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Events for January – May 2024

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Mar. 05, 2024 5pm (MATH 350)	Probability Kyle Luh & Sean O'Rourke (CU Boulder) X This will be an introductory talk on the theory of random polynomials and random matrices for the visiting graduate students. We will give a broad overview of the fields as well as some of the research directions being pursued in this department. Overview of Random Polynomials and Random Matrices
Jan. 23, 2024 3:30pm (MATH 3…	Topology Ben Spitz (UCLA) X “Classically”, Mackey and Tambara functors are equivariant generalizations of abelian groups and commutative rings, respectively. What this means is that, in equivariant homotopy theory, Mackey functors appear wherever one would expect to find abelian groups, and Tambara functors appear wherever one would expect to find commutative rings. More recently, work by Bachmann and Hoyois has garnered interest in related structures which appear in motivic homotopy theory – these motivic Mackey functors and motivic Tambara functors do not have anything to do with group-equivariance, but somehow share the same axiomatics. In this talk, I’ll introduce a general context for interpreting the notions of Mackey and Tambara Functors, which subsumes both the equivariant and motivic notions. The aim of this approach is to translate theorems between contexts, enriching the theory and providing cleaner proofs of essential facts. To this end, I’ll discuss recent progress in boosting a foundational result about norms from equivariant algebra to this more general context. Mackey and Tambara Functors Beyond Equivariant Homotopy
Jan. 25, 2024 2pm (Online)	Rep Theory Chelsea Walton (Rice University) View Online:https://cuboulder.zoom.us/j/97927772782 X This talk will be on joint work with Robert Laugwitz and Milen Yakimov (arXiv:2307.14764) that is motivated by obtaining solutions to the quantum reflection equation (qRE). To start, given a braided monoidal category C and C-module category M, we introduce a version of the Drinfeld center Z(C) of C adapted for M. We refer to this category as the "reflective center" E_C(M) of M. Just like Z(C) is a canonical braided monoidal category attached to C, we show that E_C(M) is a canonical braided module category attached to M. We will also discuss the case when C is the category of modules over a quasitriangular Hopf algebra H, and show how quantum K-matrices arise in this setting (thus yielding solutions to the qRE). Reflective centers of module categories and quantum K-matrices
CANCELED Jan. 30, 2024	Kempner Edray Goins (Pomona College) X Imagine the hands on a clock. For every complete the minute hand makes, the seconds hand makes 60, while the hour hand only goes one twelfth of the way. We may think of the hour hand as generating a group such that when we ``move'' twelve times then we get back to where we started. This is the elementary concept of a monodromy group. In this talk, we give a gentle introduction to a historical mathematical concept which relates calculus, linear algebra, differential equations, and group theory into one neat theory called ``monodromy''. We explore lots of real world applications, including why it’s so easy to get lost in parking garages, and present some open problems in the field. We end the talk with a discussion of how this is all related to solving polynomial equations, such as Abel’s famous theorem on the insolubility of the quintic by radicals. Clocks, Parking Garages, and the Solvability of the Quintic: A Friendly Introduction to Monodromy
Jan. 30, 2024 3:30pm (MATH 3…	Topology Sarah Petersen Organizational Meeting
Feb. 01, 2024 2pm (Online)	Rep Theory Thomas Creutzig (University of Alberta) View Online:https://cuboulder.zoom.us/j/97927772782 X Vladimir Kovalchuk constructed a two parameter family of VOA that have an affine sp(2) as subalgebra and in each even positive conformal weight a singlet and at each odd one an sp(2)-triplet. This structure is the third universal family of W-algebras after the W-infinity algebra and its even spin analogue and quotients of one-parameter subfamilies of Vladimir algebras are often realized by cosets of certain W-algebras of orthosymplectic type. Very much like in the W-infinity case and in the even spin case it is expected that there are quotients that are rational and lisse. I will explain that this expectation is indeed true and it is given by a novel family of rational and lisse cosets. This is joint work with Vlad Kovalchuk and Andy Linshaw. A family of rational VOAs as type C cosets.
Feb. 01, 2024 2:30pm (MATH 3…	Functional Analysis Alonso Delfín (CU Boulder) X In this talk I will discuss M(A), the multiplier algebra of a general Banach algebra A, as well as some of its properties. I will present a main result that produces an isometric representation of M(A) on a Banach space E whenever A has a contractive approximate identity (cai) and a nondegenerate isometric representation on E. This implies that the multiplier algebra of a "C*-like" L^p-operator algebra is again an L^p-operator algebra. I will then focus on some particular degenerate L^p operator algebras without cai's and say something about their multiplier algebras. Multiplier Algebras of L^p-Operator Algebras
Feb. 06, 2024 1:25pm (MATH 2…	Algebra/Logic Peter Mayr (CU Boulder) View Online:https://cuboulder.zoom.us/j/96896272260 X Let A be a finite simple non-abelian Mal'cev algebra (e.g. a group, loop, ring). We identify the Fra{\"i}ss{\'e} limit D of the class of finite direct powers of A as a filtered Boolean power of A by the countable atomless Boolean algebra B and show that D also arises as congruence classes of the countable free algebra in the variety generated by A. Further we show that filtered Boolean powers of A by B are -categorical, have the small index property, strong uncountable cofinality and the Bergman property. This is joint work with Nik Ruskuc (University of St Andrews). Filtered Boolean powers and their automorphism groups
Feb. 06, 2024 3:30pm (MATH 3…	Topology Juan Moreno X Last semester we saw the category of spectra introduced as the right place to study stable phenomena for spaces. This idea turns out to be incredibly useful largely because spectra behave very much like algebraic objects. We will spend a good part of the talk covering two instances of this: Duality - analogous to the notion of duality for vector spaces is something called Spanier-Whitehead duality for spectra. Localization - analogous to the notion of localizing abelian groups at a prime is something called Bousfield localization of spectra. With the remaining time we will see how these two notions can be used to capture important information about the stable homotopy groups of spheres. In particular we'll talk about a sort of "first approximation" to these groups: the stable image of J. Duality, Localization, and stable homotopy groups of spheres
Feb. 07, 2024 3:30pm (MATH 3…	Math Phys Markus J. Pflaum (CU Boulder) X This is an organizational meeting for the Spring 2024 semester. We will have a few external speakers, talks by graduate students on their research in Mathematical Physics and talks on the semester theme "Noncommutative geometry and topological phases of matter". Organizational Meeting\| Semester Theme: Noncommutative Geometry and Topological Phases of Matter
Feb. 08, 2024 2pm (Online)	Rep Theory Shoma Sugimoto (Tsinghua University) View Online:https://cuboulder.zoom.us/j/97927772782 X The Feigin-Tipunin algorithm is a combinatorial and iterative algorithm introduced by the speaker recently to construct and study log VA(-module)s that have the q-series valued quantum invariant of 3-mfds called homological block introduced by S. Gukov et al. as their characters. The key point is that at each step of the iteration, a geometric representation theory can be used as in the usual Feigin-Tipunin construction, so that the representation theory of the log-VA can be studied without having to examine its complicated algebraic structure. In this talk, after describing the speaker's previous work, it will be explained that if the FT algorithm can be applied to a certain lattice VOA-module, we can construct a log VA-module with the homological block of the (N+2)-Seifert 3-mfd as its character. Feigin-Tipunin algorithm
Feb. 08, 2024 3:35pm (MATH 3…	Probability Jack Morris (Cornell Tech) X How much information do the outputs of language models contain about their inputs? We investigate this problem in two scenarios, recovering text inputs from the outputs of embeddings from sentence embedders and next-token probability outputs from language models. In both cases, our methods are able to fully recover some inputs given just the model output. Inverting Language Models
Feb. 09, 2024 9:05am (Zoom)	Math Edu Lee Roberson & Jen Gensler (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/91244797970 X What we want students to take away from homework assignments and what students actually take away do not always align. To address this issue we will work to understand what meaningful learning is and use this understanding to outline our learning goals for students engaging in homework assignments. Applying this framework, we will evaluate assessment items to determine if they align with our goals and explore strategies for creating homework items that promote meaningful learning. Time permitting, we will discuss the importance of constructive feedback on homework as a component of meaningful student learning. Creating Homework that Promotes Meaningful Learning
Feb. 13, 2024 3:30pm (MATH 3…	Topology Taylor Rogers The Steenrod Algebra and Its Dual
Feb. 14, 2024 3:30pm (MATH 3…	Math Phys Andrew Hamilton (CU Boulder, Astrophysics Department) View Online:https://cuboulder.zoom.us/j/94002553301 (Password: 790356) X The grand unified group Spin(11,1) is motivated by the fact that Spin(10) chirality coincides with Dirac chirality, allowing 10 compact internal dimensions and 3+1 large spacetime dimensions to coexist in a geometry of just 11+1 dimensions. Spin(11,1) admits a 12 dimensional bosonic string theory whose 26 underlying dimensions are compactified to 12 on the 14 dimensional self-dual torus of the SU(8)xSU(8) group. The 10 dimensional geometry is a complex five-fold that becomes a Calabi-Yau five-fold after symmetry breaking to the standard model. The multiverse just got a whole lot bigger. Spin(11,1) and String Theory
Feb. 15, 2024 2pm (Online)	Rep Theory Shigenori Nakatsuka (University of Alberta) View Online:https://cuboulder.zoom.us/j/97927772782 X In physics, the webs of W-algebras are introduced as the vertex algebras associated with the (p, q)-webs of interfaces in the topologically twisted N = 4 super Yang-Mills theory. This class of vertex algebras are generalizations and extensions of the so-called vertex algebras at the corner, or equivalently the affine cosets of hook-type W-algebras in type A. Although not proven yet in general, it has recently been noticed that the webs of W-algebras actually recover the general W-algebras in type A through the ``reduction by stages". In this talk, I will present the first non-trivial examples and some interesting phenomena found in the course both in rational and irrational cases. The talk is based on a joint work with T. Creutzig, J. Fasquel, and A. Linshaw. On the structure of W-algebras in type A
Feb. 20, 2024 1:25pm (MATH 2…	Algebra/Logic John T. Baldwin (University of Illinois at Chicago) View Online:https://cuboulder.zoom.us/j/96896272260 X A block design is a set of elements and a collection of -element subsets of (called blocks) with the property that each -element subset of occurs in exactly blocks. A -Steiner system is a system. Using variants of the Hrushovski method we construct infinite block designs and Steiner systems that are a) -categorical and with more work b) have -transitive automorphism groups for prescribed . The strongly minimal Steiner -Steiner system (M,R) from Baldwin and Pao (2021) can be `coordinatized' in the sense of Ganter and Werner (1975) by a quasigroup if is a prime-power. But for the basic construction this coordinatization is never definable in (M,R) by Baldwin (2023). Nevertheless, by refining the construction, if is a prime power there is a (2,k)-variety of quasigroups which is strongly minimal and definably coordinatizes a Steiner -system. We compare these structures with Steiner systems given by Fra{\"\i}ss\'{e} constructions by Barbina and Casanovas (2019) and compare versions of the small intersection property for -categorical structures and countably saturated structures by Lascar (2002). Infinite combinatorics from finite structures
Feb. 20, 2024 3:30pm (MATH 3…	Topology Elizabeth Tatum (Universität Bonn) X In recent work, Guchuan Li, Sarah Petersen, and I have constructed models for -equivariant analogues of the integral Brown-Gitler spectra. In this talk, I will start by introducing the classical Brown-Gitler spectra and some of their applications. After that, I will discuss the -equivariant integral Brown-Gitler spectra, as well as the applications we are beginning to study. Equivariant Brown-Gitler spectra
Feb. 20, 2024 4:50pm (MATH 3…	Grad Student Seminar Alex LaJeunesse (University of Colorado) X Algebraic K-theory is a booming field that interacts with many diverse areas of mathematics. It’s universality makes it difficult to understand, but much progress has been made in recent years using a variety of “trace” methods; that is, approximating K-theory by simpler algebraic invariants. In this talk I will introduce algebraic K-theory and the related Hochschild homology, with the ultimate goal of constructing a useful map from the former to the latter. The Dennis Trace and Algebraic K-Theory
Feb. 21, 2024 3:30pm (MATH 3…	Math Phys James Woodcock (CU Boulder, Mathematics Department) Noncommutative Tori
Feb. 22, 2024 2pm (Online)	Rep Theory Ana Ros Camacho (Cardiff University) View Online:https://cuboulder.zoom.us/j/97927772782 X The study and classification of algebra objects in modular tensor categories has a strong motivation from the conformal field theoretical point of view, these objects being related to e.g. full theories [Fuchs-Runkel-Schweigert] and extensions of vertex operator algebras [Huang-Kirillov-Lepowsky, Creutzig-Kanade-McRae for superalgebras]. In this talk, I will present a classification of rigid, Frobenius algebras in the so-called Dijkgraaf-Witten categories, which we achieved using Frobenius monoidal functors. Joint work with Robert Laugwitz and Sam Hannah, based on SIGMA 19 (2023), 075, 42 pages. On the classification of rigid Frobenius algebras in Dijkgraaf-Witten categories
Feb. 23, 2024 9:05am (Zoom)	Math Edu Rebekah Jones (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/91244797970 X We will discuss the process of planning, creating and implementing active learning materials in the classroom. We’ll consider different aspects, including the purpose and style of such activities, and give concrete examples from the calculus sequence at CU. The emphasis of this talk will be on the practical aspects of this process. Developing Effective Active Learning Materials
Feb. 23, 2024 3:30pm (ECCR 2…	Kempner Chad Topaz (Williams College) X Tens of millions of people in the United States have been directly impacted by the criminal justice system, with nearly half the population affected through close familial or social ties. Alongside the direct harm inflicted by the system, an insidious challenge arises: the system's opaque nature makes pinpointing the specific loci of harm complex and elusive. Echoing the words of civil rights pioneer Ida B. Wells, who stated that "the way to right wrongs is to turn the light of truth upon them," this talk will showcase how data science can be harnessed to expose racial injustice. It will feature case studies spanning various scales and stages of the criminal justice system, including policing in the small municipality of Williamstown, Massachusetts; criminal sentencing across all 94 federal district courts; and incarceration at Rikers Island in New York during the COVID pandemic. These examples underscore the pivotal role of data science tools in fostering transparency and advancing justice. Data Science for Criminal and Social Justice
Feb. 27, 2024 1:25pm (MATH 2…	Algebra/Logic Chase Meadors (CU Boulder) View Online:https://cuboulder.zoom.us/j/96896272260 X S4-algebras (or interior algebras) provide semantics for the well-known modal logic S4, and there is a syntactic criterion characterizing when a variety of S4-algebras is locally finite in terms of its "depth" (a classical result of Segerberg and Maksimova). Since the logic MS4 (monadic S4) axiomatizes the one-variable fragment of predicate S4, it is natural to try to generalize the Segerberg--Maksimova theorem to this setting. We discuss several results in this direction. We establish that this theorem naturally extends to a family of subvarieties of MS4 containing, in particular, S4_u (S4 with a universal modality). On the other hand, we provide a translation of varieties of S5_2-algebras into varieties of MS4-algebras of depth 2 which preserves and reflects local finiteness, demonstrating that the problem of characterizing locally finite varieties of MS4-algebras is at least as hard as the corresponding problem for S5_2 (the bimodal logic of two unrelated S5 modalities), a wide-open problem. Finally, we discuss another natural subvariety of MS4 obtained by asserting a monadic analogue of Casari's predicate formula; this subvariety plays a role in obtaining a faithful provability interpretation of monadic intuitionistic predicate logic, and is expected to have a more manageable characterization of local finiteness. Local finiteness in varieties of MS4-algebras
Feb. 27, 2024 2:30pm (MATH 3…	Lie Theory Farid Aliniaeifard (UBC) X We define vertex-colourings for edge-coloured digraphs, which unify the theory of -partitions and proper vertex-colourings of graphs. Furthermore, we use our vertex-colourings to define generalized chromatic functions, which merge the chromatic symmetric and quasisymmetric functions and generating functions of -partitions. We also discuss the relations between generalized chromatic functions, Schur functions in noncommuting variables, and the well-known Stanley-Stembridge (3+1)-free conjecture. Generalized Chromatic Functions Sponsored by the Meyer Fund
Feb. 27, 2024 3:30pm (MATH 3…	Topology Courtney Hauf X This is look at operads and the definition of a permutative category. Operads and Permutative Categories
Feb. 27, 2024 4:50pm (MATH 3…	Grad Student Seminar Jon Kim (University of Colorado) X Hodge theory provides us with powerful tools for studying the cohomology of smooth manifolds through PDEs. Surprisingly, it has deep connections to complex geometry in studying bidgree differential forms on complex manifolds, specifically complex projective varieties. In this talk, I will introduce Hodge theory on complex manifolds and discuss how these connections from PDEs to complex geometry are made. We will conclude by giving a short proof of the Hodge decomposition via the existence of a Green's operator and showcase some useful consequences. Hodge Theory and The Hodge Decomposition
Feb. 28, 2024 3:30pm (MATH 3…	Math Phys Markus Pflaum (CU Boulder, Mathematics Department) Noncommutative Geometry and the Quantum Hall Effect
Feb. 29, 2024 2pm (Online)	Rep Theory Iván Angiono (Universidad Nacional de Córdoba) View Online:https://cuboulder.zoom.us/j/97927772782 X Due to a celebrated result by Deligne, symmetric tensor categories of moderate growth over (algebraically closed) fields of characteristic zero correspond to categories of representations of affine algebraic supergroups. Once we move to positive characteristic, we need to take into account the Verlinde category Ver_p: Coulembier-Etingof-Ostrik proved recently that every such symmetric tensor category is the one of representations of an algebraic group in Ver_p, under some restrictions. Thus, we wonder how to describe algebraic groups in Ver_p, which in turn correspond to pairs of usual algebraic groups and Lie algebras in Ver_p, as described by Venkatesh. This leads to the question of how to obtain Lie algebras in Ver_p. This talk is based on joint works with J. Plavnik and G. Sanmarco where we look for examples of these Lie algebras. We prove the existence of contragredient Lie algebras in symmetric tensor algebras generalizing Kac-Moody construction of Lie (super) algebras, which at the same time give a description of some examples obtained previously by 'semi simplifying' usual Lie algebras and provide new Lie algebras in Ver_p. Contragredient Lie algebras in symmetric tensor categories.
Feb. 29, 2024 3:35pm (MATH 3…	Probability Jorge Garza Vargas (Caltech) X Classical random matrix theory focuses on the study of highly structured models (e.g. Wigner and Wishart matrices) which are presented as a sequence of random matrices defined for every dimension, whose asymptotic (i.e. as the dimension goes to infinity) spectral properties must be understood in detail. However, modern problems in data and computer science require only a coarser understanding of the random matrices in question, but necessitate nonasymptotic results in settings where the models are less structured and do not necessarily belong to a prescribed sequence of matrices. In this work we provide tools for analyzing the spectral distribution of any self-adjoint noncommutative polynomial evaluated in arbitrary independent unitarily invariant Hermitian random matrices of a fixed dimension. Our results can be interpreted as a quantitative version of Voiculescu's celebrated asymptotic freeness result. This is joint work with Chi-Fang Chen and Joel Tropp. Non-asymptotic Random Matrix Theory for Unitarily Invariant Matrices
Mar. 05, 2024 10:10am (MATH …	Geometry/Analysis Ian Miller (CU Boulder) X The magnetic nonlinear Schrödinger (mNLS) equation is a variant of the well-studied nonlinear Schrödinger equation. In mNLS, identities involving motion are less well-behaved and cause difficulty for establishing blow-up results. Blow-up results have been obtained for mNLS with constant magnetic field. These results relied on the specific structure and symmetry of these potentials. In this talk we establish existence of blow-up solutions for a class of potentials with no prescribed symmetry. Blow-up results for the nonlinear Schrödinger equations with trapping magnetic potentials
Mar. 05, 2024 2:30pm (MATH 3…	Lie Theory Florencia Orosz Hunziker (University of Colorado Boulder) X Modular forms played an important role in the proof (Borcherds 1992) of the Monstrous Moonshine conjecture providing the first fundamental bridge between number theory and representation theory through vertex algebras. In this talk, we will give an overview of several ways in which Vertex algebras can be systematically be used to connect representation theory to number theory. Connections between Representation Theory and Number Theory through vertex algebras
Mar. 05, 2024 3:30pm (MATH 3…	Topology Alexander LaJeunesse X In this talk I will introduce monads and their algebras. Following a discussion of the relationship between monads and adjunctions, we will discover how every operad gives rise to a monad, and with this in hand we will spend some time trying to understand May’s comparison theorem. If time permits, we will conclude by looking at how the monadic bar construction can be used to construct canonical resolutions of algebraic objects. Monads and Operads
Mar. 06, 2024 3:30pm (MATH 3…	Math Phys Ezzeddine El Sai (CU Boulder, Department of Mathematics) Strict Deformation Quantization
Mar. 07, 2024 2pm (Online)	Rep Theory Philip Argyres (University of Cincinnati) View Online:https://cuboulder.zoom.us/j/97927772782 X The local operators of a unitary 4d N=2 SCFT in twisted Schur cohomology form a vertex operator algebra (VOA). By "local operator" we mean one associated with a point in space-time. We show that to every 4d N=2 SCFT there is associated a vertex algebra containing the VOA of local twisted Schur operators as a proper subalgebra. The new vertex operators of this larger vertex algebra are associated with certain extended operators (line, surface, etc.) in twisted Schur cohomology. Though we can compute some partial results in simple SCFTs, the structure of these extended vertex algebras is still largely mysterious. The extended vertex algebra of 4-dimensional N=2 superconformal field theories
Mar. 08, 2024 9:05am (Zoom)	Math Edu Andrew Meier & Harrison Stalvey (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/91244797970 X Although we value active learning in the form of student-led activities, we should still make an effort to utilize active learning techniques when we lecture. In this talk we will share strategies to create an engaging and active lecture, and provide examples of ways we can capitalize on students’ engagement through various styles of questioning in a lecture environment. Lecturing for Engagement
Mar. 12, 2024 2:25pm (online)	Algebra/Logic Stefano Fioravanti (Charles University Prague) View Online:https://cuboulder.zoom.us/j/96896272260 X We investigate some applications of a certain technique (that we call Freese's technique), which is a tool for identifying certain lattices as sublattices of the congruence lattice of a given algebra. In particular we gave sufficient conditions for two families of lattices (called the rods and the snakes) to be admissible as sublattices of a variety generated by a given algebra, extending a result of R. Freese and P. Lipparini. On some admissible sublattices of a congruence lattice
Mar. 12, 2024 2:30pm (MATH 3…	Lie Theory Aram Bingham (Colorado School of Mines) X Given a reductive algebraic group G with Borel subgroup B, a spherical variety X is a G-variety such that B acts with finitely many orbits. Well-known examples include toric varieties, flag varieties, symmetric varieties and their embeddings, and so-called “wonderful varieties” which play an important role in the classification of spherical varieties. In many cases, Borel subgroup orbits can be combinatorially parametrized and equipped with poset structure induced by the closure-containment (or Bruhat) order. Shellability is a much-studied property in poset topology which implies that a simplicial complex associated to a poset has the homotopy type of a wedge of spheres. It has been known since the 1980s that Bruhat orders of flag varieties are shellable, and these results have been extended to other related settings more recently. We will discuss some results, open questions, and hunches relating these two areas, based on joint work with Néstor Díaz. SOME SPECULATIONS ON SHELLABILITY AND SPHERICAL VARIETIES Sponsored by the Meyer Fund
Mar. 13, 2024 3:30pm (MATH 3…	Topology J. Peter May (University of Chicago) View Online:https://cuboulder.zoom.us/j/94002553301 (Passcode: 790356) X I defined operads in 1971, and they are ubiquitous. I defined ring spaces and ring spectra soon afterwords. The latter are the starting point of derived algebraic geometry. That was a half century ago. All of the old mathematics seems to be correct. However, answering a question Agnes Beaudry asked me in 2023 has led me to the conclusion that this idiot got the definition of ring spaces conceptually wrong. I will explain what I now think is the right way to think about ring theory operadically. The new theory is very elementary and should also have been understood a half century ago. It leads to an operadic definition of bipermutative categories that generalizes equivariantly and leads in turn to a concrete construction of commutative ring -spectra in equivariant algebraic -theory. The talk will focus on the elementary combinatorial foundations. The new theory also leads to many open questions and new directions that I may mention. Operads, Moperads, and Bioperads
Mar. 13, 2024 3:30pm (Math350)	Math Phys J. Peter May (University of Chicago) View Online:https://cuboulder.zoom.us/j/94002553301 (Passcode: 790356) X I defined operads in 1971, and they are ubiquitous. I defined ring spaces and ring spectra soon afterwords. The latter are the starting point of derived algebraic geometry. That was a half century ago. All of the old mathematics seems to be correct. However, answering a question Agnes Beaudry asked me in 2023 has led me to the conclusion that this idiot got the definition of ring spaces conceptually wrong. I will explain what I now think is the right way to think about ring theory operadically. The new theory is very elementary and should also have been understood a half century ago. It leads to an operadic definition of bipermutative categories that generalizes equivariantly and leads in turn to a concrete construction of commutative ring -spectra in equivariant algebraic -theory. The talk will focus on the elementary combinatorial foundations. The new theory also leads to many open questions and new directions that I may mention. Operads, Moperads, and Bioperads
Mar. 14, 2024 9am (Online)	Rep Theory Anne Moreau (Paris-Saclay) View Online:https://cuboulder.zoom.us/j/97927772782 On a series of simple affine VOAs at non-admissible level arising from rank one 4D SCFTs
Mar. 14, 2024 4:40pm (MATH 3…	Grad Student Seminar Nicholas Christoffersen (University of Colorado) TBD
Mar. 15, 2024 10am (KOBL 342)	Thesis Defenses Erik Knutsen (CU Boulder) PhD Thesis Defense
Mar. 19, 2024 12pm (Math 350)	Diversity Diversity Committee and Math for All (CU BOULDER) X The Diversity Committee and the Math For All Team are co-organizing a panel on the basics of academic job searches. Whether you are applying for jobs soon or in a few years, we think this information will be helpful. After the panelists share some words of wisdom on certain topics we will have a Q&A session. The panelists and their topics are: Agnès Beaudry (Research Statement Basics). Lee Roberson (Teaching Statement Basics and Teaching Portfolios). Alonso Delfín Ares de Parga (Tips on Searching for a Research-Focused Postdoc position). Rebekah Jones (Tips on Searching for a Teaching-Focused Postdoc position). Academic Job Panel
Mar. 19, 2024 2:30pm (MATH 3…	Lie Theory Dana Ernst (Northern Arizona University) X In this talk, we will discuss the architecture of braid graphs in simply-laced Coxeter systems. It turns out that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, each braid graph is a median graph (i.e., for every triple of vertices, there is a unique vertex, called the median, that belongs to shortest paths between each pair). One consequence of this result is that every braid graph can be isometrically embedded into a hypercube. For a special class of reduced expressions, called Fibonacci links, we show that the corresponding braid graphs are Fibonacci cubes. Braid graphs in simply-laced triangle-free Coxeter systems are median Sponsored by the Meyer Fund
Mar. 19, 2024 3:30pm (MATH 3…	Topology Andrew Doumont X We introduce Symplectic Manifolds and some of their fundamental techniques of study with the goal of proving Gromov's Non-squeezing Theorem. Sympletic Topology
Mar. 20, 2024 3:30pm (MATH 3…	Math Phys Robert Maier (University of Arizona and CU Boulder) View Online:https://cuboulder.zoom.us/j/94002553301 (Passcode: 790356) X Single-boson creation and annihilation operators satisfy a canonical commutation relation and generate the Weyl algebra. It is also generated by the noncommuting operators x and D (the derivative with respect to x), acting on any suitable space of functions. An ordering identity in this algebra can be viewed as an element of a two-sided ideal in the free associative algebra generated by x and D. An example of an identity is the one that rewrites a word (a specified product of x's and D's) as a linear combination of normally ordered words: ones in which D's appear to the right of x's. In the relatively easy `single annihilator' case, the coefficients are Stirling numbers, with a combinatorial interpretation. But ordering identities can be far more complicated. We introduce triangles of recursively defined generalized Stirling and Eulerian numbers, and ordering identities in which they appear as coefficients. Some of these number triangles are related by a `binomial transform' operation, but most have no known combinatorial interpretation. Combinatorial Identities in the Weyl Algebra
Mar. 21, 2024 11am (Zoom)	Algebra/Logic Mikhail V. Volkov (Ural Federal University) View Online:https://cuboulder.zoom.us/j/96896272260 X We exhibit several finite monoids each of which is not contained in any variety generated by a finitely based finite semigroup but is contained in a finitely based locally finite variety. The smallest monoid with these properties we found so far has as few as 9 elements. The result solves a 20+ years old problem and has numerous implications for the finite basis problem for finite semigroups. This is joint work with Sergey Gusev (Ekaterinburg, Russia) and Olga Sapir (Beersheba, Israel). Strongly nonfinitely based monoids
Mar. 21, 2024 2pm (Online)	Rep Theory Matthew Rupert (University of Saskatchewan) View Online:https://cuboulder.zoom.us/j/97927772782 X Quantum vertex algebras are deformations of vertex algebras introduced by Etingof and Kazhdan in 1998. They are families of vertex operators with relations deformed by a solution of the quantum Yang-Baxter equation. There is also a dual notion of quantum vertex coalgebras which deform vertex coalgebras. In this talk I will explain how to construct two distinct structures of a quantum vertex coalgebra on the Yangian associated to any simple finite-dimensional complex Lie algebra, and how to induce quantum vertex algebra structures on the dual Yangian. This talk is based on ongoing work with Alex Weekes and Curtis Wendlandt. Constructing Quantum Vertex (Co)-Algebras associated to Yangians
Mar. 28, 2024 2pm (Online)	Rep Theory David Ridout (University of Melbourne) View Online:https://cuboulder.zoom.us/j/97927772782 X Quantum hamiltonian reduction refers to a collection of functors that map the module category of a given affine vertex algebra to those of its associated W-algebras. Some of these functors are reasonably well understood and then the representation theory of the W-algebra is accessible. But some are not. Inverse quantum hamiltonian reduction is a recent discovery that there (sometimes) exist functors in the opposite direction: from a given W-algebra module category to that of another W-algebra, which may be the affine vertex algebra itself. I will give an overview of the simplest example, which connects the module categories of the Virasoro and sl2 minimal model vertex operator algebras. Inverse quantum hamiltonian reduction for beginners.
Apr. 02, 2024 1:25pm (online)	Algebra/Logic Will Brian (University of North Carolina at Charlotte) View Online:https://cuboulder.zoom.us/j/96896272260 X I'm going to talk about an old question of van Douwen: Are the shift map and its inverse conjugate in the automorphism group of P()/fin? By the late 1980's, van Douwen and Shelah proved that it is consistent they are not conjugate. Specifically, any automorphism witnessing their conjugacy would need to be nontrivial (van Douwen), but it is consistent that all automorphisms are trivial (Shelah). In this talk I'm going to discuss the recently-proven complementary result: it is consistent that the shift map and its inverse are conjugate and, in fact, it follows from CH. Does P()/fin know its right hand from its left?
Apr. 02, 2024 2:30pm (MATH 3…	Lie Theory Carl Mautner (UC Riverside) X The Schur algebra is a finite-dimensional algebra that relates the representation theory of symmetric and general linear groups. In joint work with Tom Braden we introduce a new algebra we call the Hilbert Schur algebra, which is also finite-dimensional and has the Schur algebra as a quotient. I will describe this algebra, our motivation for defining it, and some of its interesting algebraic and combinatorial properties. Hilbert Schur algebras Sponsored by the Meyer Fund
Apr. 02, 2024 3:30pm (MATH 3…	Topology Emily Montelius View Online:https://cuboulder.zoom.us/j/93245421661 X Together we will watch and discuss some introductory and overview materials covering knot theory from a graduate topology level An introduction to knot theory
Apr. 02, 2024 10:10pm (MATH …	Functional Analysis Rufus Willett (University of Hawaii) X The HK conjecture predicts an isomorphism between the homology of certain groupoids, and the K-theory of their C-algebras. On the other hand, the Baum-Connes conjecture predicts a topological computation of the K-theory of groupoid C-algebras. Both conjectures are wrong: there are counterexamples to the HK conjecture due to Scarparo, Deeley, Ortega-Scarparo, and Chaiser; and to the Baum-Connes conjecture due to Higson-Lafforgue-Skandalis. I will explain some interactions between these two conjectures, in particular focusing on the precise sense in which Baum-Connes predicts that HK is wrong, and how one might try to fix these issues. I will not assume any prior knowledge of the HK or Baum-Connes conjecture, and will focus on the case of group actions rather than general groupoids (for simplicity). Some of this is based on joint work with Christian Bönicke, Clément Dell'Aiera, and Jamie Gabe; some on joint work with Robin Deeley; and some on work of Valerio Proietti and Makoto Yamashita (that I had nothing to do with). The HK and Baum-Connes conjectures Sponsored by the National Science Foundation
Apr. 03, 2024 12:20pm (Zoom)	Math Edu Ally Graf & Patrick Newberry (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/91244797970 X Although we assess student knowledge through various methods, one common evaluation technique we use in our courses is a formal exam. In this talk we will share strategies to create an exam that showcases students' understanding of the course material, as well as how to reflect on what went well and what did not following the exam. Creating Exam Questions
Apr. 03, 2024 3:30pm (MATH 3…	Math Phys Rebecah Storms (CU Boulder, Department of Mathematics) View Online:https://cuboulder.zoom.us/j/94002553301 (Password: 790356) X I will give a brief refresh of jet bundles for those who are unfamiliar, and then I will formulate the Einstein Field Equation in this language. General Relativity
Apr. 04, 2024 9:15am (via zo…	Math Phys Albachiara Cogo (University of Tübingen) View Online:https://cuboulder.zoom.us/j/94002553301 X In this joint work with Cederbaum, Leandro and Dos Santos, we generalize to any dimension n+1 Robinson’s divergence formula used to prove the uniqueness of (3+1)-dimensional static black holes. To this end, we use a tensor first introduced by Cao and Chen for the analysis and classification of Ricci solitons. We thereby prove the uniqueness of black holes and of equipotential photon surfaces in the class of asymptotically flat (n+1)-dimensional static vacuum space-times, provided the total scalar curvature of the horizon is properly bounded from above. In the black hole case, our results recover those of Agostiniani and Mazzieri and partially re-establish the results by Gibbons, Ida, and Shiromizu, and Hwang and finally by Raulot in the case of a spin manifold; in the photon surface case, the results by Cederbaum and Galloway can also be proven. Our proof is not based on the positive mass theorem and avoids the spin assumption. Black Hole and Equipotential Photon Surface Uniqueness in (n+1)-dimensional Static Vacuum Spacetimes via Robinson's Method
Apr. 04, 2024 2pm (Online)	Rep Theory Christopher Raymond (Université Laval) View Online:https://cuboulder.zoom.us/j/97927772782 X In this talk, I want to give a pedagogical overview of Zhu's algebras and then describe an alternative approach to understanding their algebraic structure that has resurfaced in 3D-2D correspondences. First, I will introduce the approach to Zhu's algebras in physics and the resulting calculational framework. Following that, I'll describe the link between Zhu's algebras and Yangians and briefly discuss how these links are realised in physics if time permits. Zhu's algebras and related structures.
Apr. 09, 2024 3:30pm (MATH 3…	Topology Emily Montelius X In this talk I will construct the map between Algebraic and Topological K-Theory for Banach Algebras. We will complete this construction by following our Banach algebra through its general linear group construction, taking Milnor's classifying space, and then applying the plus construction. We will show that this construction agrees with the traditional definition of these K-Theories, while also returning maps between them. The comparison map between Algebraic and Topological K-Theory for Banach Algebras
Apr. 10, 2024 3:30pm (MATH 3…	Math Phys Connor McCranie (CU Boulder, Department of Mathematics) Hyperfunctions and Quantum Field Theory
Apr. 12, 2024 9:05am (Zoom)	Math Edu Joseph Timmer (University of Colorado Boulder) View Online:https://cuboulder.zoom.us/j/91244797970 X Using a rubric can make grading go smoothly and make consistency easier to attain. Creating a good rubric typically involves more than just writing out a solution to the problem. In this talk, we discuss what makes for good rubrics that ease the burden of grading and address the pros/cons of various schemes. If time permits, we will create a rubric and see these ideas in action! Creating and Adhering to Rubrics
Apr. 16, 2024 1:25pm (MATH 2…	Algebra/Logic Piotr Kawalek (TU Wien, Austria) View Online:https://cuboulder.zoom.us/j/96896272260 X In 1999, Goldmann and Russell initiated a systematic study of algorithmic complexity of equation satisfiability problems in finite groups. In this talk we present some of the most important historical results on the topic as well as some of the very recent progress. We also consider closely connected problems of satisfiability and equivalence of circuits in finite algebras from congruence modular varieties. Some of the most general results in this area were obtained through relating functions computable by short polynomials in these algebras with classes of bounded-depth modular circuits. We discuss these types of connections, which turn out to be especially useful in tractability proofs. Complexity of Satisfiability and Equivalence Problems in Finite Algebras Sponsored by the Meyer Fund
Apr. 16, 2024 3:30pm (MATH 3…	Topology Ralph Kaufmann (Purdue-University) View Online:https://cuboulder.zoom.us/j/94002553301 X We review the setup of topological invariants as they appear in physics in the interpretation of K-theory. The basic Z-valued invariant can be identified with the first Chern-class. This goes back to the TKNN integers appearing in the quantum Hall effect. More subtle invariants appear in the setting of symmetries. These can be identified as coming from Real K-theory (KR) theory and its shifted cousin, quaternionic K-theory (KQ). We will give a gentle introduction and present the mathematical and condensed matter classification. This is joint work with D. Li and E.B Kaufmann Topological Insulators and K-theory Sponsored by the Meyer Fund
Apr. 17, 2024 3:30pm (MATH 3…	Math Phys Birgit Kaufmann (Purdue-University) View Online:https://cuboulder.zoom.us/j/94002553301 X Starting from a periodic system, one can by means of Bloch theory or discretisation form an effective geometry. In the standard case, this is a bundle over the Brillouin torus, whose sections are the states. More generally, one can consider families of Hamiltonians parameterized over a base. Such geometries can be thought of as classical or C* geometries. In the latter form, they appear via quiver representations. We will present the background and tools for the analysis of these geometries. This is joint work with R. Kaufmann and S. Khlebnikov Effective geometries and their local and global invariants. Sponsored by the Meyer Fund
Apr. 18, 2024 2pm (Online)	Rep Theory Mamoru Ueda (University of Alberta) View Online:https://cuboulder.zoom.us/j/97927772782 X Creutzig-Diaconescu-Ma conjectured that there exists a homomorphism from the shifted affine Yangian to the universal enveloping algebra of an iterated W-algebra. They also conjectured that this homomorphism will induce a resolution of the generalized AGT conjecture. In this talk, I will explain how to construct a homomorphism from the affine Yangian of type A to the universal enveloping algebra of a W-algebra including the non-rectangular W-algebra. I expect that this homomorphism can be extended to the shifted affine Yangian. Affine Yangians and W-algebras
Apr. 18, 2024 3:35pm (MATH 3…	Probability Andrew Campbell (ISTA) X A universality of local eigenvalue statistics of Hermitian random matrices has been the subject of immense investigation over the past couple decades resulting in the now well established principle that local eigenvalue statistics are determined only by the symmetry class of the matrix (i.e. real symmetric vs complex Hermitian) and the shape of the local eigenvalue density. For example, generically at the edge of the spectrum the eigenvalue density of random matrices exhibits square-root decay, leading to the local eigenvalue statistics falling into the famous Tracy--Widom universality class. For non-Hermitian matrices substantially less is known, with full universality in the bulk and at the edge for a random matrix X with centered independent identically distributed entries (or IID matrix) established only quite recently. While this is quite a substantial accomplishment, it is nowhere near the generality available for Hermitian matrices. In this talk we will consider the edge statistics of a deformed IID matrix A+X, where A is some deterministic deformation of the IID matrix X. By using matrices with Gaussian entries as a middle step for comparison we will see that this model has implications for many important non-Hermitian random matrix models. This is based on joint work with Giorgio Cipolloni, Laszlo Erdos, and Hong Chang Ji. At the edge of deformed non-Hermitian matrices.
Apr. 19, 2024 2:30pm (MATH 3…	Geometry/Analysis Leandro Lichtenfelz (Wake Forest University) X The Euler equations of hydrodynamics, which model ideal fluid flows on some region , admit a beautiful description as geodesic equations on the group of volume-preserving diffeomorphisms of , as first shown by V. Arnold. A major difficulty in studying the geometry of this group is the fact that it is infinite-dimensional, so a natural idea is to try to approximate it in some sense by a finite-dimensional model. One such model was constructed by Zeitlin, based on a quantization scheme developed earlier by Hoppe. In this talk, I will give an overview of Zeitlin's model, and then describe some aspects related to its geometry, in particular conjugate points. This is joint work with Stephen Preston and Alice Le Brigant. Conjugate points on Lie groups and Zeitlin's model of hydrodynamics Sponsored by the Meyer Fund
Apr. 23, 2024 12pm (MATH 350)	Topology Topology Day : Kyle Ormsby (Reed College) X Model structures underpin the modern enterprise of abstract homotopy theory and provide a convenient framework in which we can pass from one notion of sameness (isomorphism) to a weaker one (weak equivalence). Despite their fundamental nature, model structures have historically been studied en masse or applied in specific cases, and very little is known about the totality of model structures on a given category. Homotopical combinatorics is an emerging field that remedies this situation by studying the enumerative combinatorics and structural properties of model structures on finite lattices. Specialized to a finite chain, we find rich connections with Catalan combinatorics, including (intervals in) the Tamari and Kreweras lattices. I will introduce the audience to model structures and sketch homotopical combinatorics as it currently stands, including the surprising assist its development received from equivariant infinite loop space theory. Counting model structures Sponsored by the Meyer Fund
Apr. 23, 2024 1pm (MATH 220)	Probability David Renfrew (Binghamton University) X We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up. Singularities in the spectrum of random block matrices
CANCELED Apr. 23, 2024	Algebra/Logic Paolo Marimon (TU Wien, Austria) View Online:https://cuboulder.zoom.us/j/96896272260 X We classify the operations which are minimal above the clone generated by a permutation group G acting on a set. Rosenberg (1986) showed that when G is trivial there are only five types of minimal operations. We show that for most permutation groups there are only three types of minimal operations. More generally, we carry such classification for any permutation group and specify the behaviour of the minimal operations on tuples with two or more elements in the same orbit. Our results are especially helpful for the study of constraint satisfaction problems on omega-categorical structures, where we improve and generalise results of Bodirsky and Chen (2007), and answer some questions of Bodirsky (2021). This is joint work with Michael Pinsker. Minimal operations over permutation groups
Apr. 23, 2024 2:30pm (MATH 3…	Lie Theory Thomas Creutzig (University of Alberta) X An affine Lie algebra is a central extension of the loop algebra of a finite dimensional Lie algebra. Certain modules, the vacuum modules at any complex level k, of the affine Lie algebra carry themselves an interesting algebraic structure, namely that of a vertex operator algebra (VOA). The representation theory of affine VOAs is rather rich, e.g. suitable categories of modules form ribbon categories and there are exciting connections to geometry, quantum groups, physics and much more. I want to give an overview on the state of the art of this area. Representation Theory of Affine VOAs Sponsored by the Meyer Fund
Apr. 23, 2024 3:30pm (MATH 3…	Topology Topology Day: Ping Xu (Penn State) X Dg manifolds are a useful geometric notion which unifies many important structures such as homotopy Lie algebras, foliations and complex manifolds. In this talk, we describe a Duflo-Kontsevich type theorem for dg manifolds. The Duflo theorem of Lie theory and the Kontsevich theorem regarding the Hoschschild cohomology of complex manifolds can both be derived as special cases of this Duflo--Kontsevich type theorem for dg manifolds. This is a joint work with Hsuan-Yi Liao and Mathieu Stienon. Duflo-Kontsevich type theorem for dg manifolds
Apr. 25, 2024 11:15am (MATH …	Algebraic Geometry Daniel Bragg (Utah) X We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich. A Stacky Murphy’s Law for the Stack of Curves Sponsored by the Meyer Fund
Apr. 25, 2024 12:20pm (Math …	Noncomm Geometry Dmitry Kaledin (Steklov Math Institute) X It has become accepted wisdom by now that when you localize a category with respect to a class of morphisms, what you get is not just a category but a category "with a homotopical enhancement". Typically, the latter is made precise through the machinery of "infinity-categories", or "quasicategories", but this is quite heavy technically and not really optimal from the conceptual point of view. I am going to sketch an alternative technique based on Grothendieck's idea of a "derivator". How to enhance categories, and why Sponsored by the Meyer Fund
Apr. 25, 2024 12:20pm (MATH …	Algebraic Geometry Dmitry Kaledin (Steklov Math Institute) X It has become accepted wisdom by now that when you localize a category with respect to a class of morphisms, what you get is not just a category but a category "with a homotopical enhancement". Typically, the latter is made precise through the machinery of "infinity-categories", or "quasicategories", but this is quite heavy technically and not really optimal from the conceptual point of view. I am going to sketch an alternative technique based on Grothendieck's idea of a "derivator". How to enhance categories, and why (Cross listed with Non-commutative Geometry) Sponsored by the Meyer Fund
Apr. 30, 2024 1:25pm (online)	Algebra/Logic Guenter Pilz (JKU Linz, Austria) View Online:https://cuboulder.zoom.us/j/96896272260 X Algebraists like decompositions, for doing parallel computing, etc. Some decompositions are very "successful", some others are not. We try to find the "reasons of sucess" by looking at various instants, including decompositions of semisimple rings. We also get "Generalized Discrete Fourier Transforms" for group rings and related structures. Finally, we try to solve equations of degree >4. This is joint work with Manuel Kauers. Which (sub)direct decompositions are "useful"?
Apr. 30, 2024 3:30pm (MATH 3…	Topology Sarah Petersen X This talk will describe some of the history and background of one of the oldest (now solved) problems in algebraic topology. An Introduction to the Kervaire Invariant Problem
May. 01, 2024 3:30pm (MATH 3…	Math Phys Howy Jordan (CU Boulder, Department of Mathematics) Infinite Dimensional Control Theory
May. 02, 2024 11:15am (MATH …	Algebraic Geometry Luca Schaffler (Universita Roma Tre) X Given the moduli space of hyperplanes in projective space, V. Alexeev constructed a family of compactifications parametrizing stable hyperplane arrangements with respect to given weights. In particular, there is a toric compactification that generalizes the Losev–Manin compactification for the moduli of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. In particular, we prove that in dimensions two the wall crossing corresponds to blowing up at the identity of the dense torus of the generalized Losev–Manin space. As an application, we show that any Q-factorialization of this blow-up is not a Mori dream space for a sufficiently high number of lines. This is joint work in progress with Patricio Gallardo. An explicit wall crossing for the moduli space of hyperplane arrangements Sponsored by the Meyer Fund
May. 02, 2024 12:15pm (Math …	Algebraic Geometry Farbod Shokrieh (University of Washington) X I will describe some connections between arithmetic of abelian varieties, non-archimedean/tropical geometry, and combinatorics. For example, we give formulas for (non-archimedean) canonical local heights in terms of tropical geometry. Our formulas extend classical computations of local height functions due to Néron (involving Néron models) and, in the case of elliptic curves, due to Tate (involving Bernoulli polynomials). Based on joint work with Robin de Jong. Canonical local heights and tropical theta functions Sponsored by the Simons Foundation
May. 02, 2024 3:35pm (MATH 3…	Probability Vishesh Jain (University of Illinois at Chicago) X Markov chains provide a natural approach to sample from various distributions on the independent sets of a graph. For the uniform distribution on independent sets of a given size in a graph, perhaps the most natural Markov chain is the so-called ``down-up walk''. The down-up walk, which essentially goes back to the foundational work of Metropolis, Rosenbluth, Rosenbluth, Teller and Teller on the Markov Chain Monte Carlo method, starts at an arbitrary independent set of size , and in every step, removes an element uniformly at random and adds a uniformly random legal choice. Davies and Perkins showed that there is a critical such that it is hard to (approximately) sample from the uniform distribution on independent sets for the class of graphs with vertices and maximum degree at most . They conjectured that for below this critical value, the down-up walk mixes in polynomial time. I will discuss a resolution of this conjecture, which additionally shows that the down-up walk mixes in (optimal) time . Based on joint work with Marcus Michelen, Huy Tuan Pham, and Thuy-Duong Vuong. Optimal mixing of the down-up walk on fixed-size independent sets
May. 07, 2024 10am (MATH 350)	Lie Theory Lucas Gagnon (York University) X A long-standing conjecture of Stanley and Stembridge asks whether certain symmetric function invariants of graphs are “e-positive,” i.e. expand positively in the basis of elementary symmetric functions. Efforts to resolve Stanley—Stembridge have raised the wider question of whether other symmetric functions are e-positive, particularly for symmetric invariants of other interesting objects. This talk will describe an instance of e-positivity arising in the study of modules of the unipotent upper triangular groups over a finite field. I use a collection of novel module homomorphisms to prove e-positivity, and these maps implicate a well-known algorithm from the study of Iwahori—Hecke algebras. Throughout, I will connect these results to previously known connections between the unipotent upper triangular group and symmetric functions in non commuting variables. Positivity in unipotent symmetric function invariants and Hecke traces Sponsored by the Meyer Fund
May. 07, 2024 12:10pm (MATH …	Kempner Luis David Garcia Puente (Colorado College) X Let G be a finite, connected graph. An arithmetical structure on G is an assignment of positive integers to the vertices such that, at each vertex, its label divides the sum of the labels at adjacent vertices (counted w/ multiplicity). Arithmetical structures naturally generalize the notion of the Laplacian matrix of a graph. They also appear in the context of arithmetical geometry. In this talk, I will present a general introduction to the topic, we will focus on the combinatorics of arithmetical structures on certain graph families including some counting results that relate arithmetical structures on a path graph and Catalan numbers. We will also discuss some results about the associated finite Abelian group of an arithmetical structure. What is an arithmetical structure? Sponsored by the Meyer Fund

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