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Events for the weeks of September 25th – October 7th

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Tue, Sep. 27 11am (Math 350)	Number Theory Castryck/Decru (YouTube) X In this late-breaking use of an empty seminar slot :), we will watch some subset of Castryck & Decru's YouTube talks on the subject of their break of SIDH. We'll pause and discuss as needed to sort out the details together. These attacks use genus two isogenies to completely break the SIDH variant of supersingular isogeny based cryptography. Breaking SIDH
Tue, Sep. 27 1:25pm (Zoom)	Logic Keith Kearnes (CU) X I will discuss some properties of varieties which are not Maltsev-definable relative to the class of all varieties, but which are Maltsev-definable relative to the class of varieties having a Taylor term. Relative Maltsev definability
Tue, Sep. 27 2:30pm (MATH 3…	Lie Theory Jay Taylor (University of Manchester) X Complex irreducible characters of finite groups have two main invariants that are used to measure their rationality: their character field, and their Schur indices. In this talk we’ll survey what is known about these invariants and the many questions that remain open. Rationality Questions for Finite Reductive Groups Sponsored by the Meyer Fund
Tue, Sep. 27 3:30pm (MATH 3…	Topology Howy Jordan (CU Boulder) X Fragments of logic can be internalized in appropriately structured categories, and topoi have enough structure to interpret full higher order theories. There are two caveats. First, the internal logic is constructive in general. Second, there are two natural notions of morphisms between topoi and the more common of these, geometric morphisms which act like continuous maps between spaces, only preserve a certain (quite large) fragment of logic. We'll consider in particular how this affects the real numbers. The classical constructions of the naturals, integers, and rationals go through as expected. The reals, however, split into multiple notions. Classically equivalent ways to construct the reals become distinct objects, and these correspond to interesting topologies on the real line. We consider the corresponding objects in a few categories and discuss the relevance on theories of normed algebras in topoi. https://cuboulder.zoom.us/j/97617189995 Internal Logic of Topoi, II
Wed, Sep. 28 4:40pm (MATH 3…	Grad Student Seminar Howy Jordan (CU Boulder) X Given a theory describing some class of mathematical objects (groups, rings, orders, ...) we can construct the sentences one might consider about these, model the symbols as being about an actual set with actual functions, relations, etc, and talk about how manipulating sentences to get proofs relates to truth in the models. Thanks to category theory, we can replace all of this with a lot of abstract nonsense! And who doesn't like abstract nonsense? In this talk we'll introduce some category theory and construct a category representing syntax and provability in a given theory. We'll discuss how this allows us to talk about models as a special case of structure preserving maps. As in the classical case, we get an adjunction (generalized Galois connection) between syntax and semantics. We'll also use this to engage in some shenanigans! Given a non-commutative algebraic structure, we'll pretend its commutative, even recovering classical results about commutative structures in the non-commutative case. We'll talk about how the real numbers become more subtle and interesting, with distinct characterizations giving different structures. We'll even model a theory that describes surjections of N onto R. (Don't worry, the surjections of N onto R aren't real and they can't hurt you.) Categorical Logical Shenanigans
Thu, Sep. 29 3:35pm (MATH 3…	Probability Marcus Michelen (University of Illinois Chicago) X Consider the random symmetric n x n matrix whose entries on and above the diagonal are independent and uniformly chosen from {-1,1}. How often is this matrix singular? A moment's thought reveals that if two rows or columns are equal then the matrix is singular, so the singularity probability is bounded below by 2^{-n(1 + o(1))}. Proving any sort of upper bound on the singularity probability turns out to be difficult, with results coming slowly over the past two decades. I'll discuss a recent work which shows the first exponential upper bound on this probability. Along the way, I'll discuss some tools---both old and new---which are powerful and (hopefully) interesting in their own right. This talk is based on joint work with Marcelo Campos, Matthew Jenssen, and Julian Sahasrabudhe. How often is a random symmetric matrix singular?
Tue, Oct. 4 2:30pm (MATH 3…	Lie Theory Justin Willson (CU) X Green and Xu defined the 2-root lattice of a simply laced Weyl group by looking at pairs of orthogonal roots in the root systems of simply laced Weyl groups with Y-shaped Dynkin diagrams. In this talk we will discuss the existence of elements of squared length 2 in the 2-root lattice in hyperbolic root systems as well as progress made on the conjecture that there are no such elements in the affine and finite cases. Lattice roots arising from 2-root lattices of type ADE
Wed, Oct. 5 4:40pm (MATH 3…	Grad Student Seminar Andrew Campbell (CU Boulder) X You may be surprised to learn that differentiation of polynomials is still an active and even growing area of research. We will look at some recent work of Steinerberger answering the title question for polynomials with real roots, and the surprising connection to eigenvalues of matrices. Time permitting we may discuss the challenges of the complex root case. While there are interesting connections to probability, combinatorics, and operator algebras we will try to keep the vast majority of the talk at the level of calculus, linear algebra, and undergraduate analysis. How do polynomial roots move under repeated differentiation?
Thu, Oct. 6 3:35pm (MATH 3…	Probability Leonid Petrov (University of Virginia) X Imagine two cars, slow (S) and fast (F), driving to the right on a discrete 1-dimensional lattice according to some random walk mechanism, and such that the cars cannot pass each other. We consider two systems, SF and FS, depending on which car is ahead. It is known for some time (through connections to symmetric functions and the RSK correspondence) that if at time 0 the cars are immediate neighbors, the trajectory of the car that is behind is the same (in distribution) in both systems. However, this fact fails when the initial locations of the cars are not immediate neighbors. I will explain how to recover the identity in distribution by suitably randomizing the initial condition in one of the systems. This result arises in our recent work on multiparameter stochastic systems (where the parameters are speeds attached to each car) in which the presence of parameters preserves the quantum integrability. This includes TASEP (totally asymmetric simple exclusion process), its deformations, and stochastic vertex models, which are all integrable through the Yang-Baxter equation (YBE). In the context of car dynamics, we interpret YBEs as Markov operators intertwining the transition semigroups of the dynamics of the processes differing by a parameter swap. We also construct Markov processes on trajectories which "rewrite the history" of the car dynamics, that is, produce an explicit monotone coupling between the trajectories of the systems differing by a parameter swap. Based on a joint work with Axel Saenz. Rewriting History in Integrable Stochastic Particle Systems