor on a university campus path
Institute for Advanced Study · 2025New Jersey
Autumn tree on a campus lawn
Autumn on campus
Rocky canyon landscape
Canyon country
Trees reflected in a quiet pond
Still water
Cliffside landscape in warm afternoon light
Golden cliffs
Golden mineral-colored landscape
Mineral colors
Mountain lake and pine forest
Pines by the lake
Wildflowers among green grass
Wildflowers
Open alpine meadow
High meadow
Blue lake bordered by forested hills
Blue lake
Mountain valley and evergreen forest
Mountain valley
Open meadow beneath distant mountains
Somewhere in WisconsinWisconsin
Lake and meadow beneath a blue sky
Lake and meadow
Horses crossing an open plain
Across the open plain
Elk in a broad mountain landscape
Elk in the valley
Clear lake beneath a mountain range
Clear mountain water
Alejandro Cano at a forest overlook
Forest overlook
Canal-side rooftops in Copenhagen
Copenhagen rooftops

Captions and the Öresundståg link follow the original gallery. All 42 extracted photos are included.

Get in touch

Let’s talk mathematics.

Purdue University
West Lafayette, Indiana

cano13@purdue.edu ↗