Fall 2021 Actuarial Science and Financial Mathematics Seminar: A theory of multivariate stress testing
Abstract: We present a theoretical framework for stressing multivariate stochastic models. We consider a stress to be a change of measure, placing a higher weight on multivariate scenarios of interest. In particular, a stressing mechanism is a mapping from random vectors to Radon-Nikodym densities. We postulate desirable properties for stressing mechanisms addressing alternative objectives. Consistently with our focus on dependence, we require throughout invariance to monotonic transformations of risk factors. We study in detail the properties of two families of stressing mechanisms, based respectively on mixtures of univariate stresses and on transformations of statistics we call Spearman and Kendall's cores. Furthermore, we characterize the aggregation properties of those stressing mechanisms, which motivate their use in deriving new capital allocation methods, with properties different to those typically found in the literature. The proposed methods are applied to stress testing and capital allocation, using the simulation model of a UK-based non-life insurer. About: Andreas Tsanakas is Professor in Risk Management at Bayes Business School (formerly Cass), City, University of London, and Editor-in-Chief of the Annals of Actuarial Science. His research interests range from quantitative risk management to sensitivity analysis, model uncertainty, and the role of models as decision tools in financial organisations. He has won industry research awards for his work on capital allocation, model risk and sensitivity analysis. Andreas is co-organiser of the Insurance Data Science Conference and the One World Actuarial Research Seminar. |
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Symplectic and Poisson geometry seminar: Locality for relative symplectic cohomology
The talk will be based on my recent preprint with Yoel Groman. I will start by introducing the question of locality for relative (and truncated relative) symplectic cohomologies. Then I will state our main result which involves the notion of a symplectic manifold being geometrically of finite type, e.g. cone completion of a symplectic manifold with convex boundary. I will end with examples coming from singular Lagrangian torus fibrations over complete bases and briefly mention the relevance to mirror symmetry. |
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Graph Theory and Combinatorics Seminar: Random greedy independent sets and matchings in some sparse hypergraphs of high girth
We analyze two random greedy processes on sparse random graphs and hypergraphs with fixed degree sequence. We analyze the matching process, which builds a set of disjoint edges one edge at a time, and then we analyze the independent process which builds an independent set of vertices one vertex at a time. Our results for these processes generalize and extend some results of Frieze, Wormald, Brightwell, Janson and Luczak. Using a recent result of Krivelevich, Mészáros, Michaeli, and Shikhelman, we extend our result on the matching process to any (deterministic) regular hypergraph of high girth. This talk is about joint work with Deepak Bal. |
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Probability Seminar: Heat kernel estimates and their stabilities for symmetric jump processes: stable and beyond
In this talk, we discuss the transition density of symmetric Markov processes. We focus on recent developments on Brownian-like jump process. This talk is based on joint papers with Joohak Bae, Jaehoon Kang, Jaehun Lee. |
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AWM Graduate Colloquium: Arithmetic applications of modular forms
The Fourier coefficients of modular forms have been of interest in number theory for several years. They are intimately connected with elliptic curves, class numbers for quadratic fields, and integer partitions. The last one may seem surprising at first, given that these complex-analytic functions "should" have nothing to do with that fact that one can write 4 as 4=3+1=2+2=2+1+1=1+1+1+1. We will focus mainly on integer partitions for this talk. In particular, we will discuss partition congruences, and at the end, discuss new congruences for related functions. |
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Student Cluster Algebra Seminar: Lambda lengths of tagged arcs
The geometric realization of the cluster variable is given by lambda length described in the paper by Penner which is the study of the decorated Teichmuller space. I will introduce this notion by introducing the Teichmuller space and generalized Ptolemy theorem. We will see that this lambda length of tagged arcs form a cluster exchange pattern. |
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Combinatorics Colloquium: On Submodular k-Partitioning and Hypergraph k-Cut
Submodular $k$-Partition is the following problem: given a submodular set function $f:2^V \rightarrow \mathbb{R}$ and an integer $k$, find a partition of $V$ into $k$ non-empty parts $V_1,V_2,\ldots,V_k$ to minimize $\sum_{i=1}^k f(V_i)$. Several interesting problems such as Graph $k$-Cut, Hypergraph $k$-Cut and Hypergraph $k$-Partition are special cases. Submodular $k$-Partition admits a polynomial-time algorithm for $k=2,3$ and when $f$ is symmetric also for $k=4$. The complexity is open for $k=4$ and when $f$ is symmetric for $k=5$. In recent work, motivated by this problem, we examined the complexity of Hypergraph $k$-Cut which only recently admitted a randomized polynomial-time algorithm. We obtained a deterministic polynomial-time algorithm for Hypergraph $k$-Cut as well as new insights in to Graph $k$-Cut. The ideas also led to a polynomial-time algorithm for Min-Max Symmetric Submodular $k$-Partition for any fixed $k$. The talk will discuss these results with the goal of highlighting the open problem of resolving the complexity of Submodular $k$-Partition. Based on joint work with Karthik Chandrasekharan. |
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The Calderon projector in boundary value problems
Usually elliptic operators on manifolds with boundary are not Fredholm without any restriction. When some boundary conditions are imposed it does become Fredholm. The goal will be to "parametrize" the solutions using boundary values. The Calderon projector will be a useful tool in determining which boundary conditions are Fredholm and in finding the index. |
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Graduate Student Homotopy Theory Seminar: Towards Splitting $BP<2> ⋀ BP<2>$ at Odd Primes
In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo ⋀ bo$ and $l ⋀ l$. These splittings helped make it feasible to do computations using the $bo$- and $l$-based Adams spectral sequences. I will discuss progress towards an analogous splitting for $BP<2> ⋀ BP<2>$ at odd primes. |
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Undergraduate Friday Seminar: Visualizing Number-Theoretic Fourier Series
We use Mathematica to explore Fourier series with coefficients given by number-theoretic functions, such as the Moebius function. By plotting the trajectories of such Fourier series in the complex plane, we observe a remarkable variety of behaviors, ranging from fractal-like patterns to smooth curves to chaotic and unpredictable outcomes. In certain cases, we were able to explain the observed behavior, while in others, we formulated conjectures that predict the type of pattern that can arise. |
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Number Theory Seminar: On Burgess bounds and superorthogonality
Abstract: The Burgess bound is a well-known upper bound for short multiplicative character sums, with a curious proof. It implies for example a subconvexity bound for Dirichlet L-functions. In this talk we will present two types of new work on Burgess bounds. First, we will describe new Burgess bounds in multi-dimensional settings. Second, we will present a new perspective on Burgess’s method of proof. Indeed, in order to try to improve a method, it makes sense to understand the bigger “proofscape” in which a method fits. The Burgess method hasn’t seemed to fit well into a bigger proofscape. We will show that it can be regarded as an application of “superorthogonality.” This perspective turns out to unify topics ranging across harmonic analysis and number theory. In this accessible talk, we will survey these connections, with a focus on the number-theoretic side. |
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Harmonic Analysis and Differential Equations Seminar: Reconciling modulation and dilation in time-frequency analysis
Modulation and dilation are two operators that permeate mathematical analysis. We indicate how to construct a family of modulation spaces that have a scaling symmetry and illustrate the behavior of the Schrödinger multiplier on such function spaces. This is a joint work with Tadahiro Oh, University of Edinburgh. |
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Graph Theory and Combinatorics Seminar: Small subgraphs and extensions in random graphs
Let G be a graph with several vertices v_1,..,v_r being roots. A G-extension of u_1,..,u_r in a graph H is a subgraph \hat G of H such that there exists a bijection from V(G) to V(\hat G) that preserves edges of G with at least one non-root vertex. In binomial random graphs, for sufficiently large edge probability p, the number of subgraphs isomorphic to a given graph G obeys both the law of large numbers and the central limit theorem. The maximum number of G-extensions obeys the law of large numbers as well. The talk is devoted to new results describing the limit distribution of the maximum number of G-extensions. Also, I am going to show some connections between the results on limit distributions of the numbers of small subgraphs and extensions and logical limit laws. |
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Student Cluster Algebra Seminar: Understanding Teichmuller space
Abstract: The Teichmuller space of a surface S, named after the mathematician Oswald Teichmuller, is a complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. I will show how to define Teichmuller space using the modular group and how to define a lambda length on it. Then I’ll show how to define higher Teichmuller space using this definition. |
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IRisk Lab Final Presentations
The presentation schedule is set as follows.
Each presentation lasts around 30 minutes, followed by a 5-minute Q&A. If there is any time remaining within the 40-minute window, it will be used as a short break until the next presentation starts on time. For each project detail, you can refer to https://irisklabuiuc.wixsite.com/actsi/projects. |
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IGL Open House
Join the IGL at our fall open house and poster session and see what everyone in the IGL has been working on this year! All are welcome to stop by anytime 1-4 PM. |