Department of

Mathematics


Seminar Calendar
for events the week of Wednesday, January 22, 2020.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 21, 2020

1:00 pm in 241 Altgeld Hall,Tuesday, January 21, 2020

Organizational meeting

Abstract: Just a short organizational meeting for Logic seminar and the MT/DST Seminar.

2:00 pm in 243 Altgeld Hall,Tuesday, January 21, 2020

Lichiardopol's Conjecture on Disjoint Cycles in Tournaments

Douglas B. West (Zhejiang Normal University and University of Illinois)

Abstract: In a 1981 survey on cycles in digraphs, Bermond and Thomassen conjectured that every digraph with minimum outdegree at least $2k-1$ contains $k$ disjoint cycles. In 2010, Lichiardopol conjectured a stronger property for tournaments: for positive integers $k$ and $q$ with $q\ge3$, every tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$.

Bang-Jensen, Bessy, and Thomassé [2014] proved the special case of the Bermond--Thomassen Conjecture for tournaments. This implies the case $q=3$ of Lichiardopol's Conjecture. The case $q=4$ was proved in a masters thesis by S. Zhu [2019]. We give a uniform proof for $q\ge5$, thus completing the proof of Lichiardopol's Conjecture. This result is joint work with Fuhong Ma and Jin Yan of Shandong University.

4:00 pm in 245 Altgeld Hall,Tuesday, January 21, 2020

The Helly geometry of some Garside and Artin groups

Jingyin Huang   [email] (Ohio State University)

Abstract: Artin groups emerged from the study of braid groups and complex hyperplane arrangements, and they are connected to Coxeter groups, 3-manifold groups, buildings and many others. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required. This is joint work with D. Osajda.

Wednesday, January 22, 2020

12:00 pm in 141 Altgeld Hall,Wednesday, January 22, 2020

Predictive Actuarial Analystics Using Tree-Based Models

Zhiyu Quan (University of Connecticut)

Abstract: Because of its many advantages, the use of tree-based models has become an increasingly popular alternative predictive tool for building classification and regression models. Innovations to the original methods, such as random forests and gradient boosting, have further improved the capabilities of using tree-based models as a predictive model. Quan et al. (2018) examined the performance of tree-based models for the valuation of the guarantees embedded in variable annuities. We found that tree-based models are generally very efficient in producing more accurate predictions and the gradient boosting ensemble method is considered the most superior. Quan and Valdez (2018) applied multivariate tree-based models to multi-line insurance claims data with correlated responses drawn from the Wisconsin Local Government Property Insurance Fund (LGPIF). We were able to capture the inherent relationship among the response variables and improved marginal predictive accuracy. Quan et al. (2019) propose to use tree-based models with a hybrid structure as an alternative approach to the Tweedie Generalized Linear Model (GLM). This hybrid structure captures the benefits of tuning hyperparameters at each step of the algorithm thereby allowing for an improved prediction accuracy. We examined the performance of this model vis-\`a-vis the Tweedie GLM using the LGPIF and simulated datasets. Our empirical results indicate that this hybrid tree-based model produces more accurate predictions without loss of intuitive interpretation.

2:00 pm in 447 Altgeld Hall,Wednesday, January 22, 2020

Organizational Meeting

Sungwoo Nam (Illinois Math)

Abstract: We will have an organizational meeting for this semester. This involves making a plan for this semester and possibly choose a topic for a reading seminar. If you want to speak this semester, or are interested in a reading seminar, please join us and make a suggestion.

3:00 pm in 243 Altgeld Hall,Wednesday, January 22, 2020

How do mathematicians believe?

Brian P Katz (Smith College)

Abstract: Love it or hate it, many people believe that mathematics gives humans access to a kind of truth that is more absolute and universal than other disciplines. If this claim is true, we must ask: what makes the origins and processes of mathematics special and how can our messy, biological brains connect to the absolute? If the claim is false, then what becomes of truth in mathematics? In this session, we will consider beliefs about truth and how they play out in the mathematics classroom, trying to understand a little about identity, authority, and the Liberal Arts.

3:30 pm in 341 Altgeld Hall,Wednesday, January 22, 2020

Organizational meeting

4:00 pm in 245 Altgeld Hall,Wednesday, January 22, 2020

Statistical reduced models and rigorous analysis for uncertainty quantification of turbulent dynamical systems

Di Qi   [email] (Courant Institute of Mathematical Sciences)

Abstract: The capability of using imperfect statistical reduced-order models to capture crucial statistics in turbulent flows is investigated. Much simpler and more tractable block-diagonal models are proposed to approximate the complex and high-dimensional turbulent flow equations. A rigorous statistical bound for the total statistical uncertainty is derived based on a statistical energy conservation principle. The systematic framework of correcting model errors is introduced using statistical response and empirical information theory, and optimal model parameters under this unbiased information measure are achieved in a training phase before the prediction. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the reduced-order model in various dynamical regimes with distinct statistical structures.

Thursday, January 23, 2020

11:00 am in 241 Altgeld Hall,Thursday, January 23, 2020

Heights and p-adic Hodge Theory

Lucia Mocz (University of Chicago)

Abstract: We discuss connections between p-adic Hodge theory and the Faltings height. Most namely, we show how new tools in p-adic Hodge theory can be used to prove new Northcott properties satisfied by the Faltings height, and demonstrate phenomenon which are otherwise predicted by various height conjectures. We will focus primarily on the Faltings height of CM abelian varieties where the theory can be made to be computational and explicit.

2:00 pm in 243 Altgeld Hall,Thursday, January 23, 2020

Free Banach Lattices

Vladimir Troitsky (University of Alberta)

Abstract: A free Banach lattice is the largest Banach lattice generated by a set of given cardinality. Similarly, a Banach lattice $X$ is free over a Banach space $E$ if $X$ is the largest Banach lattice which contains $E$ as a subspace and is generated by it. Equivalently, every bounded linear operator from $E$ to an arbitrary Banach lattice $Y$ extends to a lattice homomorphism from $X$ to $Y$ of the same norm. In the talk, we will discuss several methods of generating free vector and Banach lattices.

3:00 pm in 245 Altgeld Hall,Thursday, January 23, 2020

Application of Random Effects in Dependent Compound Risk Model

Himchan Jeong (University of Connecticut)

Abstract: In ratemaking for general insurance, the calculation of a pure premium has traditionally been based on modeling both frequency and severity in an aggregated claims model. Additionally for simplicity, it has been a standard practice to assume the independence of loss frequency and loss severity. However, in recent years, there has been sporadic interest in the actuarial literature exploring models that departs from this independence. Besides, usual property and casualty insurance enables us to explore the benefits of using random effects for predicting insurance claims observed longitudinally, or over a period of time. Thus, in this article, a research work is introduced with utilizes random effects in dependent two-part model for insurance ratemaking, testing the presence of random effects via Bayesian sensitivity analysis with its own theoretical development as well as empirical results and performance measures using out-of-sample validation procedures.

4:00 pm in 245 Altgeld Hall,Thursday, January 23, 2020

Semistable reduction in characteristic 0

Gaku Liu (Max Planck Institute for Mathematics in the Sciences)

Abstract: Semistable reduction is a relative generalization of the classical problem of resolution of singularities of varieties; the goal is, given a surjective morphism $f : X \to B$ of varieties in characteristic 0, to change $f$ so that it is "as nice as possible". The problem goes back to at least Kempf, Knudsen, Mumford, and Saint-Donat (1973), who proved a strongest possible version when $B$ is a curve. The key ingredient in the proof is the following combinatorial result: Given any $d$-dimensional polytope $P$ with vertices in $\mathbb{Z}^d$, there is a dilation of $P$ which can be triangulated into simplices each with vertices in $\mathbb{Z}^d$ and volume $1/d!$. In 2000, Abramovich and Karu proved, for any base $B$, that $f$ can be made into a weakly semistable morphism $f' : X' \to B'$. They conjectured further that $f'$ can be made semistable, which amounts to making $X'$ smooth. They explained why this is the best resolution of $f$ one might hope for. In this talk I will outline a proof of this conjecture. They key ingredient is a relative generalization of the above combinatorial result of KKMS. I will also discuss some other consequences in combinatorics of our constructions. This is joint work with Karim Adiprasito and Michael Temkin.

Friday, January 24, 2020

3:00 pm in 141 Altgeld Hall,Friday, January 24, 2020

Statistical inference for mortality models

Chen Ling (Georgia State University)

Abstract: Underwriters of annuity products and administrators of pension funds are under financial obligation to their policyholder until the death of counterparty. Hence, the underwriters are subject to longevity risk when the average lifespan of the entire population increases, and yet, such risk can be managed through hedging practices based on parametric mortality models. As a benchmark mortality model in insurance industry is Lee-Carter model, we first summarize some flaws regarding the model and inference methods derived from it. Based on these understandings we propose a modified Lee-Carter model, accompanied by a rigorous statistical inference with asymptotic results and satisfactory numerical and simulation results derived from a small sample. Then we propose bias corrected estimator which is consistent and asymptotically normally distributed regardless of the mortality index being a unit root or stationary AR(1) time series. We further extend the model to accommodate AR(2) process for mortality index, and, a bivariate dataset of U.S. mortality rates. Finally, we conclude by a detailed model validation and some discussions of potential hedging practices based on our parametric model.

3:00 pm in 141 Altgeld Hall,Friday, January 24, 2020

Statistical interference for mortality models

Chen Ling (Georgia State University)

Abstract: Underwriters of annuity products and administrators of pension funds are under financial obligation to their policyholder until the death of counterparty. Hence, the underwriters are subject to longevity risk when the average lifespan of the entire population increases, and yet, such risk can be managed through hedging practices based on parametric mortality models. As a benchmark mortality model in insurance industry is Lee-Carter model, we first summarize some flaws regarding the model and inference methods derived from it. Based on these understandings we propose a modified Lee-Carter model, accompanied by a rigorous statistical inference with asymptotic results and satisfactory numerical and simulation results derived from a small sample. Then we propose bias corrected estimator which is consistent and asymptotically normally distributed regardless of the mortality index being a unit root or stationary AR(1) time series. We further extend the model to accommodate AR(2) process for mortality index, and, a bivariate dataset of U.S. mortality rates. Finally, we conclude by a detailed model validation and some discussions of potential hedging practices based on our parametric model.

3:00 pm in 347 Altgeld Hall,Friday, January 24, 2020

Organizational Meeting

Kesav Krishnan (UIUC Math)

Abstract: This organizational meeting will be to decide on a schedule of speakers. All are welcome

4:00 pm in 141 Altgeld Hall,Friday, January 24, 2020

Organizational Meeting

Nachiketa Adhikari (UIUC)

Abstract: We will draft a schedule of the seminar talks this semester. Please join us and sign up if you want to speak (you don't have to decide on a topic or abstract now). As usual, there will be cookies. All are welcome!

4:00 pm in 245 Altgeld Hall,Friday, January 24, 2020

Logical and geometric tameness over the real line.

Erik Walsberg (University of California, Irvine)

Abstract: There are now a number of important and well-understood examples of logically tame first order structures over the real numbers such as the ordered field of real numbers and the ordered field of real numbers equipped with the exponential function. In these examples subsets of Euclidean space which are (first order) definable are geometrically very well behaved. Recent research had yielded general theorems in this direction. I will discuss one result in this subject: A first order structure on the real line which expands the ordered vector space of real numbers and defines a closed set X such that the topological dimension of X is strictly less then the Hausdorff dimension of X defines every bounded Borel set. Informally: An expansion of the ordered real vector space which defines a fractal is maximally wild from the viewpoint of logic. Joint with Fornasiero and Hieronymi.