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Abstract: Let X and Y be metric spaces, S be a closed subset of X, and f be a Lipschitz map from S to Y. How can we know if f can be extended to globally defined Lipschitz map preserving (or almost preserving) the Lipschitz constant of f? How do we construct such an extension? For what kind of metric spaces can this always be done?
In this series, we survey some of the key results in the study of the Lipschitz extension problem. We will start by introducing classical construction by McShane and Whitney and an existence result by Kirszbraun. Tentatively, we will explore some of the more advanced flavors, including the ball intersection property (existence), Nagata dimension and connectedness (constructive), and a recent explicit Kirszbraun formula by D. Azagra et. al.

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Abstract: I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.

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Abstract: TBA

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Abstract: We propose a PDE framework for modeling the distribution shift of a strategic population interacting with a learning algorithm. We consider two particular settings; one, where the objective of the algorithm and population are aligned, and two, where the algorithm and population have opposite goals. We present convergence analysis for both settings, including three different timescales for the opposing-goal objective dynamics. We illustrate how our framework can accurately model real-world data and show via synthetic examples how it captures sophisticated distribution changes which cannot be modeled with simpler methods.

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Abstract: I will present a paper by Perthame, Quiros and Vazquez in which they present two tumor growth models and show that the solutions of one model are an asymptotic limit of the solutions of the other model.

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Abstract: After two decades of development, superconducting circuits have emerged as a rich platform for quantum computation and simulation. When combined with superconducting qubits, lattices of coplanar waveguide (CPW) resonators can be used to realize artificial photonic materials or photon-mediated spin models. Here I will highlight the special properties of this hardware implementation that lead to these lattices naturally being described as line graphs. Elucidating this connection required combining theoretical and computational methods from both physics pure mathematics, and has lead not only to a new understanding of the physics of these devices [1,2], but also new results regarding spectral gaps of 3-regular graphs [3], and a framework for studying a new class of topologically-protected quantum error correcting codes [4].

[1] Kollár, Fitzpatrick, Houck, Nature 45, 571 (2019).
[2] Kollár, Fitzpatrick, Sarnak, Houck, Comm. Math. Phys. 376, 1909-1956 (2020).
[3] Kollár, Sarnak, Comm. AMS. 1, 1 (2021)
[4] Chapman, Flammia, Kollár, Quantum 3, 03021 (2022)

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Abstract: Ecological processes depend on the flow and balance of essential elements such as carbon, nitrogen, and phosphorus. The balance of these essential elements in ecological interactions is studied in the theory of Ecological Stoichiometry (ES). The integration of ES into population dynamics has provided insights into understanding how stoichiometric constraints affect food webs and nutrient cycles. I will present stoichiometric ordinary differential equation mathematical models that focus on the essential elements of nitrogen and carbon to investigate topics such as disease-ecosystem feedback loops and nitrogen fixation.

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Abstract: Given a probability measure on $\textrm{PGL}_2(\mathbb{R})$ under relatively weak conditions there is a unique probability measure on the projective line which is stationary under convolution with this measure. This is known as the Furstenberg measure. When the measure on $\textrm{PGL}_2(\mathbb{R})$ has finite support this measure may have fractional dimension. In this talk I will discuss some background on finding the dimensions of stationary measures and Furstenberg measures as well as sufficient conditions for them to be absolutely continuous. (Talk runs 2:00-4:00)

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Abstract: In combinatorics, the Spencer-Shelah random graph sequence is a variation on the independent-edge random graph model. We fix an irrational number $a \in (0,1)$, and we probabilistically generate the n-th Spencer-Shelah graph (with parameter $a$) by taking $n$ vertices, and for every pair of distinct vertices, deciding whether they are connected with a biased coin flip, with success probability $n^{-a}$. On the other hand, in model theory, an $R$-mac is a class of finite structures, where the cardinalities of definable subsets are particularly well-behaved. In this talk, we will introduce the notion of "probabalistic $R$-mac" and sketch the proof that the Spencer Shelah random graph sequence is an example of one.

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Abstract: In this talk we propose a domain-decomposition-based method for the solution of a PDE-coupled generalized Nash equilibrium problem. Our motivation is a problem for a multispecies biofilm community. Microbes are able to deploy different strategies in response to, and depending upon, local environmental conditions. In the setting of a microbial community, this property induces a Nash equilibrium problem because access to environmental resources is bounded. If microbes are also distributed in space, then those resources are subject to transport limitations (encoded in a PDE) and so microbial strategies at one location influence resources and, hence, microbial strategies, at another. The domain decomposition method we propose here uses overlapping subdomains, and this feature is important in the numerical solution of the PDEs.
We illustrate this in particular for a 1D problem, where the resulting ODE is stiff, but the proposed method is shown to overcome this difficulty. An example consisting of a model with two microbial species biofilm with 33 externally transported chemical concentrations is also presented.

Joint work with:
Isaac Klapper, Tianyu Zhang, Karsten Zengler, Cristal Z'{u}\~{n}iga, and Xinli Yu

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Abstract: Doodling is a creative and fundamentally human activity, resulting in doodles with intricate and often hidden implicit structure. We will treat doodles as an example for how mathematics is done — by starting with some doodles, we will ask ourselves some natural questions and see where they take us. They will lead us to some unexpected places, and to some sophisticated mathematics.

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Abstract: In the 80s, Beilinson and Lichtenbaum conjectured a general theory of motivic cohomology satisfying a list of expected properties. For example, one would expect an extension of a Grothendieck-Riemann-Roch theorem for schemes which are not necessarily smooth over a field and one would expect a relationship with syntomic cohomology in arithmetic situations. For equicharacteristic schemes, such a theory was offered last year in joint work with Matthew Morrow. I will explain aspects of this theory, emphasizing on when algebraic cycles can appear.

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Abstract: The talk focuses on one-parameter families of maps in the plane and their topology. A period-doubling cascade for a map depending on a real parameter is a path of periodic orbits, a path on which there are infinitely many period doublings, a path along which the period goes to infinity. I will begin by describing the nature of topological fixed point theorems which were long thought to be non constructive. Ideas that altered them so that they became constructive can be applied to showing the existence of period-doubling cascades in planar maps. This is smooth topology and has nothing to do with Feigenbaum's renormalization. (Talk runs 2:00 to 3:20)

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Abstract: Critical SQG (surface quasi-geostrophic) equations are aequations originating from atmospheric science which are widely studied in
relation to rapid formation of small scales in fluids. In the whole
space or on the torus, the equations have been proved to have global
smooth solutions some fifteen years ago by Caffarelli-Vasseur and,
independently, by Kiselev-Nazarov-Volberg.
The problem of existence and uniqueness of global smooth solution in
bounded domains was open until now. I will present a proof of global
regularity obtained recently with Ignatova and Q-H. Nguyen. We
introduce a new methodology of transforming the single nonlocal
nonlinear evolution equation in a bounded domain into an interacting
system of extended nonlocal nonlinear evolution equations in the whole
space. The proof then uses the method of the nonlinear maximum
principle for nonlocal operators in the extended system.

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Abstract: Network autoregressive models seek to model peer effects such as contagion and interference, in which node-level responses or behaviors may influence one another. These models are frequently deployed by practitioners in sociology and econometrics, typically in the form of linear-in-means models, in which node-level covariates and local averages of responses are used as predictors. In highly structured networks, previous work has shown that peer effects in linear-in-means models are colinear with other regression terms, and thus cannot be estimated, but this collinearity is widely believed to be ignorable, as peer effects are typically identified in empirical networks. In this work, we show a concerning negative result: under linear-in-means models, when node-level covariates are independent of network structure, peer effects become increasingly collinear with other regression terms as the network size (i.e., number of nodes) grows, and are inestimable asymptotically. We also show a narrow positive result: under certain latent space network models, some peer effects remain identified as the network size grows, albeit under rather stringent conditions. Our results suggest that linear models for peer effects are appropriate in far fewer settings than was previously believed.