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Events for January – May 2022

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Jan. 13, 2022 3pm (Zoom)	Probability Andreas Kyprianou (University of Bath) X Consider a general class of branching Markov processes with non-local offspring distribution, or a non-local general superprocess. Let k be a non-negative integer. We will show that, when the mean semigroup possesses a Perron-Frobenius type asymptotic decomposition, remarkably precise asymptotic growth results for its k-th moments and the k-th moments of its occupation measure can be established as t -> oo. The method relies on a combinatorial decomposition of the moments, correctly identifying and treating the leading order terms and using ergodic properties that follow as a result of the Perron-Frobenius assumption. This is joint work with Emma Horton and Isaac Gonzalez. Asymptotic Moments of Spatial branching processes
CANCELED Jan. 18, 2022	Topology Araminta Gwynne (MIT) TBA
Jan. 20, 2022 1pm (Zoom)	Probability Stanislav Volkov X The discrete version of the famous Bak-Sneppen model (https://en.wikipedia.org/wiki/Bak-Sneppen_model) is a Markov chain on the space of {0,1} sequences of length n with periodic boundary conditions, which runs as follows. Fix some 0 Barbay and Kenyon (2001) claimed that the fraction of zeros in the stationary distribution becomes negligible when n goes to infinity whenever p>0.54. This result is indeed correct, however, its proof is not. I shall present the rigorous proof of the Barbay and Kenyon's result, as well as some better bounds for the critical p. Rigorous bounds for the discrete Bak-Sneppen model
Jan. 25, 2022 1pm (Zoom)	Charlotte Aten (University of Rochester) X In my recent work with Semin Yoo we produced a generalization of a construction of Herman and Pakianathan which assigns to each finite noncommutative group a closed surface in a functorial manner. We give a pair of functors whose domain is a subcategory of a variety of n-ary quasigroups. The first of these functors assigns to each such quasigroup a smooth, flat Riemannian manifold while the second assigns to each quasigroup a topological manifold which is a subspace of the metric completion of the aforementioned Riemannian manifold. I will give examples of these constructions, draw some pictures, and argue that all homeomorphism classes of smooth orientable manifolds arise from this construction. I will then discuss a connection with the Evans Conjecture on partial Latin squares, give its implication for orientable surfaces, and state a related problem applicable to our construction for compact n-manifolds. Orientable smooth manifolds are essentially quasigroups
Jan. 26, 2022 5pm (MATH 350)	Grad Student Seminar Ian Miller (CU Boulder) X Sobolev spaces -- I think they are pretty neat -- I'll tell you 'bout 'em. In developing a theory of partial differential equations, we must choose a setting for solutions. For example, we could look for smooth, or analytic solutions, or solutions in $C^{k}$ -- the space of functions which are $k$ times continuously differentiable. Introducing a notion of weak solution allows us to work in spaces with more structure. Typical settings for these weak solutions are the Sobolev spaces $H^{k}$ . We will discuss the definitions of weak solutions and Sobolev spaces, as well as applications of the Riesz representation theorem and Banach contraction theorem to PDE through Sobolev spaces. Sobolov Spaces and the Question "What is a Solution to a Partial Differential Equation?"
Jan. 27, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X The Ulam lectures this semester will be around Chevalley groups and generalizations. Chevalley groups were important in leading to the theory of algebraic groups and in the classification of finite groups. They are of interest to representation theorists, algebraic geometers, number theorists and other mathematicians. Rather than give all of the lectures myself, I would like to enlist the help of the graduate student participants. In addition to the actual lectures (on Thursdays), I will also host a weekly (schedule TBD) informal working/problem session focused on finer details and working with explicit examples. The intended audience/participants are second year graduate students, but all who are interested are welcome. We will hold an organizational meeting on Thursday, Jan 27, with the intention of determining background and interest and to schedule a time for the working sessions. If you are unable to attend in person, please email michael.woodbury@colorado.edu so that I can share a Zoom link with you. To all faculty: If you have a specific application or generalization of Chevalley groups (such as a paper or papers) that you think your current or future grad students would benefit to learn, please share your ideas with me. I have many ideas myself, but I imagine the students that already do or may end up working with you would appreciate learning something particularly relevant to your field. Ulam Lectures
Jan. 27, 2022 11:15am (Math …	Noncomm Geometry Vladimir Baranovsky (UCI) X we plan to start with a brief overview of some results on characteristic classes of modules over deformation quantization (in the case of a symplectic algebraic variety). Then we apply these results and the Categorical Riemann-Roch formula to representation theory; bounding the number of simple finite dimensional modules over algebras which arise from studying resolutions of singular symplectic varieties. Deformation quantization and traces of modules
Feb. 01, 2022 1pm (Zoom)	Ross Willard (University of Waterloo) X Many years ago, Kiss proved that the commutator relation [alpha,beta]=0 can be characterized in congruence modular varieties by a simple condition involving a certain kind of 4-ary term, which is now called a Kiss term. Seven years ago, Kearnes, Szendrei and I claimed to extend this characterization to varieties having a difference term, and we used this at a key step in proving our finite basis theorem for finite algebras in varieties having a difference term and having a finite residual bound. It was recently brought to our attention that the published proof of our extension of Kiss’s result has a significant gap, bringing into question the validity of our finite basis theorem. In this talk I will sketch a new (correct) proof of this extension. This is joint work with Keith Kearnes and Agnes Szendrei. Characterizing [alpha,beta]=0 using Kiss terms
Feb. 01, 2022 3pm (MATH 350)	Topology Juan Moreno (CU Boulder) X This meeting is to kick-off a learning seminar on modular tensor categories (MTCs). The meeting will consist of a short overview of the topic after which participants will have the opportunity to sign up for talks. MTCs, and more generally fusion categories, were designed to model the physics of anyon systems which offer a potential framework for quantum computation. Anyons are particles which arise as excitations in (2+1)D topological phases of matter. Such phases (or at least some subclass of them) are believed to have a low-energy description of a (2+1)D topological quantum field theory (TQFT). The goal for this seminar is to study MTCs and their relationship with TQFTs. While their motivation is primarily in the realm of physics, these objects are interesting as stand-alone mathematical objects: (2+1)D TQFTs give rise to 3-manifold and knot/link invariants, and many important examples of MTCs come from representation theory and the theory of quantum groups. Organizational Meeting
Feb. 02, 2022 4pm (Zoom)	Diversity Diversity Committee Math Research Demystified
Feb. 03, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X The Ulam lectures this semester will be around Chevalley groups and generalizations. Chevalley groups were important in leading to the theory of algebraic groups and in the classification of finite groups. They are of interest to representation theorists, algebraic geometers, number theorists and other mathematicians. Rather than give all of the lectures myself, I would like to enlist the help of the graduate student participants. In addition to the actual lectures (on Thursdays), I will also host a weekly (schedule TBD) informal working/problem session focused on finer details and working with explicit examples. The intended audience/participants are second year graduate students, but all who are interested are welcome. I especially welcome students who may already have or want to learn some background in Lie algebras as this will be the topic for next week. This week, Thursday, Feb 3, I will give an introductory lecture. If you are unable to attend in person, please email michael.woodbury@colorado.edu so that I can share a Zoom link with you. To all faculty: If you have a specific application or generalization of Chevalley groups (such as a paper(s) or book chapter(s)) that you think your current or future grad students would benefit to learn, please share your ideas with me. I have many ideas myself, but I imagine the students that already do or may end up working with you would appreciate learning something particularly relevant to your field. Introduction to Chevalley Groups
Feb. 08, 2022 1pm (Zoom)	Dragan Masulovic (University of Novi Sad) X Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact Ramsey expansion is surprisingly short. In this talk we show that any nontrivial variety (that is, equationally defined class of algebras) enjoys various dual Ramsey properties. We develop a completely new set of strategies that rely on the fact that left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg-Moore categories for a monad. We show that finite algebras in any nontrivial variety have finite dual small Ramsey degrees, and that every finite algebra has finite dual big Ramsey degree in the free algebra on countably many free generators. As usual, these come as consequences of ordered versions of the statements. Dual Ramsey properties for classes of algebras
Feb. 08, 2022 3pm (MATH 350)	Topology Daniel Lyness (CU Boulder) X This is the first talk in the semester learning seminar on modular tensor categories and applications to topological phases of matter. In this talk, we will introduce the concept of topological field theories and give a few fundamental examples. An introduction to TQFTs
Feb. 08, 2022 5pm (MATH 350)	Grad Student Seminar Howie Jordan (CU Boulder) X Information. What is it? Where does it come from? How can I give you information about getting information? By talking at you while pizza powers your focus. (A proof of this will be constructed over the course of the talk). Information theory starts with the idea that we gain information by observing the outcome of something we were unsure about. If you learn which horse wins the race, you've gained information about the race. If you learn a politician lied, you didn't really gain any information about the politician. In this talk, I'll introduce the basic concepts and tools of classical information theory. In particular, we'll consider the notion of entropy and its origins in analyzing communication. Okay Ladies, Now Let's Get Information
Feb. 09, 2022 3:15pm (Math350)	Math Phys Howie Jordan (CU Boulder) X Information can be captured as a measure of how far a probability space is away from being a "sure thing", or equivalently, how much data is required to uniquely specify an event. A fair coin flip contains more information than an unbalanced coin. As the coin becomes more and more biased, a flip tends towards having no informational content. Classically, this leads to a rich theory including an analysis of the efficiency of encodings, compressions, and transmissions of information. It is exactly "A Mathematical Theory of Communication". Information becomes quantum when it starts being contextual and linear. Essentially, we cannot assign all events in the quantum event lattice consistent probabilities and information cannot be duplicated or destroyed. The theme for this semester's mathematical physics seminars is Quantum Information, and this first talk serves as an introduction to the foundations. We will compare the basic ingredients of quantum information theory to the classical counterpart, in particular their event structures and their notions of entropy. Over the course of the semester, we will cover such topics as decoherence in communication channels, distinguishing quantum states, information geometry, and the topological model of quantum computing. We aim to enhance our understanding by utilizing the duality between the boolean logic of subsets and the partition logic of quotients used respectively in probability and information theory, and to express these through a unifying sheaf theoretic formalism. Information, Classical and Quantum: An Introduction
Feb. 09, 2022 4pm (Math 350)	MathClub organized by Agnes Beaudry and Jonathan Quartin (Diversity Committee) X What's undergraduate math research anyways? Who is it for? It might not be what you think it's like and it might be for you, even if you haven't realized it yet! So come to the Math Club on Wednesday Feb 2 from 4 - 5 pm to learn more about undergraduate math research and the opportunities for math research experiences available to CU undergrads. And if you're wondering, does math research happen online? The answer is absolutely! Everyone is welcome! Demystifying Math Research
Feb. 10, 2022 9:05am (Math 3…	Ulam Breeann Wilson (CU Boulder) X The Ulam lectures this semester will be around Chevalley groups and generalizations. Chevalley groups were important in leading to the theory of algebraic groups and in the classification of finite groups. They are of interest to representation theorists, algebraic geometers, number theorists and many other mathematicians. As promised, I (Mike Woodbury) will not be giving all of the lectures. Our first "outside" speaker is our own Breeann Wilson who will teach us this week about the structure and classification of semisimple and simple Lie algebras over the complex field. In conjunction with and in preparation for Breeann's talk, we will also hold a problem/example session Tuesday at 4:05pm in Math 350. The goal of the Tuesday session is to see some concrete examples of Lie algebras and their root systems. Reminder to all faculty: If you have a specific application or generalization of Chevalley groups (such as a paper or papers) that you think your current or future grad students would benefit to learn, please share your ideas with me (michael.woodbury@colorado.edu). I imagine the students that already do or may end up working with you would appreciate learning something particularly relevant to your field. Structure and classification of (semi)simple Lie algebras
Feb. 15, 2022 11am (TBD)	Diversity Stephanie Roberts (CISC) Undocually workshop
Feb. 15, 2022 1pm (Zoom)	Bill De Witt (University of St Andrews, UK) X The number of subdirect powers of an finite algebraic structure is a question that has appeared at various points in recent history. The situation is known in full for groups, and to some extent in semigroups. We answer the question for unary algebras, and look at how the situation is more complicated for infinite algebras. We then discuss what these results can tell us about the general question, and how it ties into other topics such as boolean separation. The number of countable subdirect powers of finite unary algebras
Feb. 15, 2022 3pm (MATH 350)	Topology Howie Jordan (CU Boulder) X Fusion categories are complex linear categories with tensor products, duals, and satisfying simplicity conditions. These categories have a finite class of simple objects out of which all other objects are built, and these along with their duals form the representations of the pure states of particles and anti-particles in TQFTs. By adding a braiding onto the monoidal structure and twist maps compatible with the braiding we arrive at a ribbon category. A final nondegeneracy condition gives us the notion of a Modular Tensor Category (MTC). In particular, MTCs have a nondegenerate inner product structure allowing for probabilities to be assigned to states inline with traditional quantum theory. In this talk, we unpack the definitions of fusion and ribbon categories and build up to the notion of Modular Tensor Category as a nondegenerate ribbon fusion category. We will consider these axiomatically as well as skeletal versions in terms of 6j symbols and diagrammatic calculi. Fusion categories and MTCs
Feb. 16, 2022 3:20pm (MATH220)	Math Phys David Jonas (CU Boulder Chemistry Department) X Zoom Meeting ID: 940 0255 330 Password: 790356 Molecular quantum mechanics is usually understood in the context of a fast electron/slow nucleus approximation in which the fast electrons instantaneously adjust to slow motions of the nuclei. Recent experiments have shown that elementary steps in photosynthesis rely on non-adiabatic quantum mechanics. The non-adiabatic eigenfunctions that underlie this extraordinarily efficient process have remarkable nodal structures. These nodal structures were revealed by exactly factorizing the eigenfunctions into a vibrational marginal probability amplitude and an electronic conditional probability amplitude. Instead of the one-dimensional nodal curves familiar from adiabatic quantum mechanics, the two-dimensional non-adiabatic eigenfunctions from a model of photosynthetic energy transfer have zero-dimensional nodal points. Nodal points also occur for two-dimensional eigenfunctions from models that exhibit Berry’s geometric phase. We have made progress towards understanding the local geometry of these nodes by regarding the electronic probability amplitude as a vector field over an oriented surface of vibrational coordinates. For closed curves on the vibrational coordinate surface, proofs of the Poincaré-Hopf theorem establish that the electronic index around the curve equals the sum of the electronic indices of the nodes enclosed by the curve. The electronic indices of nodes are also related to the local radial exponent of the vibrational marginal probability amplitude and to Berry’s geometric phase. These results suggest that differential geometry and topology might help us to understand how photosynthesis generated the fossil fuels we burn and the oxygen we breathe. The Geometry of Nodes in Nonadiabatic Quantum Mechanics
Feb. 16, 2022 4pm (Math 350)	MathClub Agnès Beaudry (CU-Boulder) X This talk will be a friendly introduction to "categories", a concept that's everywhere in mathematics but that's often not introduced until the graduate curricula. The fact that we don't talk about it before then is just cultural, as the definitions are very basic and don't require that much math to formulate. I will give a lot of different examples, pooling from all kinds of classes you may or may not have taken. But worry not, even if you've taken very few math classes, there should be at least some examples that make sense. My hope is that, after this talk, if you hear the word "category" whispered by one of your peers, or casually dropped as a side remark by a professor, you'll feel like you know what that means. A Category? What's that?
Feb. 17, 2022 9:05am (Math 3…	Ulam Breeann Wilson (CU Boulder) X In preparation to begin studying the notes of Steinberg on Chevalley groups, Breeann will discuss the representation theory of simple Lie algebras. By the end of the talk, we hope to understand how every irreducible representation of a given simple Lie algebra is determined by a highest weight vector and that these are in correspondence to the (non-negative) integral span of certain fundamental weights. Everyone is invited to a problem session on Tuesday, February 15, in which we will continue looking at some of the specific examples of Lie algebras seen up to this point. We will also classify and construct the irreducible representations of sl(2,C). Representations of simple Lie algebras
Feb. 17, 2022 2pm (MATH 350)	Functional Analysis Robin Deeley (CU Boulder) X This talk will be the first in a series discussing KK-theory. We will follow Kasparov's paper "Equivariant KK-theory and the Novikov conjecture". In addition to organizing some of the future talks, I will discuss analytic K-homology as a warm up to Kasparov's bivariant theory. Analytic K-homology
Feb. 22, 2022 1pm (Zoom)	Jordan DuBeau (CU Boulder) X For an infinite algebra J in a countable algebraic language, we say J is Jónsson if it has no proper subalgebra of the same cardinality as J. This talk explores Jónsson algebras in a particular variety: the variety of Jónsson-Tarski algebras. When a Jónsson algebra of size $ℵ1$ was constructed in this variety, it showed that minimal varieties can contain uncountable Jónsson algebras. We will describe that construction and two further results, demonstrating exactly which cardinalities are possible for Jónsson Jónsson-Tarski algebras, and how many pairwise nonisomorphic Jónsson Jónsson-Tarski algebras exist. We discuss implications for other varieties and Jónsson algebras in general. Jónsson Jónsson-Tarski algebras
Feb. 22, 2022 3pm (MATH 350)	Topology Daniel Spiegel (CU Boulder) X The goal for this talk is to cover invariants of MTCs and provide detailed examples such as the quantum double D(G) of a finite group G. Invariants and examples of MTCs
Feb. 23, 2022 5pm (MATH 350)	Grad Student Seminar Andrew Campbell (CU Boulder) X In this talk we'll consider what can be said about the eigenvalues of A+B, given the eigenvalues of A and the eigenvalues of B. We'll begin by exploring the challenges of this problem and why the answer should look very different depending on whether or not A and B are Hermitian. We will then dive into the world of representation theory and combinatorics for a remarkable complete description of the set of all possible eigenvalues for Hermitian A and B. Finally, now that we know the set of all possible eigenvalues we'll see how free probability can give a sense of what they typically look like even for non-Hermitian A and B. What are the eigenvalues of A+B?
Feb. 24, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X Due to the delayed start last week, we were unable to hold the scheduled Ulam lecture. Breeann will thus speak during the problem session time slot (Tuesday 4:05-4:55pm) on representations of sl(2,C) and sl(3,C). On Thursday, Mike will discuss the representations of g_2. Note that beginning next week, we will start on the notes of Steinberg on Chevalley groups. These can be purchased during the month of February through the AMS for a discounted price of just under $20. The root decomposition and representation theory of the exceptional Lie algebra g_2
Feb. 24, 2022 2:30pm (MATH 3…	Functional Analysis Levi Lorenzo (CU Boulder) X As we continue our discussion of analytic K-homology, we encounter Fredholm modules in a motivating example. As seen, operator algebras allow us to recover topological spaces in the noncommutative setting, and K-homology is a valuable invariant for these noncommutative spaces. This week, we look at spaces with additional structure and attempt to recover that structure as algebraic data. Namely, if we have a manifold with a (smooth, Riemannian, Spin, Spin^c) structure, we port that information into the noncommutative world using Spectral Triples. We see how Spectral Triples open the doors to the calculus of noncommutative spaces and how a Spectral Triple determines the topological data of a space in the form of a Fredholm module and a class in K-homology. Noncommutative Spin Geometry and K-homology
Mar. 01, 2022 1pm (Zoom)	James Rooney (McMaaster University, Canada) X In their 2020 article "Deciding some Mal'tsev conditions in finite idempotent algebras" Kazda and Valeriote conjecture that for a linear strong Mal'tsev condition the associated idempotent Mal'tsev condition satisfaction problem (MCSP) will always be polynomial-time decidable. In an earlier-published 2019 article Freese, Nation and Valeriote showed that testing for a semilattice term (a nonlinear condition) is EXPTIME-complete even for idempotent algebras. While preparing my PhD thesis I investigated the hypothesis that nonlinear Mal'tsev conditions might always be EXPTIME-complete to detect. I was able to prove that there are nonlinear Mal'tsev conditions whose related idempotent MCSPs are in the class NP. Assuming that NP is not EXPTIME this provides the first examples of nonlinear Mal'tsev conditions whose idempotent MCSPs are not EXPTIME-complete. The existence of lattice terms is one such example. In this talk we briefly revisit the 2019 result of Freese, Nation and Valeriote before sketching the details of my proof that detection of lattice terms in an idempotent algebra is an NP problem. Nonlinear idempotent Mal'tsev Condition Satisfaction Problems: why semilattices are hard and lattices are easier
Mar. 01, 2022 3pm (MATH 350)	Topology Agnes Beaudry (CU Boulder) X The talk should outline the classification of unitary MTCs of rank ? 4. Towards a classification of MTCs
Mar. 02, 2022 3:20pm (Math350)	Math Phys David Aasen (Microsoft Station Q) X Topological phases can be divided into two classes corresponding to whether the microscopic degrees of freedom supporting the phase are purely bosonic (e.g., spins or qubits) or whether they include fermions (e.g., electrons). Imposing a symmetry on a topological phase enriches the classification by restricting and possibly fracturing the phase space. Fermionic topological phases additionally include an underlying fermionic particle of the system, the physical fermion. This talk will present recent results on the algebraic structure and classification of fermionic topological phases with on-site unitary symmetry using G-crossed braided tensor categories. I will emphasize the new obstructions which appear, contrast them with their bosonic counterparts, and provide a complete characterization of all symmetric unobstructed invertible fermionic phases. The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 Characterization and Classification of Fermionic Symmetry Enriched Phases Sponsored by the Meyer Fund
Mar. 02, 2022 4pm (Math 220)	MathClub Lee Roberson (CU-Boulder) X Thinking mathematically, something so powerful a book was written about it. We will explore three questions: (1) How do we learn? (2) What is mathematics? (3) What is mathematical thinking? We will then use our answers to these questions to investigate how we learn to think mathematically. How Do We Learn to Think Mathematically?
Mar. 03, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X I will discuss what the universal enveloping algebra of a semisimple Lie algebra (over C) is, and how to find the "right" basis to get an integral structure. This integral structure is the so-called Kostant-form. This is important as it will allow us to define the Chevalley groups over arbitrary fields. There is a problem/example session as usual on Tuesday from 4:05-4:55pm in Math 350. The starting point for our session this week is a worksheet discussing the construction of g_2 from its root diagram. If you would like a copy of the worksheet, please email me at michael.woodbury@colorado.edu. The universal enveloping algebra and Kostant's Z-form
Mar. 03, 2022 11:15am (Math …	Noncomm Geometry Maxim Braverman (Northeastern University) X We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of a family of Toeplitz operators on a strongly pseudoconvex domain in . This result is similar to the Boutet de Monvel's computation of the index of a single Toeplitz operator on a strongly pseudoconvex domain. As an application, we show that the bulk-boundary correspondence in a tight-binding model of topological insulators is a special case of our results. I then discuss a project of extending the Graf-Porta model for Z_2 topological insulators via the Katsnelson-Nazaikinskii construction of the "partial spectral flow". In the end, I will explain KK-theoretical extension of our main theorem to families of Toeplitz operators parametrized by an arbitrary compact manifold, obtained by Koen van den Dungen. Spectral Flow of Toeplitz operators and bulk-edge correspondence
Mar. 03, 2022 2:30pm (MATH 3…	Functional Analysis Rachel Chaiser (CU Boulder) X I will introduce the group KK(A,B) in the non-equivariant case and where the C-algebras A and B have the trivial grading. I will be following Section 2 of Kasparov's paper, primarily focusing on pages 155-156. There will be some review of Hilbert C-modules and discussion of basic examples of KK(A,B) (e.g., where A or B is the complex numbers). Defining KK(A,B)
Mar. 08, 2022 1pm (Zoom)	Monica VanDieren (Robert Morris University) X In the 1970s Saharon Shelah initiated a program to develop classification theory for non-elementary classes, and eventually settled on the setting of abstract elementary classes. For over three decades, limited progress was made, most of which required additional set theoretic axioms. In 2001, Rami Grossberg and I introduced the model theoretic concept of tameness which opened the door for stability results in abstract elementary classes in ZFC. During the following 20 years, tameness along with limit models have been used by several mathematicians to prove categoricity theorems and to develop non-first order analogs to forking calculus and stability theory, solving a very large number of problems posed by Shelah in ZFC. Recently, Marcos Mazari-Armida found applications to Abelian group theory and ring theory. In this presentation I will highlight some of the more surprising results involving tameness and limit models from the past 20 years. Twenty years of tameness
Mar. 08, 2022 2:30pm (MATH 2…	Lie Theory Breeaan Wilson (CU Boulder) Practice Session for Comprenhensive Exam
Mar. 08, 2022 3pm (MATH 350)	Topology Josh Grochow (CU Boulder) X The goal is to make concrete some of the subtleties that have come up as questions in the previous talks about the definitions of modular tensor categories, by looking at them using some small finite groups as examples. In particular, we'll only need to look at groups of order at most 8, where we can compute everything explicitly by hand. We'll see: Why there is a GL(n)'s worth of choices of how to decompose a direct sum of n isomorphic simples. What the decomposition of a tensor product of simples looks like, and a fairly explicit example of the braiding and associativity isomorphisms In what way do MTCs generalize finite groups? There's some subtlety to the phrase "Rep(G) is an MTC", and this will take us into exactly what data about Rep(G) is enough to determine the group G, which will lead to a discussion of Tannaka duality, the representation ring (a decategorification of Rep(G)), and a result of Etingof & Gelaki characterizing what happens if you forget the symmetry/braiding isomorphisms. It'll be helpful for our calculations if you've seen character tables of finite groups before, but my goal is for the story to still make sense even if you haven't seen character tables. Phenomena in modular tensor categories through the lens of small finite groups
Mar. 09, 2022 3:20pm (Math350)	Math Phys Howie Jordan (Department of Mathematics, CU Boulder) X Information theory can be said to derive from a logic of partitions or distinctions, in a manner dual to the derivation of classical probability from the more familiar logic of subsets. Then, just as we "linearize" classical subsets and probability to linear subspaces and quantum probabilities, we may linearize partition logic to obtain a logic of direct sum decompositions. We will outline the dual relationship between subsets and partitions from the perspective of categorical logic and use it to motivate a formulation of the corresponding linearized (quantum) logics. In particular, we connect this approach to the program of axiomatizing the category of complete inner product (Hilbert) spaces, which recently found its successful conclusion. The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 Categorical Partition Logic for (Quantum) Information
Mar. 10, 2022 9:05am (Math 3…	Ulam Eli Orvis (CU Boulder) X Eli will discuss the construction/definition of the Chevalley groups over arbitrary fields. This is based on Chapter 3 of Steinberg's notes. As usual, there is a problem/example session as usual on Tuesday from 4:05-4:55pm in Math 350. The starting point for our session this week is a worksheet discussing the universal enveloping algebra U=U(g) of a semisimple Lie algebra g and its Kostant form, a lattice which acts integrally on all representations of g. If you would like a copy of the worksheet, please email me at michael.woodbury@colorado.edu. The Chevalley Groups
Mar. 10, 2022 2:30pm (MATH 3…	Functional Analysis Robin Deeley (CU Boulder) X I will discuss three aspects of KK-theory. First, I will show that KK(C,C) is isomorphic to the integer via an index map. Then I will prepare us for the definition of equivariant KK-theory by discussing group actions in the C-algebraic context. Finally, if time permits, I will outline some of the issues with the construction of the Kasparov product. KK(C,C), group actions on C-algebras, and background for the product
Mar. 10, 2022 3pm (Zoom)	Probability Janos Englander (CU Boulder) X In our previous paper, the ``coin turning walk'' has been introduced on $\Z$. It is a non-Markovian process where the steps form a (possibly) time-inhomogeneous Markov chain. In this talk we follow up the investigation by introducing analogous processes in $\Z^d,\ d\ge 2$: at time $n$ the direction of the process is ``updated'' with probability $p_{n}$ ; otherwise the next step repeats the last one. We study the fundamental properties of these walks, such as transience/recurrence and scaling limits. This is joint work with S. Volkov from Lund, Sweden. The multi-dimensional coin-turning walk
Mar. 10, 2022 3pm (Zoom)	Topology Joel Ornstein (CU Boulder) X This talk will be an overview of Chen-Freedman's paper "Quantifying Homology Classes", which discusses a certain algorithm for choosing bases of homology groups. Join Zoom Meeting https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231 Quantifying Homology Classes
Mar. 14, 2022 3:30pm (Zoom m…	Thesis Defenses Jordan DuBeau (CU Boulder) X A Jónsson algebra is an infinite algebra J in a countable algebraic language that has no proper subalgebras of the same cardinality as J. We study Jónsson algebras in one particular variety in depth: the variety of Jónsson-Tarski algebras. We prove the existence, possible cardinalities, and number of Jónsson algebras in this variety. By doing so, we gain insight into Jónsson algebras in general, the varieties they can inhabit, and the cardinalities they can have. Jonsson Jonsson-Tarski Algebras
Mar. 15, 2022 8:45am (MATH 3…	Thesis Defenses Cherry Ng (CU Boulder) Equivariant Homology of Representation Spheres for the Nonabelian Group of Order 21
Mar. 15, 2022 10:30am (MATH …	Thesis Defenses Andre Davis (CU Boulder) A Computation of the Action of the Morava Stabilizer Group on the Lubin-Tate Deformation Ring
Mar. 15, 2022 1pm (Zoom)	Andrés Villaveces (Universidad Nacional de Colombia – Bogotá) X We first revisit notions of interpretability and internality in a category-theoretical language (for first order theories), reframing work of Hrushovski and Kamensky in a formalism derived from Makkai's early work. We then describe the issue of recovering the bi-intepretability class of a theory in terms of the automorphism group of a saturated model, and the role of the "Small Index Property" (SIP) - a way of recovering the topology of a group action from purely algebraic information. We then turn to abstract elementary classes, and discuss the same notions, in the opposite order: first, two situations where a Small Index Property holds (joint work with Ghadernezhad), and then some applications to the problem of interpretation and reconstruction, adapted to abstract elementary classes. On the small index property for AECs with strong amalgamation properties
Mar. 15, 2022 2:30pm (MATH 2…	Lie Theory Lucas Gagnon (CU Boulder) X Hopf algebras encode inductive structure. Two areas where this is useful are combinatorics and representation theory, and this talk will describe a Hopf algebraic connection between these areas. On the combinatorial side is the chromatic quasisymmetric function, a q-analogue of Stanley’s chromatic symmetric function; this function encodes details of the graph coloring problem and can be realized as a homomorphism between Hopf algebra structures on graphs and (quasi-)symmetric functions. On the representation theory side is a collection of GLn(Fq)-modules that I will show to be canonically equivalent to some chromatic quasisymmetric functions. Along the way, I will construct a new Hopf algebra of class functions for the unipotent upper triangular groups and use supercharacter theory to give a representation theoretic realization of the graph-to-symmetric-function Hopf algebra homomorphism. Representation theoretic Hopf algebras and q-chromatic symmetric functions
Mar. 15, 2022 3pm (MATH 350)	Topology Juan Moreno (CU Boulder) X This talk should outline the Turaev-Viro construction which takes in a spherical fusion cat- egory as input and outputs a TQFT. The related Reshetikhin-Turaev construction should also be covered. The talks are in-person with a computer pointed at the blackboard zoom option: https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231 (2+1)D TQFTs from fusion categories
Mar. 16, 2022 1:30pm (MATH 3…	Geometry/Analysis Monique Chyba & Taylor Klotz (University of Hawaii) X Copepod nauplii are tiny, larval crustaceans that are found in nearly all water environments. A recent control optimization problem has been used to model the planar motion of a two-legged version of such tiny creatures. In previous work, it was found that this planar motion strongly resembles Euler elastica. After presenting the model, and numerical simulations of these trajectories, we will show how the control system may be transformed into a simpler system utilizing symmetry. This allows us to study the geometry of the optimization problem through the lens of lagrangian classification, with the goal of rigorously demonstrating the relationship between Euler elastica and the optimal trajectories of the copepod model. Two-Legged Copepod Motion and Euler elastica.
Mar. 16, 2022 4pm (Math 220)	MathClub Rodrigo Ribeiro (CU-Boulder (postdoc)) X In this talk will see the surprising results which connect so distinct networks such as Instagram and nervous system of worms. We will also see how mathematics is used to model and investigate the evolution of social networks and other real-world networks. What do the nervous system of a worm, power-grids and Instagram have in common?
Mar. 17, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X The Ulam seminar this week will be focused on working through lingering questions from prior problem sessions. As time permits, we will also continue beyond Eli's last lecture in which the definitions of the Chevalley groups were given. Chevalley Groups: Generators and Relations
Mar. 17, 2022 3pm (Zoom)	Probability Mei Yin (University of Denver) X The topic of parking functions has wide applications in probability, combinatorics, group theory, and computer science. Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Through a combinatorial construction termed a parking function multi-shuffle, we investigate various probabilistic properties of a uniform parking function. Partially based on joint work with Richard Kenyon. Parking functions: Interdisciplinary connections
Mar. 29, 2022 1pm (Zoom)	Paolo Lipparini (Universita' di Tor Vergata, Rome, Italy) X The study of Maltsev conditions is a significant part of universal algebra, with classical characterizations of families of varieties (congruence permutable, distributive, modular...) and recent advanced results by Hobby, McKenzie, Kearnes, Kiss, among others. In particular, the interplay between distinct Maltsev conditions for congruence modular varieties has led to a refined theory for such varieties. Recall that a Maltsev condition is, roughly, a statement of the form "there are some n and terms t1,...,tn such that a certain finite set of equations hold". As we mentioned, many deep and sophisticated results are known about Maltsev conditions. On the other hand, when two conditions are compared, really little is known about the exact value of the smallest n as above. For example, a simple observation by A. Day asserts that if some variety V has k Jónsson terms witnessing congruence distributivity, then V has 2k-1 Day terms witnessing congruence modularity. About fifty years ago Day asked whether this result is best possible, but, to the best of our knowledge, an exact solution is not yet known. A deeper problem (asked by Lakser, Taylor, Tschantz in 1985) concerns the relative lengths of sequences of Day and Gumm terms characterizing congruence modularity. More recently, Kazda, Kozik, McKenzie, Moore provided still another characterization of congruence distributive and modular varieties by means of "directed" terms. Again, the exact relationships between the lengths of the sequences of terms is not known. A solution of the above problems is supposed to provide either interesting exotic examples of congruence modular and distributive varieties, or more refined structure theorems. We shall present recent results about the above Day, LTT and KKMM problems, with an unexpected application to congruence distributive varieties. Relative lengths of Maltsev conditions
Mar. 29, 2022 1pm (Math 350 …	Thesis Defenses Sarah Arpin (University of Colorado Boulder) X As quantum computers become more and more advanced, currently standard cryptographic protocol becomes weaker and weaker. Supersingular elliptic curve isogeny graphs are a promising new direction for post-quantum cryptographic protocol. The supersingular elliptic curve isogeny graph has vertices which are isomorphism classes of supersingular elliptic curves over an algebraically closed field of characteristic p, and edges which are degree-l isogenies up to equivalence. In this thesis, we study supersingular elliptic curve isogeny graphs from three directions. First, we describe the relationship between the supersingular elliptic curve isogeny graph over an algebraically closed field of characteristic p and a supersingular isogeny graph defined over the prime field. We provide a deep study and heuristics on how the supersingular elliptic curves which are defined over the prime field or which are p-power Frobenius conjugate sit inside of the supersingular elliptic curve isogeny graph. Second, we attach level structure to supersingular elliptic curves and study the endomorphism rings of these objects. We provide an equivalence of categories with objects in a quaternion algebra, and study the l-isogeny graph of supersingular elliptic curves with level structure. Third, we study oriented supersingular elliptic curve isogeny graphs, where the orientation is a choice of embedding of a quadratic field into the endomorphism ring of a supersingular elliptic curve. We study the ``volcano" and ``cordillera" structure of these oriented isogeny graphs, and we provide algorithms for walking the supersingular l-isogeny graph via paths on the oriented l-isogeny graph. Dissertation title: Supersingular Elliptic Curve Isogeny Graphs
Mar. 29, 2022 3pm (MATH 350)	Topology Juan Moreno (CU Boulder) X This talk will outline the TVBW models which use a spherical fusion category and combinatorial structures on manifolds to produce oriented (2+1)D TQFTs . Turaev-Viro-Barratt-Wesbury Models
Mar. 31, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X We will plan out speakers/topics for the remainder of the semester. organizational meeting
Mar. 31, 2022 2:30pm (Math 2…	Functional Analysis Levi Lorenzo (CU Boulder) X We set out with the intention that KK sets (which we now know have the structure of abelian groups) would be Hom sets in our new category. Accordingly, we need a way to compose an element of KK(A,B) with one of KK(B,C) to produce an element of KK(A,C). In this talk, we see how perhaps the most natural guesses for how to do so fail and introduce the Connes-Skandalis Connection which enables us to define composition. The Connes-Skandalis connection
Mar. 31, 2022 3pm (Math350)	Math Phys Anton Kapustin (Caltech) X M. Berry showed how to attach a line bundle and a connection on it to a family of quantum Hamiltonians with a non-degenerate ground state, under the assumption that the Hilbert space is finite-dimensional. The first Chern class of this line bundle is a topological invariant of the family. It is far from obvious if this construction can be generalized to quantum many-body Hamiltonians. Indeed, naive generalizations fail because ground states of different Hamiltonians typically correspond to inequivalent representations of the algebra of observables. Nevertheless, it is possible to construct such invariants by making use of a certain differential graded Lie algebra (DGLA) attached to a quantum lattice system. For example, it turns out that to any family of gapped Hamiltonians on a 1d lattice one can attach a “quantized” degree-3 cohomology class on the parameter space. In this talk I will outline a construction of this DGLA as well as the construction of higher Berry classes. The talk is based on a work in progress with Nikita Sopenko. The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 Higher Berry classes for many-body quantum lattice systems Sponsored by the Meyer Fund
Apr. 05, 2022 1pm (Zoom)	Nik Ruskuc (University of St Andrews, UK) X For a while now, Peter Mayr and I have been looking at properties of direct and subdirect products in algebra, often motivated by some well known or particularly nice results from combinatorial group theory. The topics include finite generation, finite presentability, residual finiteness, infinite subdirect powers, etc. A fairly rich landscape has emerged over the years. Perhaps unsurprisingly the most general results can be obtained in the context of congruence permutable or modular varieties. This then leaves semigroups outside, and I have been working on such questions in parallel with some of my PhD students. In this talk I will try to sketch this landscape, not so much by means of a systematics introduction, but a few selected strands, results and comparisons. Direct and subdirect products in combinatorial algebra, groups and semigroups Sponsored by the Meyer Fund
Apr. 05, 2022 3pm (MATH 350)	Topology Marvin Qi (CU Boulder) X This talk should cover symmetries of MTCs and the process of gauging these symmetries to create new MTCs. The emerging notion of G-crossed braided fusion categories should also be noted. The talks are in-person with a computer pointed at the blackboard zoom option: https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231 Categorical symmetries and gauging
Apr. 06, 2022 3:20pm (Math35…	Math Phys Daniel Spiegel (CU Boulder) X We study the norm topology on the pure state space of a C*-algebra A. This is a rather strong topology but it is nonetheless interesting because it corresponds to the topology of the projective Hilbert space of any irreducible representation A. We prove that any continuous map into this space is locally equivalent to composition of a fixed pure state with a norm-continuous family of inner automorphisms of A. This follows as a corollary of our main result: the operators obtained from the Kadison transitivity theorem can be chosen to depend continuously on the initial data. Finally, we will discuss how the aforementioned inner automorphisms can be used to construct local trivializations giving the unitary group of A the structure of a principal bundle over any path component of the pure state space. The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 The continuous Kadison transitivity theorem
Apr. 06, 2022 4pm (Math 220)	MathClub Padi Fuster Aguilera (CU-Boulder (postdoc)) X In general, it is hard to find solutions to differential equations. How can we obtain information without being able to solve them explicitly? We will talk about some geometric and topological methods that can help us knowing more about their behavior. These ideas are very visual and somewhat simple but applying them can become a tricky business! I know that it exists, but I don’t know what it is!
Apr. 06, 2022 5pm (MATH 350)	Grad Student Seminar Eli Orvis (CU Boulder) X Number theorists are interested in solving equations over specified rings (Z, for example), while geometers are interested in the shape of solution sets. Deep insights during the 20th century brought these fields together and initiated the field of "arithmetic geometry." In this talk, we'll give an introduction to the circle of ideas that relates algebraic number theory, commutative algebra, and algebraic geometry. In particular, we will motivate the group of divisors of an algebraic curve by recognizing it as the parallel of a natural object in the number-theoretic setting. P.S. Courses in all three subjects listed in the penultimate sentence will (hopefully) be offered next year, so I encourage anyone interested in these subjects to attend this talk as a "warm-up"! Algebra and Numbers and Curves, Oh My!
Apr. 07, 2022 9:05am (Math 3…	Ulam Breeann Wilson (CU Boulder) X Recap of the generators and relations for Chevalley groups leading to the Bruhat decomposition. Will use the example of SL(n) throughout as motivating example. Problem Session on Tuesday, 4/12, to finish up loose ends, as needed. Bruhat Decomposition
Apr. 07, 2022 2:30pm (MATH 3…	Functional Analysis Zachary Garvey (CU Boulder) X TBD The Kasparov product
Apr. 08, 2022 1:30pm (MATH 3…	Noncomm Geometry Yi Wang (Stony Brook) X Let M be a closed oriented smooth manifold, and LM be the free loop space of M. It has been known since the seminal work of Chas and Sullivan that loop operations produce rich algebraic structures on the ordinary homology and S^1-equivariant homology of LM, and there are similar structures on Hochschild cohomology and cyclic cohomology of algebras. In this talk, I will describe a chain level refinement of the gravity algebra structure on the (negative) S^1-equivariant homology of LM, and relate it to structures on the algebraic side, in the spirit of Deligne’s conjecture. This result is literally an application of a theorem of Ben Ward to a construction of Kei Irie; the key ingredient is a Jones’ type isomorphism between the S^1-equivariant homology of an S^1-space X and the cyclic homology of certain cocyclic chain complex associated to X. Chain level gravity algebra structure in string topology
Apr. 12, 2022 12pm (TBD)	Diversity Patricia Gonzalez (Assistant Dean of Inclusive Practice, CU) Anti-Blackness Within Non-Black Communities
Apr. 12, 2022 1pm (Zoom)	Thomas Kucera (University of Manitoba) X This is work motivated by questions at the intersection of algebra and model theory, and using advanced techniques of model theory. Baldwin and Shelah (Algebra Universalis, 1983) studied saturated free algebras. Pillay and Sklinos (Bull. Symb. Logic 2015), following the lead of this paper, studied "almost indiscernible theories", taking the opportunity to refine the statements of the major results and improve the proofs. We extend these results to large infinite contexts, both in the size of the language and the kinds of tuples allowed in a "basis"; and return to examples and applications in algebra, in particular in the theory of modules. The theory develops by noting various analogies. The model-theoretic concept 'indiscernible sequence' generalizes 'linearly independent set' in a vector space, 'free (generating) set' of an algebra, 'algebraic independence' in an algebraically closed field, and similar concepts. 'Saturated model' generalizes concepts such as 'injective envelope of a module', 'algebraic closure of a field', and similar constructions. A complete first-order theory is "almost indiscernible" if it has a (sufficiently large) saturated model which lies in the algebraic closure of an indiscernible set (of sequences). Requiring that a saturated model be generated by an indiscernible set imposes strong structural constraints, but nonetheless there are natural motivating examples. I start with some history and motivation from algebra, then I will give an overview of the main model theoretic concepts and techniques, motivating them as much as possible by examples from algebra. I'll state the new technical structural results for almost indiscernible theories in our more general context, with no more than informal 'hand-waving' about the proof techniques. Then I will present some consequences for free algebras and for theories of modules, including structure theorems and some examples. I conclude with a list of open questions. This is joint work with Anand Pillay. Saturated free algebras and almost indiscernible theories: an overview
Apr. 12, 2022 2:30pm (MATH 3…	Lie Theory Richard Green (CU Boulder) X A "2-root'' is a symmetrized tensor product of orthogonal roots of a Kac--Moody algebra. We study the 2-roots of the Weyl group W of a simply laced Y-shaped Dynkin diagram with three branches of arbitrary finite lengths. The symmetric square of the reflection representation of W has a natural codimension-1 submodule with a canonical basis of 2-roots. We will discuss some of the remarkable properties of this basis. (This is joint work with Tianyuan Xu.) 2-roots for simply-laced Weyl groups, Part I
Apr. 12, 2022 3pm (MATH 350)	Topology Sam Gunningham (Montana State University) X Skein modules are linear spaces spanned by knots and links in a given 3-manifold, modulo certain skein relations. They were defined about 30 years ago independently by Przytycki and Turaev and have been extensively studied in the subsequent years. In this introductory talk I will propose a new role for skein modules: as (a component of) the state space of a certain 4-dimensional topological quantum field theory, which according to the work of Kapustin and Witten, encodes the mathematical features of the geometric Langlands program. This realization leads to some surprising conjectures (which can be directly verified in some key cases), relating two different flavors of skein modules on a given closed 3-manifold. This is joint work with David Ben-Zvi, David Jordan, and Pavel Safronov. Langlands duality for 3-manifolds Sponsored by the Meyer Fund
Apr. 13, 2022 3:20pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder Department of Mathematics and CTQM) X The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 The geometry of state spaces in quantum theory
Apr. 13, 2022 5pm (MATH 350)	Grad Student Seminar Levi Lorenzo (CU Boulder) X Math Studs, Come on Down!! You're the next contestants on......."Is This a Groupoid??" Recorded live in 350! Come play everyone's second favorite math game where, unlike responses to my happy hour emails, the answer is always yes! In this talk we'll explore the wide, mysterious, and hilariously-named world of groupoids. Tristan Bice once quipped: "What's a category??...it's a groupoid without inverses," and while I promise my jokes won't be a lick better than his, I will draw a lot of pictures :) . No knowledge of dynamical systems will be assumed for this talk. (Not actually recorded) Is This a Groupoid??
Apr. 14, 2022 9:05am (Math 3…	Ulam Joel Ornstein (CU Boulder) X I will discuss $θ$ -groups introduced by Vinberg in the 1970's and more recent constructions of Jordan classes induced by group action. This talk will explore graded reductive Lie algebras and representations of group actions arising from the Lie algebra grading. theta-groups and Jordan classes
Apr. 14, 2022 2:30pm (MATH 3…	Functional Analysis Rachel Chaiser (CU Boulder) X TBD Properties of the Kasparov product
Apr. 19, 2022 1pm (Zoom)	Michael Levet (CU Boulder) X The Weisfeiler—Leman (WL) algorithm is a key combinatorial subroutine in Graph Isomorphism, that (for fixed $k ≥ 2$ ) computes an isomorphism invariant coloring of the k-tuples of vertices. Brachter & Schweitzer (LICS 2020) recently adapted WL to the setting of groups. Using a classical Ehrenfeucht-Fra\"iss\'e pebble game, we will show that Weisfeiler—Leman serves as a polynomial-time isomorphism test for several families of groups previously shown to be in $P$ by multiple methods. These families of groups include: (1) Coprime extensions $H$ by $N$ , where $H$ is $O(1)$ -generated and the normal Hall subgroup $N$ is Abelian (Qiao, Sarma, & Tang, STACS 2011). (2) Groups without Abelian normal subgroups (Babai, Codenotti, & Qiao, ICALP 2012). In both of these cases, the previous strategy involved identifying key group-theoretic structure that could then be leveraged algorithmically, resulting in different algorithms for each family. A common theme among these is that the group-theoretic structure is mostly about the action of one group on another. Our main contribution is to show that a single, combinatorial algorithm (Weisfeiler-Leman) can identify those same group-theoretic structures in polynomial time. We also show that Weisfeiler—Leman requires only a constant number of rounds to identify groups from each of these families. Combining this result with the parallel WL implementation due to Grohe & Verbitsky (ICALP 2006), this improves the upper bound for isomorphism testing in each of these families from $P$ to $TC0$ . This is joint work with Joshua A. Grochow. Weisfeiler—Leman for Group Isomorphism: Action Compatibility
Apr. 19, 2022 2:30pm (MATH 3…	Lie Theory Richard Green (CU Boulder) X A "2-root'' is a symmetrized tensor product of orthogonal roots of a Kac--Moody algebra. We study the 2-roots of the Weyl group W of a simply laced Y-shaped Dynkin diagram with three branches of arbitrary finite lengths. The symmetric square of the reflection representation of W has a natural codimension-1 submodule with a canonical basis of 2-roots. We will discuss some of the remarkable properties of this basis. (This is joint work with Tianyuan Xu.) 2-roots for simply-laced Weyl groups, Part II
Apr. 19, 2022 3pm (MATH 350)	Topology Peter May (University of Chicago) X Groups can be described as quotients of free groups by relations. This basic fact has a categorical generalization called the Beck monadicity theorem. As I will explain, that is a statement about monads associated to "nice" adjoint pairs of functors, in this case the free functor from sets to groups and the forgetful functor from groups to sets. It is a lovely result when it holds, but adjoint functors are ubiquitous and they are usually not "nice". In homotopy theory, it is very important to understand "homotopical monadicity theorems" which say when something like the conclusion of the monadicity theorem holds "up to homotopy''. I have worked on this question for over half a century, but I have only reached conceptual understanding of what I have been doing all along in the last few months, in joint work with Hana Kong and Foling Zou. The understanding leads to new applications. I will try to explain the basic ideas in as elementary a way as I can. Homotopical Monadicity Theorem Sponsored by the Meyer Fund
Apr. 20, 2022 4pm (Math 220)	MathClub Joe Timmer (CU-Boulder) X Lots of people play games and part of the enjoyment of them is to not to play them optimally. In this talk, we will discuss ways mathematics can be used to analyze games, develop strategies, and show that games can still be fun, even if there is an optimal way to play them! How to Ruin Fun Games with Math
Apr. 21, 2022 9:05am (Math 2…	Ulam Michael Woodbury (CU Boulder) X We will discuss the general notion of reductive groups and see how (split) reductive groups are determined by their root data. Several examples of such groups and their dual groups will be considered. Reductive groups and Root Data
Apr. 26, 2022 2:30pm (MATH 2…	Lie Theory Mandi Schaeffer Fry (Metropolitan State University of Denver) X For a prime p, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with p’-degree and the corresponding set for the normalizer of a Sylow p- subgroup. Navarro’s refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro–Späth–Vallejo have reduced the McKay–Navarro conjecture to certain “inductive” conditions on finite simple groups. I’ll discuss joint work with Lucas Ruhstorfer, in which we prove that these inductive McKay–Navarro (also called the inductive Galois–McKay) conditions hold when p = 2 for several groups of Lie type. The inductive McKay—Navarro conditions for the prime 2 and some groups of Lie type.
Apr. 26, 2022 3pm (MATH 350)	Topology Charlie Stahl (CU Boulder) X This talk should cover the Levi-Wen string net construction and the relationship to the Turaev-Viro construction. Depending on the presenter this could be a very physics-leaning talk. The talks are in-person with a computer pointed at the blackboard zoom option: https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231 String nets
Apr. 27, 2022 3:20pm (Math350)	Math Phys Howie Jordan (CU Boulder) X The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356 Topos approach to quantum information theory
Apr. 27, 2022 5pm (MATH 350)	Grad Student Seminar Ben Kitchen X A directed graph (or digraph) is a graph whose edges are ordered pairs of vertices. On a graph or digraph, a color change rule is a process by which a set of filled vertices can cause other vertices to become filled. The throttling number minimizes the sum of the size of the initial filled set and the amount of time it takes to fill every vertex. In this thesis, the positive semidefinite (PSD) color change rule is adapted to digraphs, and throttling for the PSD color change rule is studied specifically. The behavior of the PSD throttling number when the order (or direction) of an edge is changed is analyzed, and bounds for the PSD throttling number of a digraph are discussed. These bounds are later shown to be tight for trees. We then look further into what happens if the direction of every edge is changed simultaneously (the resulting digraph is called the reversal). We demonstrate that a PSD throttling process on a digraph can also be reversed to obtain a PSD throttling process on the digraph's reversal. Finally, we use this fact to study the behavior of the sum of the throttling number of a digraph and the throttling number of its reversal. Positive Semidefinite Throttling on Directed Graphs
Apr. 28, 2022 9:05am (Math 3…	Ulam Michael Woodbury (CU Boulder) X After completing the discussion from last week on the examples PGL(2) and SL(2), I will discuss parabolic subgroups. We will look at them from two points of view: (1) Lie theoretic: via certain subsets of roots (2) Geometric: as stabilizers of certain subspaces I will then explain how these two points of view interact in the case of the exceptional group of G_2. This will make use of a very useful, but perhaps(?) little known/used basis of the octonians. Parabolic subroups
Apr. 28, 2022 2:30pm (MATH 3…	Functional Analysis Robin Deeley (CU Boulder) X TBD Poincare duality in KK-theory and examples
Apr. 28, 2022 3pm (Zoom)	Probability Vishesh Jain (University of Illinois at Chicago) X Let $M_{n}$ be an $n \times n$ random matrix whose entries are i.i.d. copies of a discrete random variable $ξ$ . It has been conjectured that the dominant reason for the singularity of $M_{n}$ is the event that a row or column of $M_{n}$ is zero, or that two rows or columns of $M_{n}$ coincide (up to a sign). I will discuss joint work with Ashwin Sah (MIT) and Mehtaab Sawhney (MIT), towards the resolution of this conjecture. Singularity of discrete random matrices
Apr. 28, 2022 4pm (Zoom)	Topology Lucas Gagnon (CU Boulder) X This talk will be an overview if the paper An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson by Julianna S. Tymoczko. See https://arxiv.org/abs/math/0503369. Join Zoom Meeting https://cuboulder.zoom.us/j/98979344859?pwd=K3ZrTWt4Wkk1cllOQ1pkcW4vZ1FCZz09 Meeting ID: 989 7934 4859 Passcode: 787231 An introduction to equivariant cohomology and homology
May. 16, 2022 9am (Zoom)	Probability Vlad-Dumitru Margarint (CU Boulder) X The Schramm-Loewner Evolution (SLE) is a family of random planar fractal curves, introduced by Schramm in 2000. These random fractal curves are proved to describe the scaling limits of a number of discrete models that are of great interest in planar Statistical Physics. In this mini-course, I will give an overview of some introductory aspects of this theory, focusing the exposure on the intuition behind the results rather than their proofs. I will also give a short sketch of the proof of Rohde-Schramm Theorem on the existence of the SLE trace, an overview of a technique used to approximate uniformly the SLE traces, and a short description of a simultaneously growing Multiple SLE model, for N=2 curves and beyond. A Mini-course on Schramm-Loewner Evolution
May. 18, 2022 9am (Zoom)	Probability Vlad-Dumitru Margarint (CU Boulder) X The Schramm-Loewner Evolution (SLE) is a family of random planar fractal curves, introduced by Schramm in 2000. These random fractal curves are proved to describe the scaling limits of a number of discrete models that are of great interest in planar Statistical Physics. In this mini-course, I will give an overview of some introductory aspects of this theory, focusing the exposure on the intuition behind the results rather than their proofs. I will also give a short sketch of the proof of Rohde-Schramm Theorem on the existence of the SLE trace, an overview of a technique used to approximate uniformly the SLE traces, and a short description of a simultaneously growing Multiple SLE model, for N=2 curves and beyond. A Mini-course on Schramm-Loewner Evolution
May. 20, 2022 9am (Zoom)	Probability Vlad-Dumitru Margarint (CU Boulder) X The Schramm-Loewner Evolution (SLE) is a family of random planar fractal curves, introduced by Schramm in 2000. These random fractal curves are proved to describe the scaling limits of a number of discrete models that are of great interest in planar Statistical Physics. In this mini-course, I will give an overview of some introductory aspects of this theory, focusing the exposure on the intuition behind the results rather than their proofs. I will also give a short sketch of the proof of Rohde-Schramm Theorem on the existence of the SLE trace, an overview of a technique used to approximate uniformly the SLE traces, and a short description of a simultaneously growing Multiple SLE model, for N=2 curves and beyond. A Mini-course on Schramm-Loewner Evolution
May. 25, 2022 1:30pm (Math 3…	Diversity Howy Jordan (UC Boulder) X You're invited to a show n tell n monch time with Howy! What we now call the autism spectrum is a cluster of social, cultural, and developmental disabilities popularized by Hans Asperger and Leo Kanner. Because of the cultural and social components, the presentation of autism varies widely from person to person, including gendered characteristics, which has led to widespread misunderstanding and misdiagnosis. In this talk we'll discuss the most common identifying characteristics of the autism spectrum, co-diagnostics, and autism's place in the modern neurodiversity and disability rights movements. Food will be served: expect Papa John's pizza and Krispey Kreme donuts. This is a joint seminar with the Gagnon Student Seminar. A hybrid option will be available over Zoom. Neurodiversity & Autism Self-Advocacy Networks
May. 25, 2022 1:30pm (Math 3…	Grad Student Seminar Howy Jordan (UC Boulder) X You're invited to a show n tell n monch time with Howy! What we now call the autism spectrum is a cluster of social, cultural, and developmental disabilities popularized by Hans Asperger and Leo Kanner. Because of the cultural and social components, the presentation of autism varies widely from person to person, including gendered characteristics, which has led to widespread misunderstanding and misdiagnosis. In this talk we'll discuss the most common identifying characteristics of the autism spectrum, co-diagnostics, and autism's place in the modern neurodiversity and disability rights movements. Food will be served: expect Papa John's pizza and Krispey Kreme donuts. This is a joint seminar with the Diversity Committee. A hybrid option will be available over Zoom. Neurodiversity & Autism Self-Advocacy Networks

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Events for January – May 2022

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MathClub

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Grad Student Seminar

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Functional Analysis

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Math Phys

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Noncomm Geometry

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Thesis Defenses

Thesis Defenses

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Geometry/Analysis

MathClub

Ulam

Probability

Thesis Defenses

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Ulam

Functional Analysis

Math Phys

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Math Phys

MathClub

Grad Student Seminar

Ulam

Functional Analysis

Noncomm Geometry

Diversity

Lie Theory

Topology

Math Phys

Grad Student Seminar

Ulam

Functional Analysis

Lie Theory

Topology

MathClub

Ulam

Lie Theory

Topology

Math Phys

Grad Student Seminar

Ulam

Functional Analysis

Probability