| Office | 3957 Math Sciences Building |
| Office Hours | Thursdays 1-2 |
| sdqunell at math .ucla .edu |
I am occasionally co-organizing various representation theory participating seminars. The topic this spring is monstrous moonshine. A syllabus and links to the recordings will be made available soon. The topic this summer was quiver varieties via Kirillov. A syllabus and links to the recordings are in the following Overleaf.
I am currently a co-organizer of the weekly GSO seminar. Every week, graduate students present a mathematical topic of their choice to their peers. We have had a wide variety of topics in the past, from ergodic theory to infinite hat problems to blindfolded integration. There is also free pizza. Send me an email if you are interested in giving a talk!
Below are talks that I have given while at UCLA. More detailed notes are available upon request.
| Title & Links | Description |
| Trace Invariants of Categorified Quantum Groups; Video | In higher representation theory, one seeks to produce categorical analogues of important algebras and modules. To obtain ordinary representation theory, one usually applies the Grothendieck group functor. This makes ordinary representation theory a "shadow" of the richer categorical picture. Inspired by tools in geometric topology, Lauda et al. introduced the new trace method of decategorifying. In this talk, I introduced the main objects and tools of higher representation theory. I also explored the relationship between the K group and trace group methods in a few specific cases of interest. Given as part of the K-theory seminar in Winter 2024. |
| Chow Rings of Classifying Spaces | The earliest counterexamples to the integral Hodge conjecture involved spaces that can be constructed as orbit spaces of a finite group acting freely on some variety. This suggests that the Chow rings of such varieties are quite interesting. These Chow rings can be computed from an appropriately defined Chow ring of the corresponding classifying space. In this talk, I defined the Chow ring of a classifying space, as well as do a few basic computations. I also introduced cycle maps and explained some of the more advanced strategies for computing these Chow rings generally. Given as part of the motivic cohomology seminar in Winter 2024. |
| How Would Aliens do Math? | Many of the most common mathematical ideas are those that we find intuitive or obvious. But if there were an alien species that had differently structured brains, what sorts of ideas would they find intuitive, and what sorts of problems would they want to solve? In this talk, I explored the forces that have guided the development of human mathematics and considered how they might be different for aliens. I also imagined some examples of extraterrestrial societies that would have a truly alien point of view. Given as part of the GSO seminar in Winter 2024. |
| Hecke Convolution and Correspondences; Video | To finish "geometrizing" the ordinary Langlands program, we need to describe the geometric version of the convolution product in the Hecke algebra. This comes from a diagrammatically defined convolution of perverse sheaves. This categorical perspective also provides a new "canonical" basis for the Hecke algebra. In this talk, I "sheafized" the Hecke algebra and showed how to recover it via the Grothendieck group of an appropriate category. I also gave a few applications. Given as part of the geometric Langlands seminar in Fall `23. |
| The Siegel Modular Variety | The most basic example of a Shimura variety comes from the Siegel modular variety. This variety is defined with respect to symplectic vector spaces, which are real vector spaces with extra complex structure. In this talk, I defined the Siegel modular variety and its associated Shimura data. I will also provide several ways to understand this construction through both symplectic geometry and algebraic group theory. Given as part of the Shimura varieties seminar in Fall 2023. |
| Triangulated Categories and Injective Resolutions | Triangulated categories are best understood by their prototypical example; "chain complexes up to homotopy". These categories give a natural setting for many homological constructions such as derived functors. The categories of spectra that we see later in the seminar are best understood through this lens. In this talk, I will introduce triangulated categories, give some examples, and show how they can be used to study derived functors. Given as part of the Adams spectral sequence seminar in Fall 23. |
| Why Scissors Can't Beat Rock | Not all of Hilbert's problems are as famous as the Riemann Hypothesis. Hilbert's third problem was solved less than a year after its announcement, and so it is not as well known today. Even so, Dehn invariants are still being explored by modern topologists. The full proof is unique due to these novel invariants and the relative simplicity. In this talk, I covered the motivation and proof of Hilbert's third problem, as well as some related results. Given as part of the Fall 2023 GSO seminar. |
| Hall Algebras; Video | A classic question in representation theory is to ask how much info of an object can be gleaned from its categories of representations. In the quiver case, we know that the indecomposables are in bijection with the associated positive root lattice for the underlying graph. The Hall Algebra construction allows us to recover the full multiplicative structure on the positive enveloping algebra. I discussed how this isomorphism could be obtained, as well as some extensions and applications to other settings. Given as part of the summer '23 quiver varieties seminar. |
| Intro to Symplectic Geometry; Video | Many of the quiver varieties that we will study later in the seminar have natural symplectic or Poisson structures on them. In this talk, I introduced symplectic manifolds and various constructions on them. I also discussed Hamiltonian reduction and how it will lead to resolutions of singularities for our quiver varieties. Given as part of the quiver varieties seminar in Summer 2023. |
| Intro to the Cobordism Hypothesis | The cobordism hypothesis is a problem motivated by physics and driving much of the development of higher category theory. It states that extended topological quantum field theories are determined by their values at a single point. The infinity-categorical formulation is arguablty the most natural and is one of the best examples of an application of infinity categories outside of pure homotopy theory. In this talk, I gave the background and statement of the problem, as well as arguments for why infinity categories are the right tool for the job. Given in the infinity categories seminar in Spring 2023. |
| Towards Infinity Yoneda | How much information does the comma infinity category construction remember about the elements of a given infinity category? The goal of the Yoneda lemma is to prove that it essentially remembers everything we are interested in. Before we can prove our desired equivalence, we need to establish that the assignment of an element to its comma category is appropriately functorial. In this talk, we explained how this is done in terms of the Cartesian fibrations from last week. Jointly given with Emmy Van Rooy in the Spring 2023 infinity categories seminar. |
| The Spectrum of a Tambara Functor | If a Tambara functor is an "equivariant ring", then what is the correct notion of an "equivariant ideal"? What about a prime ideal? In this talk, I introduced some basic ring-theoretic definitions and then went through some computations for the spectrum of the Burnside Mackey functor. Given as part of the equivariant algebra participating seminar in Spring 2023. |
| Preprojective Representations and the AR quiver | Having finished the classification of finite type quivers, we ask what can be said about quivers of infinite type. To address this question, we need tools for constructing large classes of indecomposables. The Coxeter functor from before is exactly the tool we shall use. In this talk, we review properties of the Coxeter functor and what representations can be obtained from it. We also introduce the preprojective AR quiver as a means of visualizing the action of the reflection functors on the indecomposable representations. Given as part of the quiver representations seminar in Spring 2023. |
| Green and Tambara Functors | Many of the standard Mackey functors conveniently have a ring structure assigned to each subgroup, like the cup product in cohomology or the tensor product in the representation ring. Many also have a very natural "norm" map which functions like a multiplicative version of the transfer. How then should we define a ring-like object amongst the Mackey functors? In this talk, I introduced the two most standard approaches, the Green and Tambara functors. I also gave examples and tried to explain the motivation behind each. Given as part of the equivariant algebra participating seminar in Spring 2023. |
| Designing Utilitarian Dating Apps | What if a dating app was designed in accordance with mathematical principles? What would be the optimal way of pairing up users, and what pragmatic considerations would a designer have to make? In this presentation, I will try very hard to convince you that my undergrad research project is not just some niche graph theory question. Given as part of the GSO seminar in Spring 2023. |
| Infinity Cosmoi and Enrichment | Quasi-categories let us realize many key homotopical properties via combinatorial arguments, but this style of proof will become intractable with more sophisticated constructions. Moreover, there are many other competing notions of infinity categories that deserve consideration. The theory of infinity cosmoi allows us to compare different models for infinity categories. I will introduce infinity cosmoi, as well as the homotopy technology required to work with them. Given as part of the infinity categories seminar in Spring 2023. |
| Introduction to Quivers and Representations | Quiver representations allow us to learn about quivers via the structure of appropriate module categories and vice versa. Even the basic examples let us recover interesting results from linear algebra. I covered the basic definitions as well as a few key properties about kQ-mod that make the whole theory tractable. I also give the general philosophy and direction of the seminar. Given as part of the quiver representations seminar in Spring 2023. |
| Homological Algebra for Quivers | Many of the ideas from earlier in this reading group, like Gorenstein rings, and tilting objects, can be demonstrated nicely for quiver algebras (or quotients thereof). In this talk, I explain some of the basic tricks for identifying indecomposable modules over these algebras, as well as go through several examples. Most importantly, I explain the cover art of the seminar's text, Krause's Homological Theory of Representations. |
| Gorenstein Rings | Much of modular representation theory works out so nicely because kG-mod is Frobenius exact; the projective and injective objects coincide. One proof of this fact uses that kG itself is an injective module. Gorenstein rings generalize this notion by allowing the base ring to have finite self-injective dimension, and the subcategory of reasonable "Gorenstein projective" modules will also be Frobenius exact. In this talk, I introduce these rings and describe their singularity categories. I also introduce cotorsion pairs and orthogonal decomposition of the module category. Given as part of a reading group/seminar for Krause's Homological Theory of Representations in Winter 2023. |
| Brown Representability for Triangulated Categories | Classically, the Brown representability theorem shows that well-behaved functors on pointed categories of spaces are representable. This result can be generalized to functors on triangulated categories admitting all coproducts and that are generated by well-behaved subsets of objects. In this talk, I introduced homotopy colimits in trinagulated categories and proved the theorem in the case that our ambient category is compactly generated. I also discussed other viable generating conditions and some immediate corollaries. At the end, I showed how the triangulated version subsumess the classical one when working in a homotopy category of pointed CW spectra. Given as part of a reading group/seminar for Krause's Homological Theory of Representations in Winter 2023 |
| Applied Category Theory and Un-abstracting the Nonsense | Category theory has a reputation for being forbiddingly abstract and impractical for questions ourside of algebra. Within the past 5-10 years, however, mathematicians have found several clever functors between abstract "syntax" categories and computationally feasible "semantics" categories, opening up a connection between categorical tools and applied ideas. In this talk, I reviewed the most well studied problem in applied category theory, the DisCoCat model of natural language processing, as well as made the case for why "Applied Category Theory" is not an oxymoron. Given in the GSO seminar Winter 2023. |
| 5 Surprising Proofs of the Fundamental Theorem of Algebra (You Won't Believe Number 4!) | Many different areas of math have their own specialized proofs of this classic theorem, and I review some of the most bizarre. Given in the GSO seminar Fall 2022. |
| Formal Group Laws | Formal group laws are used to study the arithmetic of elliptic curves. This talk is a review of the basic definitions and tools . I roughly follow Silverman chapter 4. Given in the number theory participating seminar on elliptic curves and the Mordell-Weil theorem in Winter 2022. |
| Functional Equation of the Riemann Zeta Function | Many of the interesting properties of the Riemann Zeta function can be deduced from the functional equation that it satisfies. Similar types of L-functions have this kind of functional equation. The goal of the talk is to introduce the language and analysis of L functions and functional equations before moving on to more general Artin L functions. Roughly follows Riemann's original paper "On the Number of Prime Numbers less than a Given Quantity". Given as part of the number theory participating seminar on class field theory in Fall 2021. |
| The Four Squares Problem | A classical problem in number theory is to compute r(n,k), the number of ordered k-tuples of integers for which the sum of squares of coordinates is equal to n. A famous result of Jacobi is that r(n,4) can be computed exactly and is always positive when n is. Jacobi's technique was to study the symmetries of certain modular forms. This presentation covers the needed tools within modular forms to prove this result, as well as related estimates for more general r(n,k). Given as part of the number theory participating seminar on modular forms in Fall 2021. |