Symplectic and Poisson Geometry Seminar: Minimal Lagrangian tori and action-angle coordinates
In this talk I will report on joint work with Rosa Sena-Dias studying minimal Lagrangian submanifolds appearing as the fibers of the moment map on a toric Kahler manifold. I will use action-angle coordinates to answer the following questions: How many such minimal Lagrangian tori exist? Can their stability, as critical points of the area functional, be inferred from the ambient geometry? Which sets of such Lagrangian submanifolds can be made minimal with respect to some toric Kahler metric. If time permits I will also make some comments on Lagrangian mean curvature flow and Hamiltonian stationary Lagrangians. Please contact jpalmer5@illinois.edu for Zoom link. |
|||||||||||||||
Graph Theory and Combinatorics Seminar: Shadow of 3-uniform hypergraphs under a minimum degree condition
We prove a minimum degree version of the Kruskal-Katona theorem for triple systems. Given $d> 1/4$ and a triple system on $n$ vertices with minimum |
|||||||||||||||
AWM Graduate Colloquium: Characterizing generic elements of Polish groups
Given a group acting on a space, we may want to characterize the "usual" behavior of an element. We're often used to "usual" meaning "almost everywhere", with respect to some measure, but for many groups this is not a feasible option. In this talk I will introduce the right notion of "usual", and use this to characterize what elements "usually" look like, for some important groups. Contact na17@illinois.edu for Zoom details. |
|||||||||||||||
Math Department Retirees' Luncheon
Abstract: We've changed the venue for the Annual Retirees' Luncheon from Silvercreek to a Zoom meeting at 4 p.m (happy hour). Watch your email for more information coming soon! Or email currid@illinois.edu if you have any questions. |
|||||||||||||||
Mathematics Colloquium: Lamplighter metric spaces and their embeddings into $L_1$
Note: Zoom link will open at 3:45 p.m. Understanding how a group or a graph, viewed as a geometric object, can be faithfully embedded into certain Banach spaces is a fundamental topic with applications to geometric group theory and theoretical computer science. In this joint work with Florent Baudier, Pavlos Motakis and Andras Zsak we observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the $L_1$-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on a $n$-point metric space embeds bi-Lipschitzly into $L_1$ with distortion $O(\log n)$. In particular, for every finite group $G$ the lamplighter group $H = \mathbb{Z}_2\wr G$ bi-Lipschitzly embeds into $L_1$ with distortion $O(\log\log|H|)$. In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into $L_1$ of the lamplighter group over the $d$-dimensional infinite lattice $\mathbb Z^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite $d$-dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces. |
|||||||||||||||
Number Theory Seminar: Hilbert spaces and low-lying zeros of L-functions.
Abstract: Given a family of L-functions, there has been a great deal of interest in estimating the proportion of the family that does not vanish at special points on the critical line. Conjecturally, there is a symmetry type associated to each family which governs the distribution of low-lying zeros (zeros near the real axis). Generalizing a problem of Iwaniec, Luo, and Sarnak (2000), we address the problem of estimating the proportion of non-vanishing in a family of L-functions at a low-lying height on the critical line (measured by the analytic conductor). We solve the Fourier optimization problems that arise using the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type), and we can explicitly construct the associated reproducing kernels. If time allows, we will also address the problem of estimating the height of the "lowest" low-lying zero in a family for all symmetry types. In this context, a new Fourier optimization problem emerges, and we solve it by establishing a connection to the theory of de Branges spaces of entire functions and using the explicit reproducing kernels we constructed. This is joint work with Emanuel Carneiro (ICTP) and Andrés Chirre (NTNU). |
|||||||||||||||
Probability Seminar: Fluctuation results for size of the vacant set for random walks on discrete torus
We consider a random walk on the $d\ge 3$ dimensional discrete torus starting from vertices chosen independently and uniformly at random. In this talk, we discuss the fluctuation behavior of the size of the range of the random walk trajectories at a time proportional to the size of the torus. The proof relies on a refined analysis of tail estimates for hitting time. We also discuss related results and open problems. |
|||||||||||||||
Student Cluster Algebra Seminar: A brief introduction to cluster scattering diagrams
We will begin by defining scattering diagrams, which were first introduced in two dimensions by Kontsevich and Soibelman and then in arbitrary dimension by Gross and Siebert as a tool for constructing mirror spaces. Using concrete examples, we will then walk through the cluster scattering diagram construction given by Gross, Hacking, Keel, and Kontsevich and give definitions of cluster varieties, broken lines, theta functions, and other relevant objects. If time allows, we will then briefly sketch how cluster scattering diagrams are used to prove important results for ordinary cluster algebras, including positivity and the existence of the theta basis. |
|||||||||||||||
Actuarial Science and Financial Mathematics Seminar: Improving automobile insurance claims frequency prediction with telematics car driving data
Abstract: Novel navigation applications provide a driving behavior score for each finished trip to promote safe driving, which is mainly based on experts' domain knowledge. In this paper, with automobile insurance claims data and associated telematics car driving data, we propose a supervised driving risk scoring neural network model. This one-dimensional convolutional neural network takes time series of individual car driving trips as input and returns a risk score in the unit range of (0, 1). By incorporating credibility average risk score of each driver, the classical Poisson generalized linear model for automobile insurance claims frequency prediction can be improved significantly. Hence, compared with non-telematics-based insurers, telematics-based insurers can discover more heterogeneity in their portfolio and attract safer drivers with premiums discounts. About: Dr. Guangyuan Gao is an Associate Professor in the School of Statistics at the Renmin University of China. He obtained his Phd in Statistics at the Australian National University in 2016. Before that, he obtained his bachelor in Engineering at Tongji University in Shanghai. His research interests include non-life insurance claims reserving, mortality forecasting, telematics car driving data analysis, automobile insurance pricing, Bayesian statistics, etc. His research work has been published in ASTIN, SAJ, IME, NAAJ, etc. He is in charge of a National Natural Science Foundation of China. |
|||||||||||||||
Algebra, Geometry & Combinatorics: Naruse hook formula for linear extensions of mobile posets
Abstract: Linear extensions of posets are important objects in enumerativevand algebraic combinatorics that are difficult to count in general. Families of posets like Young diagrams of straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula from 2014. In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine $d$-complete posets and border strip skew shapes. We give a Naruse type hook-length formula to count linear extensions of such posets by proving a major index $q$-analogue. We also give an inversion index $q$-analogue of the Naruse formula for mobile tree posets. |
|||||||||||||||
Localization and Laplacians
Abstract: Localization can be broadly described as a phenomenon where certain integrands turn out to only be supported near special points. The classic result of this form is stationary phase. I will talk about different instances of localization and the analysis involved. I'll begin with stationary phase and move onto Atiyah Bott localization, the Duistermaat-Heckman theorem, and Witten deformation. For more on Witten deformation, check out the Graduate Geometry and Topology Seminar this week. |
|||||||||||||||
Graduate Student Homotopy Theory Seminar: Connections between motivic and classical homotopy theory
In the first part of this talk we will motivate the construction of the stable motivic homotopy category over the complex numbers through a non-standard construction of the stable homotopy category. After some comparisons between the motivic and classical theory, we will introduce the Chow t-structure and explain how it relates to some chromatic theory. |
|||||||||||||||
Undergraduate Friday Seminar: Outreach & Animation: Making Mathematics Come to Life!
Using the free, online graphing tool Desmos to educate, animate, and share beautiful ideas with any audience, regardless of their mathematical background. For some examples of what we'll talk about, you can view some of the animations here. |
|||||||||||||||