The standard Kleiman-Bertini transversality theorems say that if a variety is homogeneous with respect to the action of an algebraic group, then this action moves any two subvarieties into transverse position. I will describe refinements which treat cases where the action is not transitive, along with an application to the positivity of cohomology and K-theory classes of subvarieties of a generalized flag variety.

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]]>From a mathematical viewpoint, general relativity (GR) lies at the intersection of differential geometry and partial differential equations. It is the study of manifolds whose curvature obeys certain properties (dictated by the Einstein equations), and also of various quantities of interest on these manifolds. These quantities usually have physically significant interpretations, and it is interesting to study how these quantities are affected by the underlying spacetime curvature. Since GR is a nonlinear theory, the quantities themselves also influence the underlying curvature. This talk will be an introduction to the mathematical side of GR, viewed from the lens of differential geometry and analysis.

]]>What does it mean to make art (music, poetry, painting, design) in a world in which human creativity is being mined by generative artificial intelligence?

How can A.I. expand our creative horizons? How are artists experimenting with artificial intelligence today? Can art help us define what it truly means to be human? How might research and creative practice at the University of Michigan help answer these questions and more? Join us on March 5, 2024 to add your voice to the discussion

The University of Michigan Arts Initiative, the Michigan Institute for Data Science (MIDAS), Arts Research Incubation and Acceleration (ARIA), and the Michigan Center for Applied and Interdisciplinary Mathematics (MCAIM) jointly offer this symposium, bringing together writers, poets, visual artists and designers, and other creatives alongside scientists, engineers, humanists, philosophers, and other thinkers to investigate the process and products of the collaboration of AI and human artists, to showcase ongoing arts research efforts at the University of Michigan, and to explore research partnerships and other collaborative opportunities going forward.

U-M faculty members who are working or planning to work on projects / ideas related to AI for creative arts are also cordially invited to join our research discussion session (lunch and right after). We will facilitate the exchange of ideas, collaboration, and connecting projects with funding sources.

Please visit our event page for more information on sessions and speakers.

In this meeting, we will continue our discussion of Reinholtz's book, Equitable Teaching and Engaging Mathematics Teaching: A Guide to Disrupting Hierarchies in the Classroom. In our first meeting of the semester, we started discussion of the text, discussing the material through the beginning of chapter 3 (roughly, sections 1.1, 2.3.5-2.4, and 3.1-3.2). For this meeting, we will discuss sections 3.3 through the end of chapter 4. Meetings of the LCIT occur in East Hall room 4866.

]]>In this talk, we will consider the mapping class group of a compact surface with n punctures, and its pure subgroup consisting of mapping classes that fix the punctures pointwise. We will describe two stability phenomena that the (co)homology of these families of groups (indexed by the number n of punctures) exhibit. When the surface has non-empty boundary, the `puncture' homological stability of the family of mapping class groups is closely related with the representation stability of the sequence of homology groups of the pure mapping class groups. If time permits, we will explain this relation and how it could be used to obtain new results.

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]]>Given a discrete group G and a family F of subgroups of G there is a G-CW complex that classifies G-CW complexes with isotropy contained in the family F. Such space is unique up to G-equivariant homotopy and is often called the classifying space of G for the family F. For the trivial family it is the universal cover of a K(G,1) space, and more generally, classifying spaces for families play an important role in the classification of manifolds with a given fundamental group G. In this talk, we will introduce these notions and survey what is known about classifying spaces for some families of subgroups of the mapping class group of an orientable surface of finite type. Time permitting, we will discuss recent joint work with Porfirio León Álvarez and Luis Jorge Sánchez Saldaña in this topic.

]]>Exceptional vector bundles are a class of rigid bundles commonly found on Fano varieties with close connections to geometry and homological algebra. Foundational results of Drezet and Le Potier from the 1980s show that on the projective plane exceptional bundles have a deep and elaborate relationship with stable bundles in general. In this talk I will explore how these results in some ways extend and in other ways fail to extend to three-dimensional projective space, and discuss several interesting open problems concerning exceptional bundles.

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]]>Abstract: Granular matter, being an assembly of discrete particles, has complex mechanical behaviors emerging from the interactions of these particles, which often have a disordered yet non-trivial spatial arrangement. Unlike crystalline materials, the packing structure in a disordered material is often hard to describe mathematically, which prohibits us from understanding the deformation from a structure-property point of view. In this presentation, I will first present experimental results of the deformation of a layer of granular particles floating at an air-oil interface, through which I can demonstrate the elasto-plastic nature of deformation in the quasi-static regime. Based on the experimental results, a machine learning-based modeling framework was developed based on the interplay between elasticity, packing structure, and quasi-localized rearrangements of particles. The model can capture a ductile-to-brittle transition observed in the experimental system due to the change of particle properties.

In the second part of the talk, I will demonstrate the implications of the complex mechanical behaviors of granular materials for locomotion. In this problem, granular matter can be considered as a soft and yielding medium that interacts with a deforming body. I will show experimentally that a scallop-like swimmer with reciprocally flapping wings generates locomotion in granular matter, which is often not possible in Newtonian liquids at low Reynolds numbers. We use X-ray imaging and discrete element method simulations to reveal the microscopic picture of how the wings interact with surrounding particles. The locomotion is enabled by a prolonged hysteresis in the material response that originates from a combination of jamming-induced material rigidity and plastic deformation of the free surface. Cooperative effects are observed when the two wings are in close proximity, which potentially involves interaction of zones with jammed particles as well as heap building on the free surface.

Contact: Silas Alben

Cluster varieties are geometric objects corresponding to cluster algebras; they have many open subsets called cluster tori. These tori cover almost all of the cluster variety, but not quite all of it; the "deep locus" is the part of the cluster variety which is not in any cluster torus. In joint work with Marco Castronovo, Mikhail Gorsky and José Simental Rodríguez, we conjecture a description of the deep locus, and prove it for braid varieties on 2 and 3 strands. In this talk, I will explain our conjecture, and I will make clear what combinatorial problem we'd need to solve in order to prove this result for all braid varieties. I will not assume that the audience has seen cluster varieties or braid varieties before and, indeed, I hope that this talk will serve as a good introduction to those topics.

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]]>A representation of a group G is said to be rigid, if it cannot be continuously deformed to a non-isomorphic representation. If G happens to be the fundamental group of a complex projective manifold, rigid representations are conjectured (by Carlos Simpson) to be of geometric origin. In this talk I will outline the basic properties of rigid local systems and discuss several consequences of Simpson‘s conjecture. I will then outline recent progress on these questions (joint work with Hélène Esnault) and briefly mention applications to geometry and number theory such as the recent resolution of the André-Oort conjecture by Pila-Shankar-Tsimerman.

]]>I will talk about an embedding of generic FI-modules into a non-semisimple pre-Tannakian category which interpolates the categories of representations of general affine groups which are iterated extensions of representations of general linear groups. I will also discuss an analogous interpolation of representations of semidirect products of symplectic groups with Heisenberg groups.

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]]>Lloyd Shapley’s cooperative value allocation theory is a central concept in game theory that is widely applied in various fields to assess individual contributions and allocate resources. The Shapley value formula and his four defining axioms form the foundation of the theory.

We interpret the Shapley value as an expectation of a certain stochastic path integral, with each path representing a general coalition process. As a result, the value allocation is naturally extended to all partial coalition states. Furthermore, the new allocation scheme can be readily generalized by path-integrating various edge flows, which we refer to as the f-Shapley value. Finally, by employing Hodge theory on graphs, we show how to compute the stochastic path integral via the graph Poisson equation.

I will report on joint work in progress with Wyss and Ziegler. The first part of this talk will be a gentle introduction to p-adic integration, focusing on the case of quotient singularities where it is closely related to orbifold cohomology. In the second half we will compute p-adic integrals for singular moduli spaces of objects in certain abelian categories (e.g. representations of a symmetric quiver, meromorphic Higgs bundles, etc) and see that p-adic integration can be related to BPS cohomology arising in the theory of Donaldson-Thomas invariants.

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Contact: Robert Krasny

This is a joint meeting of the dynamics seminar + the geometry seminar

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]]>Understanding the geometry of the Mandelbrot set has been a central pillar of holomorphic dynamics over the past four decades. Much of its structure is now understood, but a critical question remains unresolved: is the Mandelbrot set locally connected? The first major break- through towards this conjecture was achieved by Yoccoz in the nineties, who proved that the Mandelbrot set is locally connected at all parameters which are not infinitely quadratic-like renormalizable. A key ingredient in Yoccoz’s work is the PLY-inequality, which bounds the diameter of certain subsets, called limbs, of the Mandelbrot set. These limbs are naturally labeled by the rational numbers, and the PLY-inequality asserts that the p/q-limb of the Mandelbrot set has size O(1/q). Milnor conjectured that O(1/q2) is the correct scale. For any N ≥ 1, the main result of this thesis is to verify Milnor’s conjecture for all p/q-limbs where a finite continued fraction of p/q has uniformly bounded length. Our strategy relies on careful analysis of the bifurcation of parabolic fixed points; we also further develop some of the classical theory in this area. We introduce parabolic and near-parabolic renormaliza- tion operators for maps which have parabolic fixed points of arbitrary multiplier and there perturbations, constructing invariant classes for these operators. We provide an alternative definition to the parabolic towers introduced by Epstein and construct a dynamically natural topology on the space of all parabolic towers. We also study the dynamics of Lavaurs maps, constructing analogues of polynomial external rays for these functions showing that these rays arise as the Hausdorff limits of polynomial external rays.

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]]>Order collaring, the automatic conversion of default market orders into limit orders with 5% spread over prior prices, has been utilized at Robinhood to protect retail investors from trading at unfavorable prices. In this paper, we provide empirical evidence that this policy harms retail traders in the form of higher trading costs. Using two quasi-experiments involving Robinhood’s trading hours and the discontinuity around 5% spread, we find that Robinhood customers have higher likelihood of paying extreme spreads over close prices. Further, the policy is associated with extreme price movements in stocks. We estimate that the economic loss of the retail traders due to order collaring is on the order of millions of dollars per day.

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]]>Let F be a pseudo-Anosov homeomorphism of a hyperbolic surface S. In this talk, we’ll describe joint work with Tarik Aougab and Dave Futer that predicts the number of fixed points of F, up to constants that depend only on the surface S. If F satisfies a mild condition called “strongly irreducible,” then the logarithm of the number of fixed points of F is coarsely equal to its translation length on the Teichmuller space of S. Without this condition, there is still a coarse formula involving subsurface projections of F’s invariant laminations.

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Contact: AIM Seminar Organizers

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]]>We consider the graphon mean-field system introduced by Bayraktar et al. in Bayraktar, Chakraborty, Wu (AAP 2023)

which is the large-population limit of a heterogeneously interacting diffusive particle system.

The interaction is of mean-field type with weights characterized by an underlying graphon function.

Via continuous observations of the trajectories of the finite-population particle system,

we build plug-in estimators of the particle densities, drift coefficients, and graphon interaction weights of the mean-field system.

Our estimators for the densities and drifts are direct results of kernel interpolation on the empirical data, and a deconvolution method leads to an estimator of the underlying graphon function

We prove that the estimator converges to the true graphon function as the number of particles tends to infinity when all other parameters are properly chosen.

Besides, we also conduct a minimax analysis on the plug-in estimator of the particle densities within a particular class of particle systems, which justifies its pointwise optimality.

Joint work with Erhan Bayraktar

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Contact: Guanhua Sun

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]]>I will introduce and discuss a remarkable class of algebraic varieties, called braid varieties. These include all open Richardson and positroid varieties, and are closely related to augmentation varieties for Legendrian links. The topology of braid varieties is related to various link invariants such as HOMFLY polynomial and Khovanov-Rozansky homology, while their coordinate ring has a cluster structure.

The talk is based on joint works with Roger Casals, Mikhail Gorsky, Ian

Le, Linhui Shen and Jose Simental.

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]]>Given a plane curve singularity C, one can define an algebraic variety called the compactified Jacobian of C. We introduce a class of "generic" curves, and describe the homology of the corresponding compactified Jacobians in terms of combinatorics of non-coprime rational q,t-Catalan numbers. All notions will be introduced in the talk, this is a joint work with Mikhail Mazin and Alexei Oblomkov.

]]>Mean field games (MFGs) study strategic decision-making in large populations where individual players interact via specific mean-field quantities. They have recently gained enormous popularity as powerful research tools with vast applications. For example, the Nash equilibrium of MFGs forms a pair of PDEs, which connects and extends variational optimal transport problems. This talk will present recent progress in this direction, focusing on computational MFG and engineering applications in robotics path planning, pandemics control, and Bayesian/AI sampling algorithms. This is based on joint work with the MURI team led by Stanley Osher (UCLA).

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]]>A prediction market is a market where people can trade based on the outcomes of future events. It is widely used in sports games, elections, and the pricing of digital options. In math finance, prediction markets can be modeled by the so-called win martingales, continuous time martingales that end up with Bernoulli distributions. In this talk, choosing specific divergences as objective functionals, we will solve a class of optimal win martingale. In some cases, we will get explicit formulas of optimizers, and make connections between Schrödinger and filtering problems. Based on the joint work with Julio Backhoff.

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Bio: Boyce Griffith is a Professor in the Department of Mathematics and of Biomedical Engineering at the University of North Carolina, where he is also an Adjunct Professor of Applied Physical Sciences and Associate Chair for Research in the Department of Mathematics. His research group focuses on the development and application of numerical methods for simulating fluid-structure interaction with a particular focus on models of the heart and its valves. Their core approach is based on extensions of the immersed boundary method fluid-structure interaction.

Contact: S. Alben

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]]>In many real-life policy making applications, the principal (i.e., governor or regulator) would like to find optimal policies for a large population of interacting agents who optimize their own objectives in a game theoretical framework. With the motivation of finding optimal policies for large populations, we start with introducing continuous time Stackelberg mean field game problem between a principal and a large number of agents. In the model, the agents in the population play a non-cooperative game and choose their controls to optimize their individual objectives while interacting with the principal and the other agents in the society through the population distribution. The principal can influence the resulting mean field game Nash equilibrium through incentives to optimize her own objective. Therefore, Stackelberg mean field game problems are by their nature bi-level problems where we have an optimal control problem at the principal level and a Nash equilibrium problem at the population level. This bi-level nature creates many efficiency challenges for the implementation of numerical approaches. For this reason, we will analyze how to rewrite this bi-level problem as a single-level problem and propose a deep learning approach to solve it. Then we will briefly discuss the convergence of the numerical solution where we utilize the single level problem to the solution of the original problem. We will conclude by demonstrating some applications such as the systemic risk model for a regulator and many banks and an optimal contract problem between a project manager and a large number of employees.

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