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Events for April – August 2022

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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
Jun. 08, 2022 11am (Math 220)	Number Theory Uri Shapira (Technion) X * HYBRID: Speaker will be on zoom. Join by zoom or join the watch party in Math 220 * https://cuboulder.zoom.us/j/91928134679 The passcode is the j-invariant of the elliptic curve with CM by the Gaussians (or email Kate). Abstract: In the last three and a half decades, starting with the pioneering solution of the Oppenheim conjecture on the values of quadratic forms by Margulis, the interplay between the ergodic theory of homogeneous spaces and number theory solidified. Moreover, exciting new connections between the fields keep arising even today. The aim of my talk will be to demonstrate this with a circle of problems which might be best described as problems in the "geometry of integral vectors". To each integral vector v in Z^d one can associate several natural geometric and arithmetic objects. For example: (1) The "direction" of v, namely its radial projection to the unit sphere. (2) The residue class of v modulo a fixed integer k. (3) The "orthogonal lattice" to v, defined by intersecting Z^d with the hyperplane orthogonal to v. Each of these objects resides in a natural probability space which allows one to ask statistical questions as the integral vector varies. I will survey classical and recent results describing the limit statistics of these objects as the integral vector varies in certain sets. For example: (1) A ball of radius T as T goes to infinity (Schmidt). (2) A sphere of radius T as T goes to infinity (Linnik/Duke/Aka - Einsiedler - S) (3) The integral points on certain non-compact quadratic surfaces (S - Sargent). (4) The integral points approximating an irrational line in R^d (S - Weiss). Focussing on one of the above examples, I will try to explain how the corresponding problem can be casted as a question in homogeneous dynamics. Applying ergodic theory to number theory
Jul. 11, 2022 5pm (CASE E-340)	Topology Inna Zakharevich (Cornell University) X Like many mathematicians, I enjoy analyzing how my favorite things work (often to extreme degrees). In this talk we'll play around with some of the mathematics of origami. How can we fold paper? More fundamentally, what does it even mean to fold it? And once it's folded, what questions can we ask? In this talk we'll mess around with solving polynomials, the difference between local and global properties, and maybe even get to (flat-)fold some paper. The Mathematics of Origami Sponsored by the National Science Foundation
Jul. 12, 2022 5pm (CASE E-340)	Topology John Lind (California State University Chico) X I will present a new type of generative music based on the "abelian sandpile model" - a discretized model for the flow of heat built using rich combinatorial structures. The music is created by encoding the mathematics into an algorithm which then plays synthesizers according to the rules laid out by the dynamical system. I will explain how this works, and we will listen to examples. As we will see, the mathematics behind the music is regulated by a discrete form of the Laplacian operator and an analogy between graphs and Riemann surfaces. Abelian Sandpile Music Sponsored by the National Science Foundation
Jul. 14, 2022 5pm (CASE E-340)	Topology Jonathan Campbell (Center for Communications Research, La Jolla) X In any given dimension, there are a finite number of possible crystal structures that can exist. In this talk I'll show how group cohomology can be used to classify such crystal structures (and I'll define group cohomology!) We will work out some of the classification for 2-dimensional crystal structures, the so-called "wall paper" groups. Group Cohomology and Crystals Sponsored by the National Science Foundation
Aug. 30, 2022 3:30pm (MATH 3…	Topology Howy Jordan (CU Boulder) X The open sets of a topology space have a nice algebraic structure, and it turns out that much of classical topology can be formulated in terms of this structure without reference to any underlying set of points. Like many such algebraic lattice structures, we can regard it as a system of propositions in logic with meet (intersection) and join (union) in the lattice corresponding to logical "and" and to logical "or". From this perspective, a topological space is a propositional theory with models of the theory given by the points of the space. With suitable separation properties, the models are entirely determined by the open sets. In many familiar cases, we can give the lattice of opens first and then derive the points from this. In general however, such a lattice might have no points at all. In logical terms, some propositional theories can be consistent but still fail to have any set-based models. Such generalized lattices of opens are often called locales, and the corresponding theories are those of geometric propositional logic. In this talk, we'll consider the fundamental relationship between locales and traditional topology, discuss why one might be interested in point-free topology, and consider examples of consistent theories which fail to have models in the usual sense. Locales and Geometric Propositional Logic
Aug. 31, 2022 3:20pm (Math35…	Math Phys Daniel Spiegel (CU Boulder) X The theme of this semester’s mathematical physics seminar is quantum information theory. To get us started, I’ll introduce density operators, which play a fundamental role in the theory and serve as a tool for understanding subsystems of a composite quantum system. After defining what a density operator is, I’ll cover some of their basic operations and results, including the partial trace, Schmidt decomposition, and purification. Zoom access: https://cuboulder.zoom.us/j/8063594807 Introduction to Density Operators
Aug. 31, 2022 4:30pm (MATH 3…	Grad Student Seminar Chase Meadors (CU Boulder) X A "Galois Connection" is a formalism that abstracts the relationship between certain algebraic structures I won't mention and automorphism groups studied in classical Galois theory. Galois connections are ubiquitous in Logic, and most notably are used to define the relationship between the "Syntax" and "Semantics" of a system of logic. We will take a look at Galois connections and see several interesting examples of the kind just mentioned (with pictures!) Galois Theory (but I never mention fields)

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