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12:00pm - Zoom ID 989 6529 0738. Passcode: the last four digits of the zoom ID in the reverse order - Probability and Analysis Webinar Joe Neeman - (UT Austin) Large deviations for triangle densities Take a uniformly random graph with a fixed edge density e. Its triangle density will typically be about e^3, and we are interested in the large deviations behavior: what's the probability that the triangle density is about e^3 - delta? The general theory for this sort of problem was studied by Chatterjee-Varadhan and Dembo-Lubetzky, who showed that the solution can be written in terms of an optimization over certain integral kernels. This optimization is difficult to solve explicitly, but Kenyon, Radin, Ren and Sadun used numerics to come up with a fascinating and intricate set of conjectures regarding both the probabilities and the structures of the conditioned random graphs. We prove these conjectures in a small region of the parameter space.
Joint work with Charles Radin and Lorenzo Sadun |
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2:00pm to 3:00pm - 510R - Combinatorics and Probability Olya Mandelshtam - (University of Waterloo) The multispecies zero range process and modified Macdonald polynomials Over the last couple of decades, the theory of interacting particle systems has found some unexpected connections to orthogonal polynomials, symmetric functions, and various combinatorial structures. The asymmetric simple exclusion process (ASEP) has played a central role in this connection. Recently, Cantini, de Gier, and Wheeler found that the partition function of the multispecies ASEP on a circle is a specialization of a Macdonald polynomial $P_{\lambda}(X;q,t)$. Macdonald polynomials are a family of symmetric functions that are ubiquitous in algebraic combinatorics and specialize to or generalize many other important special functions. Around the same time, Martin gave a recursive formulation expressing the stationary probabilities of the ASEP on a circle as sums over combinatorial objects known as multiline queues, which are a type of queueing system. Shortly after, with Corteel and Williams we generalized Martin's result to give a new formula for $P_{\lambda}$ via multiline queues.
The modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ are a version of $P_{\lambda}$ with positive integer coefficients. A natural question was whether there exists a related statistical mechanics model for which some specialization of $\widetilde{H}_{\lambda}$ is equal to its partition function. With Ayyer and Martin, we answer this question in the affirmative with the multispecies totally asymmetric zero-range process (TAZRP), which is a specialization of a more general class of zero range particle processes. We introduce a new combinatorial object in the flavor of the multiline queues, which on one hand, expresses stationary probabilities of the mTAZRP, and on the other hand, gives a new formula for $\widetilde{H}_{\lambda}$. We define an enhanced Markov chain on these objects that lumps to the multispecies TAZRP, and then use this to prove several results about particle densities and correlations in the TAZRP. |
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2:00pm to 3:00pm - RH 510R - Mathematical Physics Olya Mandelshtam - (U Waterloo) The multispecies zero range process and modified Macdonald polynomials Over the last couple of decades, the theory of interacting particle systems has found some unexpected connections to orthogonal polynomials, symmetric functions, and various combinatorial structures. The asymmetric simple exclusion process (ASEP) has played a central role in this connection. Recently, Cantini, de Gier, and Wheeler found that the partition function of the multispecies ASEP on a circle is a specialization of a Macdonald polynomial $P_{\lambda}(X;q,t)$. Macdonald polynomials are a family of symmetric functions that are ubiquitous in algebraic combinatorics and specialize to or generalize many other important special functions. Around the same time, Martin gave a recursive formulation expressing the stationary probabilities of the ASEP on a circle as sums over combinatorial objects known as multiline queues, which are a type of queueing system. Shortly after, with Corteel and Williams we generalized Martin's result to give a new formula for $P_{\lambda}$ via multiline queues.
The modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ are a version of $P_{\lambda}$ with positive integer coefficients. A natural question was whether there exists a related statistical mechanics model for which some specialization of $\widetilde{H}_{\lambda}$ is equal to its partition function. With Ayyer and Martin, we answer this question in the affirmative with the multispecies totally asymmetric zero-range process (TAZRP), which is a specialization of a more general class of zero range particle processes. We introduce a new combinatorial object in the flavor of the multiline queues, which on one hand, expresses stationary probabilities of the mTAZRP, and on the other hand, gives a new formula for $\widetilde{H}_{\lambda}$. We define an enhanced Markov chain on these objects that lumps to the multispecies TAZRP, and then use this to prove several results about particle densities and correlations in the TAZRP. |
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4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Eun-Jae Park - (Yonsei University) Staggered discontinuous Galerkin methods on polygonal meshes Recently, polygonal finite element methods have received considerable attention. This is because general meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. In this talk, a new computational paradigm for discretizing PDEs is presented via the staggered Galerkin method on general meshes. |
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4:00pm - NS2 1201 - Differential Geometry Hongyi Sheng - (UC Irvine) Deformations of the scalar curvature and the mean curvature In Riemannian manifold $(M^n, g)$, it is well-known that its minimizing hypersurface is smooth when $n\leq 7$, and singular when $n\geq 8$. This is one of the major difficulties in generalizing many interesting results to higher dimensions, including the Riemannian Penrose inequality. In particular, in dimension 8, the minimizing hypersurface has isolated singularities, and Nathan Smale constructed a local perturbation process to smooth out the singularities. However, Smale’s perturbation will also produce a small region with possibly negative scalar curvature. In order to apply this perturbation in general relativity, we constructed a local deformation prescribing the scalar curvature and the mean curvature simultaneously. In this talk, we will discuss how the weighted function spaces help us localize the deformation in complete manifolds with boundary, assuming certain generic conditions. We will also discuss some applications of this result in general relativity. |
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2:00pm to 3:00pm - Rowland Hall 510R - Combinatorics and Probability Roman Vershynin - (UCI) Mathematics of synthetic data and privacy An emerging way to protect privacy is to replace true data by synthetic data. Medical records of artificial patients, for example, could retain meaningful statistical information while preserving privacy of the true patients. But what is synthetic data, and what is privacy? How do we define these concepts mathematically? Is it possible to make synthetic data that is both useful and private? I will tie these questions to a simple-looking problem in probability theory: how much information about a random vector X is lost when we take conditional expectation of X with respect to some sigma-algebra? This talk is based on a series of papers joint with March Boedihardjo and Thomas Strohmer, mainly this one: https://arxiv.org/abs/2107.05824 |
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9:00am to 10:00am - Zoom - Inverse Problems Yi-Hsuan Lin - (National Yang Ming Chiao Tung University) Simultaneous recovery inverse problems for nonlinear and nonlocal equations |
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10:00am to 10:50am - https://uci.zoom.us/j/94710132565 - Number Theory Ofir Gorodetsky - (Oxford University) Sums of two squares are strongly biased towards quadratic residues Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than primes of the form 1 mod 4. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic density sense. Our understanding of this is conditional on the generalized Riemann Hypothesis as well as Linear Independence of the zeros of L-functions. We investigate similar questions for sums of two squares in arithmetic progressions. We find a significantly stronger bias than in primes, which happens for almost all integers in a natural density sense. Because the bias is more pronounced, we do not need to assume Linear Independence of zeros, only a Chowla-type Conjecture on non-vanishing of L-functions at 1/2.
We'll aim to be self-contained and define all the notions mentioned above during the talk. We shall review the origin of the bias in the case of primes and the work of Rubinstein and Sarnak. We'll explain the main ideas behind the proof of the bias in the sums-of-squares setting.
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11:00am to 11:50am - Zoom ID: 949 5980 5461, Password: the last four digits of ID in the reverse order - Harmonic Analysis Gabriel Rivière - (Université de Nantes) Meromorphic continuation of Poincaré series Poincaré series are natural functions that arise in Riemannian geometry |
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4:00pm - Zoom ID: 930 9003 8527 - Colloquium Jinchao Xu - (Penn State University) Piecewise Polynomials by Neural Networks and Finite Elements Polynomials and piecewise polynomials are most commonly used function classes in analysis and applications. In this talk, I will use these function classes to motivate and to present some remarkable properties of the function classes given by (both shallow and deep) neural networks. In particular, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions. Meeting ID: 930 9003 8527 Join Zoom Meeting https://uci.zoom.us/j/93090038527?pwd=cGJaT0g2U1JKUHEvT1FuS1RaSldsdz09
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