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Events for the weeks of February 22nd – March 6th

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Tue, Feb. 24 12:10pm (MATH …	Kempner Liviu Suciu (The Pennsylvania State University) X I plan to discuss several related arithmetic and analytic dualities governed by the Möbius function and the sawtooth function. While these dualities—whether pointwise (twisted or untwisted), or functional in the L^2 or continuous setting—are formally valid, identifying conditions under which they become genuine identities is a subtle and challenging problem. It draws on deep results in analytic number theory, including Vinogradov-type bounds for exponential sums over primes (Davenport; de la Bretèche–Tenenbaum), as well as Tauberian theory (Wintner; Ingham). Arithmetic and analytic dualities governed by the Mobius and sawtooth functions
Tue, Feb. 24 1:25pm (online)	Algebra/Logic Zoltan Vidnyanszky (ELTE Budapest) View Online:https://cuboulder.zoom.us/j/96896272260 X Descriptive combinatorics is a field concerned with graph theoretic problems on nice (definable) infinite graphs. In this first half of the talk, I will introduce the field of descriptive combinatorics and discuss some of the basic results of the area. Then, I will turn to the descriptive version of complexity problems, in particular, the CSP Dichotomy: it turns out that the finitary complexity landscape is only partially reflected in the descriptive context. CSPs in the descriptive context
Tue, Feb. 24 2:30pm (MATH 3…	Lie Theory Richard Green (CU) X The E8 root system is a highly symmetric configuration of vectors, corresponding to the 240 spheres that touch a given one in the densest possible packing of spheres in 8-dimensional Euclidean space. Despite this eightness, some combinatorial features of the root system are much easier to understand in terms of 9-dimensional coordinates. We will use this point of view to gain insight into a certain tightly structured partially ordered set, by using the combinatorics of perfect matchings and Fano planes. This is based on work in progress with Tianyuan Xu (University of Richmond). The ninefold symmetry of the E8 root system
Wed, Feb. 25 2:10pm (Math350)	Math Phys Howy Jordan (CU Boulder) View Online:https://cuboulder.zoom.us/j/97049173677 The spectral theorem and projection valued measures II
Wed, Feb. 25 4:40pm (MATH 3…	Grad Student Seminar Daniel Lyness (CU Boulder) X Tori are cool. Varieties are dope. Fans are… related somehow? Prepare to dive into the world of Toric Varieties, which are varieties with a torus embedded inside. We’ll see how lots of our favourite examples of varieties are actually toric varieties, and also how a Toric Variety can be defined from the combinatorial data of a fan (the type used in the 18th century on a hot day). I’ll define everything, but having taken AG1 or knowing what Spec(R) is for a ring R will be an advantage. What even is a Toric Variety?
Thu, Feb. 26 11:15am (MATH …	Algebraic Geometry Lior Alon (MIT) X Abstract: The Poisson summation formula tells us that a lattice and its dual lattice are related through Fourier transform: the Fourier transform of the counting measure of one is the counting measure of the other. The search for non-periodic analogues of Poisson summation—discrete sets whose Fourier transform is again a discrete measure—has appeared independently across number theory, harmonic analysis, and crystallography. In the physical setting, the Fourier transform of the counting measure of an atomic structure is exactly the diffraction pattern observed experimentally. The term quasicrystal refers to a non-periodic set whose diffraction is a pure point measure, and the discovery of such materials by Dan Shechtman in 1984 profoundly altered crystallography and led to a Nobel Prize. A major mystery that remained unresolved until very recently was whether one could find a quasicrystal whose Fourier transform is supported on a discrete (rather than merely countable) set. These objects are now called Fourier quasicrystals (FQs), and until 2020 only the trivial periodic examples were known. Kurasov and Sarnak constructed the first genuinely non-periodic one-dimensional FQ using Lee–Yang polynomials, multivariate polynomials whose zeroes avoid prescribed regions of the complex plane. In our recent work, we show that every one-dimensional Fourier quasicrystal arises from the Kurasov–Sarnak construction via Lee–Yang polynomials, yielding a complete classification in dimension one. We then extend this framework to all dimensions by introducing Lee–Yang varieties, high-codimension algebraic varieties whose geometric and analytic properties govern the Fourier structure of the associated quasicrystals. The talk will introduce Fourier quasicrystals, Lee–Yang polynomials, and Lee–Yang varieties, and explain the geometric mechanism of generating summation formulas from algebraic varieties and how it is related to real-rootedness of trigonometric equations. No prior background will be assumed. Joint work with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov. Fourier Quasicrystals and Lee–Yang Varieties
Thu, Feb. 26 3:30pm (Math 3…	Probability Lior Alon (MIT) X Motivated by Berry’s random wave model for chaotic domains, high-energy eigenfunctions are expected, at the scale of the wavelength, to behave like random waves. Accordingly, the size of nodal sets is determined to first order by the Weyl law, while its second-order fluctuations are expected to be universal. In analogy with spectral statistics, we refer to these fluctuations of the nodal set size as nodal statistics. Beginning with work of Berry, and later of Blum–Gnutzmann–Smilansky and Bogomolny–Schmit, it was conjectured that in chaotic systems nodal statistics are asymptotically Gaussian. While striking mathematical results support this picture on the sphere (Nazarov–Sodin), arithmetic symmetries on the torus prevent chaoticity and lead to a breakdown of universality, as shown by Wigman and collaborators and by Marinucci, Rossi, and Peccati. In this talk I will follow Smilansky’s insight and focus on nodal statistics in graph-based models. For quantum graphs and discrete operators on graphs, Berkolaiko’s nodal magnetic theorem relates nodal statistics to the stability of eigenvalues under magnetic perturbations. This connection yields Gaussian limiting statistics for quantum graphs with disjoint cycles (Alon–Band–Berkolaiko), and, combined with Morse inequalities, extends to random discrete operators on finite graphs with disjoint cycles and to complete graphs with strong on-site disorder (Alon–Goresky). I will conclude with recent joint work on nodal statistics for random matrices (Alon–Mikulincer–Urschel), showing that nodal statistics for GOE matrices obey a semicircle law rather than the conjectured Gaussian behaviour. If time permits, I will briefly discuss ongoing work on the Rosenzweig–Porter model, where adding a random on-site potential to a GOE matrix leads to nodal statistics interpolating from semicircle behaviour at low disorder to Gaussian behaviour at strong disorder, suggesting a phase transition between universality classes and a connection to localization–delocalization phenomena. Nodal statistics from quantum graphs to random matrices: universality and phase transitions
Wed, Mar. 4 2:10pm (Math350)	Math Phys Markus J. Pflaum (CU Boulder) View Online:https://cuboulder.zoom.us/j/97049173677 TBA
Wed, Mar. 4 3:30pm (MATH 1…	Diversity Siddhant Agrawal, Robin Deeley, Sarah Petersen, Joseph Timmer (CU Boulder) X From adjunct positions to postdocs to faculty, there are a variety of academic jobs, and the application process for those jobs can be daunting. Join our panelists, Professors Siddhant Agrawal and Robin Deeley, Teaching Professor Joseph Timmer, and Postdoctoral Scholar Sarah Petersen, who will share their insights into navigating the academic job market. Designed for those entering the job market in the near future, this session aims to replace uncertainty with strategy and actionable guidance. Pizza will be served! Academic Job Panel

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