Charlotte Aten math research

Universal Algebra and Lattice Theory - Lecture 2: Examples of algebras

Charlotte Aten math research — Sun, 16 Aug 2026 22:20:35 GMT

This is the second in a series of talks about universal algebra and lattice theory. I give examples of algebras, including magmas, semigroups, monoids, groups, rings, modules, semilattices, lattices, quasigroups, Latin squares, and n-ary magmas. The webpage for this series of talks can be found at https://aten.cool/2020ua.html. (Typo near the end: I acutally gave an example of an (n-1)-ary magma, so I was off by one.)

Higher-dimensional book-spaces (Charlotte Aten at PALS 2024)

Charlotte Aten math research — Sun, 16 Aug 2026 22:13:49 GMT

This is my talk at the University of Colorado Boulder's PALS seminar in the fall of 2024. I discuss the construction of a family of topological lattices, called book spaces, whose underlying spaces are simplicial complexes of a particular form.

PALS: https://math.colorado.edu/algebralogic/index.html
Slides: https://aten.cool/documents/slides/aten_PALS_2024.pdf
Preprint: https://arxiv.org/abs/2409.12923

Discrete neural nets and polymorphic learning (Charlotte Aten at PALS 2023)

Charlotte Aten math research — Sun, 16 Aug 2026 22:12:33 GMT

This is my talk at the University of Colorado Boulder's PALS seminar in the fall of 2023. I discuss my work with neural nets whose activation functions are polymorphisms of certain finite graphs.

PALS: https://math.colorado.edu/algebralogic/index.html
Slides: https://aten.cool/documents/slides/aten_PALS_2023.pdf
Preprint: https://arxiv.org/abs/2308.00677

Invariants of structures (Charlotte Aten at University of Denver Algebra and Logic Seminar 2022)

Charlotte Aten math research — Sun, 16 Aug 2026 22:11:24 GMT

This is a talk that I gave for the University of Denver Algebra and Logic Seminar in the fall of 2022. I talked about one part of my thesis, which concerns a categorification of the notion of mathematical structure appearing in Bourbaki's textbook on set theory. You can find a link to my slides below if you'd like to follow along with the video. I'll also include a link to my thesis, which contains more examples.

Title: Invariants of structures
Speaker: Charlotte Aten, University of Denver
Abstract: I will be discussing one part of my PhD thesis, in which I provide a categorification of the notion of a mathematical structure originally given by Bourbaki in their set theory textbook. The main result is that any isomorphism-invariant property of a finite structure can be computed by counting the number of isomorphic copies of small substructures it contains. A special case of this theorem is the classical result of Hilbert about elementary symmetric polynomials generating the algebra of all symmetric polynomials.

Slides: https://aten.cool/documents/slides/aten_du_algebra_logic.pdf
Thesis: https://aten.cool/documents/papers/thesis.pdf
Preprint: https://arxiv.org/abs/2402.18063

A multi-linear geometric estimate

Charlotte Aten math research — Sun, 16 Aug 2026 22:10:08 GMT

This is my talk at Virginia Tech's analysis and mathematical physics seminar for the fall of 2021, which is about joint work with Alex Iosevich in which we generalize a geometric estimate on level sets of non-degenerate bilinear forms originally made in a 2008 paper of Hart and Iosevich to the case of multi-linear forms. I discuss applications to particular sets of points in the plane over a finite field which I whimsically named omphaloi after the mythical Greek center of the world as well as the relationship between this work and the sum-product conjecture.

Slides: https://aten.cool/documents/slides/aten_geometric_estimate.pdf
Preprint: https://arxiv.org/abs/2112.00810

The Hart and Iosevich paper: https://arxiv.org/abs/0705.4256
Alex Iosevich: https://alexiosevich.com/

Orientable smooth manifolds are essentially quasigroups (Charlotte Aten at BUGCAT 2021)

Charlotte Aten math research — Sun, 16 Aug 2026 22:08:59 GMT

This is my talk at Binghamton University's annual BUGCAT conference for 2021. I discuss the contents of a preprint Semin Yoo and I recently posted on the arXiv, which concerns the manufacture of (pseudo)manifolds from alternating n-quasigroups. I give an overview of our basic constructions and argue that every orientable triangulable manifold can be produced in this manner.

Corrections:
8:15 I meant 0-skeleton, not 1-skeleton.
25:30 The last bullet of the slide is missing an important part. The correct statement is on the slides linked below.

BUGCAT: https://seminars.math.binghamton.edu/BUGCAT/index.html
Slides: https://aten.cool/documents/slides/BUGCAT_2021_quasigroup_manifolds.pdf
Semin Yoo: https://sites.google.com/site/seminmathematics/
Preprint: https://arxiv.org/abs/2110.05660

My Hawai'ian Earring (SUMS Math Talk)

Charlotte Aten math research — Sun, 16 Aug 2026 22:08:23 GMT

This is a talk I gave for the Society for Undergraduate Math Students at the University of Rochester in the spring of 2021. In 2018 I was invited to a conference on universal algebra and lattice theory in Hawai'i. I designed an earring for the conference which depicts an object in lattice theory called the free distributive lattice on three generators. In this talk I show some of the linear algebra and coding involved in writing a Python script which uses the Sage computer algebra system to create an stl file suitable for printing the earring on a 3d printer. Please note: This talk is not about the topological space called a Hawai'ian earring.

Slides: https://aten.cool/documents/slides/SUMS_Hawaiian_earring.pdf
Code: https://git.aten.cool/caten/FreeDistributiveLattice
Sage: https://www.sagemath.org/
RepRap Prusa i3: https://reprap.com/wiki/Prusa_i3
Blender: https://www.blender.org/

Multiplayer rock-paper-scissors (Charlotte Aten at BUGCAT 2020)

Charlotte Aten math research — Sun, 16 Aug 2026 22:07:02 GMT

This is my talk at Binghamton University's annual BUGCAT conference for 2020. I discuss some of my recently-published work on versions of the game rock-paper-scissors which allow for many simultaneous players and many more items to choose among. I describe how these games can be viewed as algebraic structures and show that they generate the varieties generated by hypertournament algebras.

Paper: https://aten.cool/documents/papers/RPS.pdf
Slides: https://aten.cool/documents/slides/aten_bugcat_2020.pdf
BUGCAT: https://seminars.math.binghamton.edu/BUGCAT/index.html

Multiplayer rock-paper-scissors (Charlotte Aten at NY Combinatorics Seminar 2021)

Charlotte Aten math research — Sun, 16 Aug 2026 22:05:46 GMT

This is my talk on multiplayer versions of rock-paper-scissors for the spring 2021 New York Combinatorics Seminar.

Paper: https://aten.cool/documents/papers/RPS.pdf
Slides: https://aten.cool/documents/slides/NY_combinatorics_seminar_2021_spring_aten.pdf
New York Combinatorics Seminar: https://www.nycombinatorics.org/seminar

Multiplayer rock-paper-scissors (Charlotte Aten at PALS 2021)

Charlotte Aten math research — Sun, 16 Aug 2026 22:04:35 GMT

This is my talk on multiplayer versions of rock-paper-scissors for the spring 2021 PALS seminar at CU Boulder.

Paper: https://aten.cool/documents/papers/RPS.pdf
Slides: https://aten.cool/documents/slides/PALS_RPS.pdf
Panglobal Algebra and Logic Seminar: https://math.colorado.edu/algebralogic/about.html

Universal Algebra and Lattice Theory - Lecture 5: Posets and lattices

Charlotte Aten math research — Sun, 16 Aug 2026 22:03:00 GMT

This is the fifth in a series of talks about universal algebra and lattice theory. I discuss the basics of posets and lattices, as well as some connections between topology, probability, and lattice theory. I also give a short history of the early stages of lattice theory. The 1935 paper of Birkhoff I mention can be found at https://math.hawaii.edu/~ralph/Classes/619/birkhoff1935.pdf. The webpage for this series of talks can be found at https://aten.cool/2020ua.html.

Universal Algebra and Lattice Theory - Lecture 6: The distributive and modular laws

Charlotte Aten math research — Sun, 16 Aug 2026 22:01:46 GMT

This is the sixth in a series of talks about universal algebra and lattice theory. I introduce the distributive and modular laws and describe their relationship to each other and to duality in lattices. I proceed to present Dedekind and Birkhoff's results characterizing these two laws and end with a little note about graph theory. The webpage for this series of talks can be found at https://aten.cool/2020ua.html.

Universal Algebra and Lattice Theory - Lecture 3: Homomorphisms, subalgebras, and products

Charlotte Aten math research — Sun, 16 Aug 2026 22:00:34 GMT

This is the third in a series of talks about universal algebra and lattice theory. I discuss homomorphisms, subalgebras, and products in more detail, as well as the corresponding class operators H, S, and P. I define a variety of algebras and give some basic results on subuniverses generated by a set of elements. The webpage for this series of talks can be found at https://aten.cool/2020ua.html.

Universal Algebra and Lattice Theory - Lecture 4: Congruences and quotients

Charlotte Aten math research — Sun, 16 Aug 2026 21:59:18 GMT

This is the fourth in a series of talks about universal algebra and lattice theory. I discuss relations, kernels, and congruences, as well as their relationship with the normal subgroups of group theory. I also define quotient algebras and prove the Fundamental Homomorphism Theorem, which generalizes the usual isomorphism theorem for groups and rings to all algebras. I give some basic results on congruences generated by a set of pairs. The webpage for this series of talks can be found at https://aten.cool/2020ua.html.

Universal Algebra and Lattice Theory - Lecture 1: Introduction and background

Charlotte Aten math research — Sun, 16 Aug 2026 20:54:13 GMT

This is the first in a series of talks about universal algebra and lattice theory. I tell some of the story of how I came to study universal algebra, define the basic objects in this area of mathematics, and give some context for their study. The webpage for this series of talks can be found at https://aten.cool/2020ua.html.