The Axiom of Choice, now used throughout non-foundational mathematics with minimal hesitation, has an extremely interesting history and many non-trivial consequences.

First formulated explicitly by Ernst Zermelo to solve the so-called Well-Ordering Problem, this axiom sparked debates throughout the mathematical community. David Hilbert, arguably one of the greatest mathematicians of his time, described Choice as the axiom "most attacked up to the present in the mathematical literature;" and in Gregory Moore's words "if the Axiom's most severe constructivist critics prevailed, mathematics would be reduced to a collection of algorithms."

Modern mathematics, however, severely relies on the Axiom, even if not in its full strength. Commonly used mathematical principles such as Zorn's Lemma, the trichotomy of cardinals, the right-invertibility of surjective functions, and the existence of a basis for arbitrary vector spaces are all consequences of, and, in fact, equivalent to, the Axiom of Choice.

Through this inaugural talk of the Student Logic and History of Math Seminar, we explore some of the history, philosophy, and mathematics surrounding the Axiom of Choice.

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Neural-network quantum states (NQS) is an exciting area of quantum-inspired machine learning, wherein in a fully classical neural network is used to efficiently learn the optimal state vector for a quantum unsupervised learning problem, bypassing the need to model the full exponentially large state space. In theory, NQS architectures are broadly applicable, and have been demonstrated successfully on several fundamental applications. We seek to further explore the practical utility of NQS as a black box solver for various applications in the scientific computing domain. We first present NQS as an explicit avenue for de-quantizing variational quantum eigensolvers, demonstrating with the variational quantum linear solver (VQLS). We then explore the utility of VQLS and its de-quantization, VNLS, as black box solvers for a Newton-based linear complementarity solver, used to model the time evolution of a rudimentary soil system. We introduce a new retentive network (RetNet) autoregressive NQS ansatz to solve electronic ground state calculations in computational quantum chemistry, and we finish with a scaling law analysis of the RetNet architecture and its immediate autoregressive predecessors as applied to electronic structure calculations.

Hybrid Defense:

5822 East Hall

https://umich.zoom.us/j/92606555957

In their 2018 paper, Thomas Lam and Pavel Galashin define polypositroids, the polytopal analog of positroids. However, they also go one step further and observe that the definition of a polypositroid is secretly based on the type A root system. In this talk, we’ll discuss how polypositroids can be extended to arbitrary finite root systems and examine how the parameterization of polypositroids can be generalized too.

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]]>Gröbner bases are computational and combinatorial tools in algebra and geometry. In this talk, we'll introduce them and relate their existence to the noetherianity property. We'll then give an adaptation of Gröbner bases for non-noetherian rings that takes into account natural equivariant structures of the rings and describe how such rings exhibit "noetherianity up to symmetry".

]]>Lorentzian polynomials serve as a bridge between continuous and discrete convex analysis, with tropical geometry providing the critical link. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions, leading for example to a short proof of Mason's conjecture on the number of independent sets of a matroid. This lecture series will explore the intricate relationships among Grassmannians over hyperfields, dequantization processes, and the theory of Lorentzian polynomials. In ongoing collaborative work with Matt Baker, Mario Kummer, and Oliver Lorscheid, we extend the connection between Lorentzian polynomials and discrete convex functions to matroids over triangular hyperfields, as introduced by Viro. This extension deepens our understanding of the space of Lorentzian polynomials, revealing a complex interplay among analysis, combinatorics, and geometry.

The three lectures in this series are designed to be accessible to a broad audience and appropriate for a Department Colloquium.

Among his many honors, Prof. Huh is a recipient of the Fields Medal (2022), the MacArthur Fellowship (2022), and the New Horizons in Mathematics Prize (2019). He received his PhD in Mathematics from U-M in 2014.

Murai observed an interesting stability pattern in the resolutions of symmetric monomial ideals. I will explain my preliminary attempt at providing a structural explanation to Murai's results by proving finiteness theorems for their local cohomology modules. Along the way, I will also prove the Le--Nagel--Nguyen--Römer conjectures for sequences of GL_n-equivariant k[x_1, x_2, ..., x_n]-modules when k is an infinite field of characteristic p > 0.

]]>We introduce and define the ring of periods, a subring of the complex numbers defined by Kontsevich and Zagier consisting of "numbers of arithmetic origin." This ring includes all algebraic numbers, but also some transcendental numbers, such as pi. We give examples of periods and state some of the major conjectures involving this ring, including deep connections with arithmetic geometry and PDE's.

]]>Lorentzian polynomials serve as a bridge between continuous and discrete convex analysis, with tropical geometry providing the critical link. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions, leading for example to a short proof of Mason's conjecture on the number of independent sets of a matroid. This lecture series will explore the intricate relationships among Grassmannians over hyperfields, dequantization processes, and the theory of Lorentzian polynomials. In ongoing collaborative work with Matt Baker, Mario Kummer, and Oliver Lorscheid, we extend the connection between Lorentzian polynomials and discrete convex functions to matroids over triangular hyperfields, as introduced by Viro. This extension deepens our understanding of the space of Lorentzian polynomials, revealing a complex interplay among analysis, combinatorics, and geometry.

The three lectures in this series are designed to be accessible to a broad audience and appropriate for a Department Colloquium.

Among his many honors, Prof. Huh is a recipient of the Fields Medal (2022), the MacArthur Fellowship (2022), and the New Horizons in Mathematics Prize (2019). He received his PhD in Mathematics from U-M in 2014.

To answer the question posed in the title, we first need to define what we mean by "better." In this talk, we will look at the tournament designs of football and table tennis from an optimal stopping perspective. In particular, we consider the problem of finding the optimal scheme for a knock-out tournament with 2^n players, aiming at determining the top player. In each game in the tournament, we observe a real-time score, modeled by a Brownian motion with drift where the drift reflects the players' relative abilities. We can stop observing the game when the outcome seems clear and decide who advances. However, the longer a match is played, the more cost one needs to pay. We formulate and solve a stopping problem to minimise the probability of eliminating the best player while keeping the cost of observation low. The result will tell us how to smartly distribute the time cost across tournament games, and thus, reveals which sport has a superior design. Additionally, we discuss a few variants of the problem and some possible generalisations.

]]>Lorentzian polynomials serve as a bridge between continuous and discrete convex analysis, with tropical geometry providing the critical link. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions, leading for example to a short proof of Mason's conjecture on the number of independent sets of a matroid. This lecture series will explore the intricate relationships among Grassmannians over hyperfields, dequantization processes, and the theory of Lorentzian polynomials. In ongoing collaborative work with Matt Baker, Mario Kummer, and Oliver Lorscheid, we extend the connection between Lorentzian polynomials and discrete convex functions to matroids over triangular hyperfields, as introduced by Viro. This extension deepens our understanding of the space of Lorentzian polynomials, revealing a complex interplay among analysis, combinatorics, and geometry.

The three lectures in this series are designed to be accessible to a broad audience and appropriate for a Department Colloquium.

Among his many honors, Prof. Huh is a recipient of the Fields Medal (2022), the MacArthur Fellowship (2022), and the New Horizons in Mathematics Prize (2019). He received his PhD in Mathematics from U-M in 2014.

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]]>Draisma recently proved that finite length polynomial representations of the infinite general linear group GL are topologically GL-noetherian, i.e., the descending chain condition holds for GL-stable closed subsets. The scheme-theoretic variant of this theorem is a major open problem in the area. I will briefly outline the rich history of this problem and provide a negative answer in characteristic two.

]]>The SIAM student chapter at the University of Michigan is hosting the 2024 SIAM Student Mini-Symposium in Applied Mathematics on September 15th, 2024.

Sponsored by the Michigan Center for Applied and Interdisciplinary Mathematics, this event will allow students from different disciplines in the area to see what is being done in the field and promote interest in applied mathematics in general. This event is open to all undergraduate and graduate students in the University of Michigan.

More information and registration (required) are available on the event's web site: https://sites.google.com/umich.edu/2024mcaim/home

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]]>The study of structural periodicity in engineered materials, such as metamaterials, phononic crystals, and metasurfaces, has opened new frontiers in wave control. Our research group explores the fundamental principles and advanced techniques for harnessing structural periodicity to manipulate wave propagation for diverse applications in sensing, energy harvesting, and space technology. By leveraging the unique properties of these periodic structures, we demonstrate how they can be used to enhance ultrasonic sensing and nondestructive testing, improve the efficiency of energy harvesters, and develop innovative solutions for space applications. The first part of my talk will focus on gradient index phononic crystal (GRIN-PC) lenses conforming pipe-like structures. Conformal GRIN-PC lenses are designed by tailoring unit cell geometry according to a specific refractive index profile. We explore the focusing of multi-mode guided waves at the desired locations (i.e. sensor nodes) along the pipe structure to address the attenuation problem in long-range pipelines. Then, I will explain how we exploit the negative refraction property of phononic crystals for designing a super lens. Unlike GRIN-PC lenses, which have at least minimum wavelength resolution as their natural limit, negative refraction-based PC lenses can potentially overcome the diffraction limit, which is highly favorable in medical imaging or other applications requiring localized wave intensity in areas smaller than a square wavelength. The second part of my talk will deal with reconfigurable metasurfaces for full wavefront control with an emphasis on energy harvesting of low frequency elastic waves. We fully analyze and design the elastic metasurfaces by tailoring the phase gradient of individual unit structures for different wave functions and present theoretical findings along with experimental validation. The last part of my talk will highlight the potential of metamaterials in space applications, such as novel in-space manufacturable extended solar arrays and antennas with high precision and mass efficiency.

Contact: Silas Alben

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]]>This paper investigates the asymptotic behavior of the linear-quadratic stochastic optimal control problems. By establishing a connection between the ergodic cost problem and the so-called cell problem in the homogenization of Hamilton-Jacobi equations, we reveal the turnpike properties of the linear-quadratic stochastic optimal control problems from various perspectives.

]]>We will give a gentle introduction to the world of big mapping class groups. These arise as groups of topological symmetries of: Infinite-type surfaces (dimension 2), locally finite, infinite graphs (dimension 1), and closed subsets of Cantor sets, i.e. second countable Stone spaces (dimension 0). These groups are all uncountable and come equipped with a topology that makes them into non-locally compact (aka "huge") Polish groups. We will survey some of the rich and interesting interplay between the topological, geometric, and algebraic aspects of these groups. Our main motivation will be to propose these groups as a fertile testing ground for extending the tools of geometric group theory beyond finitely (or compactly) generated groups to general "huge" groups; however, we will also mention potential connections to other fields of geometry, topology, and dynamics.

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]]>We examine the stationary relaxed singular control problem within a multi-dimensional framework for a single agent, as well as its Mean Field Game (MFG) equivalent. We demonstrate that optimal relaxed controls exist for both maximization and minimization cases. These relaxed controls are defined by random measures across the state and control spaces, with the state process described as a solution to the associated martingale problem. By leveraging findings from \cite{kur-sto}, we establish the equivalence between the martingale problem and the stationary forward equation. This allows us to reformulate the relaxed control problem into a linear programming problem within the measure space. We prove the sequential compactness of these measures, thereby confirming the feasibility of achieving an optimal solution. Subsequently, our focus shifts to Mean Field Games. Drawing on insights from the single-agent problem and employing Kakutani--Glicksberg--Fan fixed point theorem, we derive the existence of a mean field game equilibria.

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]]>The US political arena is generally a bleak mix of distressing and aggravating, particularly in the current highly polarized climate, but that distress only increases the need to honestly and scientifically understand the forces at play in ideological space, and the polarization offers emergent simplicity and thus a unique opportunity for mathematical modeling. This talk will share some recent and ongoing attempts to visualize and understand the current political landscape, as well as some quantitative patterns in political psychology and the modern information ecosystem. These recently-gathered data inform a mechanistic dynamical model of ideological drift, allowing theory to extrapolate the complex implications of micro-scale data to macro-scale outcomes---while iteratively improving and suggesting further data-gathering efforts to corner remaining uncertainty. The first waves of results from this perspective provide some remarkable insights, providing some hope of understanding and productively informing political messaging and algorithmic design for a more reasonable political future.

]]>Quivers and their mutations play a fundamental role in the theory of cluster algebras. We focus on the problem of deciding whether two given quivers are mutation equivalent to each other. Our approach is based on introducing an additional structure of a cyclic ordering on the set of vertices of a quiver. This leads to new powerful invariants of quiver mutation. These invariants can be used to show that various quivers are not mutation acyclic, i.e., they are not mutation equivalent to an acyclic quiver. This talk is partially based on joint work with Sergey Fomin [arXiv:2406.03604].

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]]>In this talk I will present a principal-agent problem in continuous time with multiple lump-sum payments (contracts) paid at different deterministic times. Based on the approach introduced in Cvitanić-Possamai-Touzi, we reduce the non-zero sum Stackelberg game between the principal and agent to a standard stochastic optimal control problem. We apply our result to a benchmark model for which we investigate how different inputs (payment frequencies, payment distribution, discount factors, agent's reservation utility, renegotiation) affect the principal's value. This is a joint work with Erhan Bayraktar, Ibrahim Ekren, and Liwei Huang.

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]]>Rogue waves or freak waves are spatially-localized disturbances of a background field that are also temporally localized. In the setting of the focusing nonlinear Schrödinger equation, which is a universal model for the complex amplitude of a wave packet in a general one-dimensional weakly nonlinear and strongly dispersive setting that includes water waves and nonlinear optics as special cases, a special exact solution exhibiting rogue-wave character was found by D. H. Peregrine in 1983. Since then, with the help of complete integrability, Peregrine’s solution has been generalized to a family of solutions of arbitrary “order” where more parameters appear in the solution as the order increases. These parameters can be adjusted to maximize the amplitude of the rogue wave for a given order. This talk will describe several recent results concerning such maximal-amplitude rogue wave solutions in the limit that the order increases without bound. For instance, it turns out that there is a limiting structure in a suitable near-field scaling of the peak of the rogue wave; this structure is a novel exact solution of the focusing nonlinear Schrödinger equation — the “rogue wave of infinite order” — that is also connected with the hierarchy of the third Painlevé equation. This is joint work with Deniz Bilman and Liming Ling.

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Contact: Evgeniy Khain (Oakland University)

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]]>This is a one day conference that travels around the Midwest; this Fall, we are hosting it in Ann Arbor.

The speakers will be

Chris Eur (Carnegie Mellon University)

Patricia Klein (Texas A&M University)

Matt Larson (Princeton / Institute for Advanced Study)

Jianping Pan (North Carolina State University)

There will also be a poster fair; you can sign up to present a poster!

To register for ALGECOM, please fill out the google poll at

https://forms.gle/BVe3MHfckc8kXTkn9

If you are applying for financial support, please fill out the form before October 1 in order to be considered. However, please do fill out the form if you think you will come, even if you are local and don't need support; it is helpful to us to know who will be coming.

The ALGECOM organizers are

David Speyer (U Michigan)

Peter Tingley (Loyola)

Alex Yong (UIUC)

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]]>Mean field games model the strategic interaction among a large number of players by reducing the problem to two entities: the statistical distribution of all players on the one hand and a representative player on the other. The master equation, introduce by Lions, models this interaction in a single equation, whose independent variables are time, state, and distribution. It can be viewed as a nonlinear transport equation on an infinite dimensional space. Solving this transport equation by the method of characteristics is essentially equivalent to finding the unique Nash equilibrium. When the equilibrium is not unique, we seek selection principles, i.e. how to determine which equilibrium players should follow in practice. A natural question, from the mathematical point of view, is whether entropy solutions can be used as a selection principle. We will examine certain classes of mean field games to show that the question is rather subtle and yields both positive and negative results.

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Contact: Robert Krasny.

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]]>It will initially be considered the asymptotic behavior of the solution of a mean-field system of Backward Stochastic Differential Equations with Jumps (BSDEs), as the multitude of the system equations grows to infinity, to independent and identically distributed (IID) solutions of McKean–Vlasov BSDEs. This property is known in the literature as backward propagation of chaos. Afterwards, it will be provided the suitable framework for the stability of the aforementioned property to hold. In other words, assuming a sequence of mean-field systems of BSDEs which propagate chaos, then their solutions, as the multitude of the system equations grows to infinity, approximates an IID sequence of solutions of the limiting McKean–Vlasov BSDE. The generality of the framework allows to incorporate either discrete-time or continuous-time approximating mean-field BSDE systems.

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