Department of

Mathematics


Seminar Calendar
for Mathematics Colloquium events the year of Thursday, December 7, 2017.

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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 17, 2017

4:00 pm in 245 Altgeld Hall,Tuesday, January 17, 2017

Gaussian Risk Models with Financial Constraints

Lanpeng Ji (University of Applied Sciences of Western Switzerland)

Abstract: In classical risk theory, the surplus process of an insurance company is modeled by the compound Poisson or the general compound renewal risk process. For both applied and theoretical investigations, calculation of the ruin probabilities for such models is of particular interest. In order to avoid technical issues and to allow for dependence among the claim sizes, these risk models are often approximated by the classical Brownian motion (or diffusion) (e.g., [1,2]) or the fractional Brownian motion risk model (e.g., [3,4]). Calculation of ruin probabilities and other ruin related quantities for Brownian motion and more general Gaussian risk models has been the subject of study of numerous contributions (e.g., [4-7]). This talk focuses on the most recent findings for Gaussian risk models with financial constraints such as inflation, interest and tax. In particular, three Gaussian risk models and their ruin probabilities will be discussed in detail. Finally, some future research directions on this topic will also be discussed. References: [1] Iglehart, D. L. 1969. Diffusion approximations in collective risk theory, Journal of Applied Probability 6: 285–292. [2] Grandell, J. 1991. Aspects of Risk Theory. New York: Springer. [3] Michna, Z. 1998. Self-similar processes in collective risk theory, J. Appl. Math. Stochastic Anal., 11(4): 429-448. [4] Asmussen, S. and Albrecher, H. 2010. Ruin Probabilities. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second edition. [5] Cai, J., Gerber, H.U. and Yang, H.L. 2006. Optimal dividends in an Ornstein–Uhlenbeck type model with credit and debit interest, North American Actuarial Journal 10 (2): 94–119. [6] Debicki, K. 2002. Ruin probability for Gaussian integrated processes, Stochastic Process. Appl., 98(1): 151-174. [7] Husler, J. and Piterbarg, V.I. 2008. A limit theorem for the time of ruin in a Gaussian ruin problem, Stochastic Process. Appl., 118(11): 2014-2021. See video of talk at https://www.youtube.com/watch?v=dYUhaq2j4CQ&feature=youtu.be

Thursday, January 19, 2017

4:00 pm in 245 Altgeld Hall,Thursday, January 19, 2017

Risk sharing and risk aggregation via risk measures

Haiyan Liu (University of Waterloo)

Abstract: In this talk, we discuss two problems in risk management using the tools of risk measures. In the first part of the talk, we address the problem of risk sharing among agents using a two-parameter class of quantile-based risk measures, the so-called Range-Value-at-Risk (RVaR), as their preferences. We first establish an inequality for RVaR-based risk aggregation, showing that RVaR satisfies a special form of subadditivity. Then, the risk sharing problem is solved through explicit construction. Comonotonicity and robustness of the optimal allocations are investigated. We show that, in general, a robust optimal allocation exists if and only if none of the risk measures is a VaR. Practical implications of our main results for risk management and policy makers will be discussed. In the second part of the talk, we study the aggregation of inhomogeneous risks with a special type of model uncertainty, called dependence uncertainty, evaluated by a generic risk measure. We establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.

Friday, January 20, 2017

4:00 pm in 245 Altgeld Hall,Friday, January 20, 2017

Inference for Mortality Models and Predictive Regressions

Liang Peng (Department of Risk Management and Insurance, Georgia State University)

Abstract: Forecasting mortality is of importance in managing longevity risks for insurance companies and pension funds. Some widely employed models are the so-called Lee-Carter model and its extensions. First we show that the proposed two-step inference procedure in Lee and Carter (1992) can not detect the true dynamics of the mortality index except when the index follows from a unit root AR(1) process. Second we propose a new method to test whether the index does follow from a unit root AR(1) model and then apply the new test to some mortality data to show that a blind application of an existing R package leads to different conclusions.

Testing for predictability of asset returns has been a long history in econometrics. Recently, based on a simple predictive regression, Kostakis, Magdalinos and Stamatogiannis (2015) proposed a Wald test derived from the IVX methodology for stock return predictability and Demetrescu (2014) showed that the local power of the standard IVX-based test could be improved in some cases when a lagged predicted variable is added to the predictive regression on purpose. Therefore an interesting question is whether a lagged predicted variable does appear in the model. Here we propose unified tests for testing the existence of a lagged predicted variable in a predictive regression and the predictability regardless of whether the predicting variable is stationary or nearly integrated or unit root. We further apply the proposed tests to some real data sets in finance.

Tuesday, January 24, 2017

4:00 pm in 245 Altgeld Hall,Tuesday, January 24, 2017

A New Approach for Buffering Portfolio Returns in Investment-Linked Annuities

Daniel Linders (Technical University of Munich)

Abstract: This paper introduces a new class of investment-linked annuity contracts. To reduce payout volatility, we gradually adjust cash flows to portfolio returns. This contrasts with standard investment-linked annuity contracts in which cash flows immediately incorporate portfolio returns. To build a realistic risk-management framework, we consider a general financial market. Our framework allows to use various non-Gaussian distributions which incorporate stylized facts about portfolio returns. Furthermore, we show how to price and hedge the liabilities of our new annuity contract.

Wednesday, January 25, 2017

4:00 pm in 245 Altgeld Hall,Wednesday, January 25, 2017

Degenerate diffusions and heat kernel estimates

Jing Wang (J.L. Doob Research Assistant Professor, University of Illinois at Urbana-Champaign)

Abstract: In this talk we will look at degenerate hypoelliptic diffusion processes and the small time behaviors of their transition densities. Diffusion processes play important roles in modeling risky assets in financial mathematics and actuarial science. The small time estimates of their transition densities are particularly useful for pricing options with short maturities. In this talk we will introduce the degenerate diffusion processes that are characterized by their levels of degeneracy. The ones of weaker degeneracy -- also called strong Hörmander's type -- are closely related to sub-Riemannian geometry. An important example is the Brownian motion process on a sub-Riemannian manifold. In general, small time asymptotic estimates are available for a subelliptic heat kernel on the diagonal and out of cut-locus. In special cases such as for Brownian motions on sub-Riemannian model spaces, we can obtain explicit expressions for their transition densities (heat kernels) and hence small time asymptotic estimates, particularly on the cut-loci. In the second part of the talk, we will study the strictly degenerate case-diffusion processes that are of weak Hörmander's type. Namely the hypoellipticity is fulfilled with the help of the drift term. This type of processes are particularly interesting in financial mathematics for pricing Asian options. We obtain large deviation properties for nilpotent diffusion processes of weak Hörmander's type.

Friday, January 27, 2017

4:00 pm in 245 Altgeld Hall,Friday, January 27, 2017

A Marked Cox Model for IBNR Claims: Theory and Application

Dameng Tang (University of Waterloo)

Abstract: Incurred but not reported (IBNR) loss reserving is a very important issue for Property & Casualty (P&C) insurers. To calculate IBNR reserve, one needs to model claim arrivals and then predict IBNR claims. However, factors such as temporal dependence among claim arrivals and exposure fluctuation are often not incorporated in most of the current loss reserving models, which greatly affect the accuracy of IBNR predictions. In this talk, I will present a new modelling approach under which the claim arrival process together with the reporting delays follows a marked Cox process. The intensity function of the Cox process is governed by a hidden Markov chain. I will show that the proposed model is versatile in modeling temporal dependence, can incorporate exposure fluctuation, and can be interpreted naturally in the insurance context. The associated reported claim process and IBNR claim process remain to be a marked Cox process with easily convertible intensity function and marking distribution. The specific structure of the intensity function allows for generating discretely observed claim processes, which is critical for data fitting purposes. Closed-form expressions for both the autocorrelation function (ACF) and the distributions of the numbers of reported claims and IBNR claims are derived. I will then present a generalized expectation-maximization (EM) algorithm to fit the model to data and to estimate the model parameters. The proposed model is examined through simulation studies and is applied to a real insurance claim data set. We compare the predictive distributions of our model with those of the over-dispersed Poisson model (ODP), a stochastic model that underpins the widely used chain-ladder method. The results show that our model can yield more accurate best estimates and more realistic predictive distributions. This is joint work with Andrei Badescu and Sheldon Lin.

Monday, January 30, 2017

4:00 pm in 156 Henry Admin Bldg,Monday, January 30, 2017

Discounted Sums at Renewal Times

Daniel Dufresne (Director of the Centre for Actuarial Science, University of Melbourne)

Abstract: Actuarial models usually include discounting, to take the time value of money into account. Mathematically this has proved difficult when amounts are paid at random times, for instance in risk theory. We assume that i.i.d. amounts {C(k)} are paid at renewal times {T(k)}. Of practical interest is the distribution of Z(t), the discounted value of claims occurring over the period [0,t]. New results on how to find the distribution of Z(t) will be presented. An important tool is sampling the process {Z(t)} at an independent exponential time, which leads to explicit distributions of Z(t) in specific cases. Joint work with Zhehao Zhang.

Thursday, February 2, 2017

4:00 pm in 245 Altgeld Hall,Thursday, February 2, 2017

Random matrices, d-bar problems, and approximation theory

Ken McLaughlin (Colorado State)

Abstract: Some surprising questions in analysis arise in the interconnections of the topics in the title. We will encounter zeros of the Taylor approximants of exp(z), and other analytic functions. We will consider questions of support of equilibrium charge distributions in the plane. Semi-classical analysis of d-bar problems will provide merriment along the way.

Thursday, February 9, 2017

4:00 pm in 245 Altgeld Hall,Thursday, February 9, 2017

Tales of Elliptic fibrations

Mboyo Esole (Northeastern Math)

Abstract: Elliptic curves have been part of mathematics since ancient Greece and beyond. When an elliptic curve moves over a variety, it draws a fibration called a genus one fibration. The cases of surfaces have been explored by Kodaira, Neron, and others in the early 1960s. During the second string revolution, elliptic fibrations have played a central role in describing non-perturbative effects in string theory and M-theory. Ever since, ideas from physics have inspired new points of view on elliptic fibrations, providing a rich set of ideas to explore their geometry using tools from representation theory, hyperplane arrangements, intersection theory, and birational geometry. In this colloquium, I will review these ideas and present new results.

Thursday, February 16, 2017

4:00 pm in 245 Altgeld Hall,Thursday, February 16, 2017

Weyl law for the Volume Spectrum

André Neves (University of Chicago)

Abstract: The volume spectrum was introduced by Gromov in the 70’s. Recently, with Liokumovich and Marques, we proved a Weyl Law for the volume spectrum that was conjectured by Gromov. I will talk about how a better understanding of the volume spectrum would help in answering some well known questions for minimal surfaces or volume of nodal sets.

Thursday, February 23, 2017

4:00 pm in 245 Altgeld Hall,Thursday, February 23, 2017

From Picard to Rickman: Mappings in spatial quasiconformal geometry

Pekka Pankka (University of Helsinki)

Abstract: One of the classical theorems in complex analysis is the Picard’s theorem stating that a non-constant entire holomorphic map from the complex plane to the Riemann sphere omits at most two points. From the conformal point of view, two dimensional geometry is special in this sense. Namely, by classical Liouville’s theorem from the same era, every conformal map from a domain of the n-sphere to the n-sphere is a restriction of a Möbius transformation for n>2. In particular, Picard’s theorem holds trivially in higher dimensions. An alternative for the overly rigid spatial conformal geometry is a so-called quasiconformal geometry; heuristically, instead of preserving the angles we allow them to distort by a bounded amount. In this talk, I will discuss the role of Picard’s theorem in quasiconformal geometry, which takes us from complex analysis to geometric topology.

Thursday, March 16, 2017

4:00 pm in 245 Altgeld Hall,Thursday, March 16, 2017

The Topological Closure of Algebraic and Semi-Algebraic Flows on Complex and Real Tori

Sergei Starchenko (Notre Dame)

Abstract: Let $A$ be a complex abelian variety and $\pi\colon \mathbb{C}^n\to A$ be the covering map. In this talk we consider the topological closure $\pi(X)$ of an algebraic subvariety $X$ of $\mathbb{C}^n$ and describe it in terms of finitely many algebraic families of cosets of real subtori. We also obtain a similar description when $A$ is a real torus and $X$ is a semi-algebraic subset of $\mathbb{R}^n$. This is joint work with Y. Peterzil.

Thursday, March 30, 2017

4:00 pm in 245 Altgeld Hall,Thursday, March 30, 2017

Projections and Curves in Infinite-Dimensional Banach Spaces

Bobby Wilson (MIT and MSRI)

Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections including the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss some related questions. This is joint work with Marianna Csornyei and David Bate.

Thursday, April 6, 2017

4:00 pm in 245 Altgeld Hall,Thursday, April 6, 2017

On the transport property of Gaussian measures under Hamiltonian PDE dynamics

Tadahiro Oh (Edinburgh)

Abstract: In probability theory, the transport property of Gaussian measures have attracted wide attention since the seminal work of Cameron and Martin '44. In this talk, we discuss recent development on the study of the transport property of Gaussian measures on spaces of functions under nonlinear Hamiltonian PDE dynamics. As an example, we will discuss the case for the 2-d cubic nonlinear wave equation, for which we introduce a simultaneous renormalization of the energy functional and its time derivative to study the transport property of Gaussian measures on Sobolev spaces. This talk is based on a joint work with Nikolay Tzvetkov (Université de Cergy-Pontoise).

Tuesday, April 11, 2017

4:00 pm in 245 Altgeld Hall,Tuesday, April 11, 2017

Harmonic analysis techniques in several complex variables

Loredana Lanzani (Syracuse University)

Abstract: This talk concerns the application of relatively classical tools from real harmonic analysis (namely, the $T(1)$-theorem for spaces of homogenous type) to the novel context of several complex variables. Specifically, I will present recent joint work with E. M. Stein on the extension to higher dimension of Calderón's and Coifman-McIntosh-Meyer's seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset{\mathbb C}$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel $H(w, z) = \tfrac{1}{2\pi i}(w-z)^{-1}dw$ is that it is holomorphic as a function of $z\in D$. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogue of $H(w, z)$. This is because geometric obstructions arise (the Levi problem), which in dimension one are irrelevant. A good candidate kernel for the higher dimensional setting was first identified by Jean Leray in the context of a $C^\infty$-smooth, convex domain $D$: while these conditions on $D$ can be relaxed a bit, if the domain is less than $C^2$-smooth (never mind Lipschitz!) Leray's construction becomes conceptually problematic. In this talk I will present (i) the construction of the Cauchy-Leray kernel and (ii) the $L^p(bD)$-boundedness of the induced singular integral operator under the weakest currently known assumptions on the domain's regularity -- in the case of a planar domain these are akin to Lipschitz boundary, but in our higher-dimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the so-called "$T(1)$-theorem technique" from real harmonic analysis. Time permitting, I will describe applications of this work to complex function theory -- specifically, to the Szego and Bergman projections (that is, the orthogonal projections of $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).

Thursday, April 20, 2017

4:00 pm in 245 Altgeld Hall,Thursday, April 20, 2017

Tarski numbers

Mark Sapir (Centennial Professor, Vanderbilt University, and George A. Miller Visiting Professor, Univ. of Illinois)

Abstract: It is known since Hausdorff, Banach and Tarski that one can decompose a 2-sphere into 4 pieces, move the pieces using rotations of the sphere, and obtain two spheres of the same radius (assuming the Axiom of Choice). Thus a sphere with the group of rotations acting on it has a paradoxical decomposition with 4 pieces. In general if we have a group G acting on a set X, then the Tarski number of X is the minimal number of pieces in a paradoxical decomposition of X. For example, if G acts on itself by left multiplication, then we can talk about the Tarski number of G. I will show how to use Golod-Shafarevich groups to prove that the set of possible Tarski numbers of groups is infinite. I will also show how to use l_2-Betti numbers of groups and cost of group actions to construct groups with Tarski numbers 5 and 6. Note that 4, 5 and 6 are the only numbers that are currently known to be Tarski numbers of groups. This is a joint work with Gili Golan and Mikhail Ershov.

Tuesday, April 25, 2017

4:00 pm in 314 Altgeld Hall,Tuesday, April 25, 2017

Cobordisms: old and new

Ulrike Tillmann (Oxford University)

Abstract: Cobordims have played an important part in the classification of manifolds since their invention in the 1950s. In a different way, they are fundamental to the axiomatic approach to Topological Quantum Field Theory. In this colloquium style talk I will explain how recent results have shed new light on both of them.

The Tondeur Lectures in Mathematics will be held April 25-27, 2017. A reception will be held following the first lecture from 5-6 pm April 25 in 239 Altgeld Hall.

Wednesday, April 26, 2017

4:00 pm in 245 Altgeld Hall,Wednesday, April 26, 2017

Classifying spaces of bordism categories and a filtration of Thom's theory

Ulrike Tillmann (Oxford University)

Abstract: We describe a refinement of a theorem with Galatius, Madsen and Weiss which describes the classifying space of bordism categories. In particular this can be interpreted to give evidence for the cobordism hypothesis for invertible TQFTs.

The Tondeur Lectures in Mathematics will be held April 25-27, 2017. A reception will be held following the first lecture from 5-6 pm April 25 in 239 Altgeld Hall.

Thursday, April 27, 2017

4:00 pm in 245 Altgeld Hall,Thursday, April 27, 2017

Operads from TQFTs

Ulrike Tillmann (Oxford University)

Abstract: Manifolds give rise to interesting operads, and in particular TQFTs define algebras over these operads. In the case of Atiyah's 1+1 dimensional theories these algebras are well-known to correspond to certain algebras. Surprisingly, independent of the dimension of the underlying manifolds, in the topologically enriched setting the manifold operads detect infinite loop spaces. We will report on joint work with Basterra, Bobkova, Ponto, Yeakel.

The Tondeur Lectures in Mathematics will be held April 25-27, 2017. A reception will be held following the first lecture from 5-6 pm April 25 in 239 Altgeld Hall.

Thursday, May 4, 2017

4:00 pm in 245 Altgeld Hall,Thursday, May 4, 2017

The L^2 metric on Hitchin’s moduli space

Rafe Mazzeo (Stanford)

Abstract: The much-studied moduli space of solutions to Hitchin’s equations on a Riemann surface carries a natural complete Weil-Petersson type metric. The large-scale structure of this metric is only now being revealed, first through an ambitious set of physics conjectures by Gaiotto-Neitzke-Moore, and now through techniques of geometric analysis. I will discuss this whole picture and report on recent work with Swoboda-Weiss-Witt.

Thursday, August 31, 2017

4:00 pm in 245 Altgeld Hall,Thursday, August 31, 2017

Symplectic non-squeezing for the cubic nonlinear Schrodinger equation on the plane

Monica Visan (UCLA)

Abstract: A famous theorem of Gromov states that a finite dimensional Hamiltonian flow cannot squeeze a ball inside a cylinder of lesser radius, despite the fact that the ball has finite volume and the cylinder has infinite volume. We will discuss an infinite-dimensional analogue of Gromov's result, in infinite volume. Specifically, we prove that the flow of the cubic NLS in two dimensions cannot squeeze a ball in $L^2$ into a cylinder of lesser radius. This is joint work with R. Killip and X. Zhang.

Thursday, September 7, 2017

4:00 pm in 245 Altgeld Hall,Thursday, September 7, 2017

Random groups and surfaces

Moon Duchin (Tufts)

Abstract: I'll survey some of the beautiful history of the study of random objects in geometry, topology, and group theory. The focus will be the exchanges between work on random groups and work on random surfaces, including some very recent results and current research topics.

Thursday, September 14, 2017

4:00 pm in 245 Altgeld Hall,Thursday, September 14, 2017

Probabilistic and combinatorial methods in the study of prime gaps

Kevin Ford (Illinois)

Abstract: We will describe how new bounds for the largest gaps between consecutive primes have utilized tools from several different areas, including number theory (efficient prime detecting sieves), probability (randomized congruence system coverings, concentration arguments) and probabilistic combinatorics (hypergraph covering). In particular, we will outline the recent breakthroughs of the speaker together with Ben Green, Sergei Konyain, James Maynard and Terence Tao. We will also describe new work with Konyagin, Maynard, Carl Pomerance and Tao which provides new bounds on consecutive composite values of integers in other sequences, e.g. polynomial sequences.

Thursday, September 21, 2017

4:00 pm in 245 Altgeld Hall,Thursday, September 21, 2017

The Man Who Knew Infinity: the Movie, the Man, and the Mathematics

George Andrews (Penn State)

Abstract: In the spring of 2016, the motion picture, The Man Who Knew Infinity, was released. It is now available on DVD. The movie tells the life story of the Indian genius, Ramanujan. In this talk, I hope to start with the trailer from the movie. Then I shall provide a brief discussion of Ramanujan's life with a glimpse of the mathematics contained in his celebrated Lost Notebook (of which Bruce Berndt and I have just concluded preparing the fifth and final volume explicating the many assertion therein). The bulk of the talk will consider the path from the Rogers-Ramanujan identities to current open problems and how computer algebra assists in their study.

Wednesday, October 4, 2017

4:00 pm in 245 Altgeld Hall,Wednesday, October 4, 2017

Modular forms, physics, and topology

Dan Berwick-Evans (Illinois)

Abstract: Modular forms appear in a wide variety of contexts in physics and mathematics. For example, they arise in two dimensional quantum field theories as certain observables. In algebraic topology, they emerge in the study of invariants called elliptic cohomology theories. A long-standing conjecture suggests that these two appearances of modular forms are intimately related. After explaining the ingredients, I’ll describe some recent progress. 

Thursday, October 5, 2017

4:00 pm in 245 Altgeld Hall,Thursday, October 5, 2017

From Physics to Mathematics and back again: an exploration of generalized Kähler geometry

Marco Gualtieri (University of Toronto)

Abstract: In 1984, physicists Gates, Hull and Rocek, experts in the then-burgeoning field of supersymmetry, realized that their physical model required the existence of a unknown geometric structure, involving a Riemannian metric with a pair of compatible complex structures. Until relatively recently, we lacked a basic understanding of the features of this generalization of Kähler geometry. It was only after Hitchin's introduction of the concept of a generalized complex structure that we were able to prove the conjectures made by the physicists and provide various examples of what is now known as a generalized Kähler structure. I will explain the basic features of this fascinating geometric structure and outline the many relations we have discovered to other parts of geometry, including twistor theory, Poisson geometry, and Dirac geometry.

Tuesday, October 10, 2017

4:00 pm in 314 Altgeld Hall,Tuesday, October 10, 2017

Lecture I. Set theory and trigonometric series

Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017. A reception will follow this first lecture from 5-6 pm in 239 Altgeld Hall. Alexander Kechris will present: "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.

Wednesday, October 11, 2017

4:00 pm in 245 Altgeld Hall,Wednesday, October 11, 2017

Lecture II. The complexity of classification problems in ergodic theory

Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017.Alexander Kechris will present "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.

Thursday, October 12, 2017

4:00 pm in 245 Altgeld Hall,Thursday, October 12, 2017

Lecture III: Descriptive graph combinatorics

Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017. Alexander Kechris will present "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.

Thursday, October 19, 2017

4:00 pm in 245 Altgeld Hall,Thursday, October 19, 2017

Discrete Geometry of polygons and Soliton Equations

Gloria Mari Beffa (University of Wisconsin-Madison)

Abstract: The relation between the discrete geometry of surfaces and completely integrable systems has been well stablished in the last few decades, through work of Bobenko, Suris and many others. The recent introduction of discrete moving frames by Mansfield, Mari-Beffa and Wang, and the study of the pentagram map by Richard Schwartz and many others, has produced a flurry of work connecting the discrete geometry of polygons to some completely integrable systems in any dimension, including connections to Combinatorics and the study of the role that the background geometry has in the generation of algebraic structures that often describe integrability. In this talk I will review definitions and background, and will describe recent advances in the proof of the integrability of discretizations of Adler-Gel’fand-Dikii systems (generalized KdV), aided by the use of the geometry of polygons in RPm.

Thursday, October 26, 2017

4:00 pm in 245 Altgeld Hall,Thursday, October 26, 2017

Planar graphs and Legendrian surfaces

Emmy Murphy (Northwestern)

Abstract: Associated to a planar cubic graph, there is a closed surface in R^5, as defined by Treumann and Zaslow. R^5 has a canonical geometry, called a contact structure, which is compatible with the surface. The data of how this surface interacts with the geometry recovers interesting data about the graph, notably its chromatic polynomial. This also connects with pseudo-holomorphic curve counts which have boundary on the surface, and by looking at the resulting differential graded algebra coming from symplectic field theory, we obtain new definitions of n-colorings which are strongly non-linear as compared to other known definitions. There are also relationships with SL_2 gauge theory, mathematical physics, symplectic flexibility, and holomorphic contact geometry. During the talk we'll explain the basic ideas behind the various fields above, and why these various concepts connect.

Thursday, November 9, 2017

4:00 pm in 245 Altgeld Hall,Thursday, November 9, 2017

Applications of functoriality

Emily Riehl (Johns Hopkins University)

Abstract: Abstract: This talk will introduce the concept of a functor from category theory assuming no prior acquaintance and highlight some applications of this notion. In the first part we will conceive of a functor as a bridge between two different mathematical theories. As a prototypical example we will consider the fundamental group construction from algebraic topology and explain how its functoriality leads to a proof of the Brouwer fixed point theorem. In the second part, we will explain how the search for a functorial clustering algorithm lead to a breakthrough in topological data analysis and speculate how a similar functors might be used to define combinatorial models of metric spaces.

Thursday, November 16, 2017

4:00 pm in 245 Altgeld Hall,Thursday, November 16, 2017

Primes with missing digits

James Maynard (Institute For Advanced Study, Princeton)

Abstract: Many famous open questions about primes can be interpreted as questions about the digits of primes in a given base. We will talk about recent work showing there are infinitely many primes with no 7 in their decimal expansion. (And similarly with 7 replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most $X^{1-c}$ elements less than $X$) which is typically very difficult. The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.

Thursday, November 30, 2017

4:00 pm in 245 Altgeld Hall,Thursday, November 30, 2017

Modeling the evolution of drug resistance in cancer

Jasmine Foo (School of Mathematics, University of Minnesota-Twin Cities)

Abstract: Despite the effectiveness of many therapies in reducing tumor burden during the initial phase of treatment, the emergence of drug resistance remains a primary obstacle in cancer treatment. Tumors are comprised of highly heterogeneous, rapidly evolving cell populations whose dynamics can be modeled using evolutionary theory. In this talk I will describe some mathematical models of the evolutionary processes driving drug resistance in cancer, and demonstrate how these models can be used to provide clinical insights. In particular I will describe several branching process models of tumor evolution and analyze the timing tumor recurrence. These models will be applied to study the impact of dosing schedules and the tumor microenvironment on the emergence of drug resistance in lung cancer.

Tuesday, December 5, 2017

4:00 pm in 245 Altgeld Hall,Tuesday, December 5, 2017

Two Problems in Risk Management with Basis Risk

Jingong Zhang (University of Waterloo)

Abstract: Basis risk occurs naturally in a variety of finance and insurance applications, and introduces additional complexity to risk management. The theme of this presentation is to study risk management in the presence of basis risk under two settings: index insurance design and dynamic longevity hedge. In the first part of the talk, we study the problem of index insurance design under an expected utility maximization framework. We formally prove the existence and uniqueness of optimal contract for general utility functions, and obtain analytical expressions of the optimal indemnity function for exponential utility and quadratic utility functions. Our method is illustrated by a numerical example where weather index insurance is designed for protection against the adverse rice yield using temperature and precipitation as the underlying indexes. When compared to the linear-type contracts that have been advocated in the literature, the empirical results show that our proposed index-based contract is more efficient at reducing farmers’ basis risk. In the second part of the talk, from a pension plan sponsor’s perspective, we study dynamic hedging strategies for longevity risk using standardized securities in a discrete-time setting. The hedging securities are linked to a population which may differ from the underlying population of the pension plan, and thus basis risk arises. Drawing from the technique of dynamic programming, we develop a framework which allows us to obtain analytical optimal dynamic hedging strategies to achieve the minimum variance of hedging error. Extensive numerical experiments show that our hedging method significantly outperforms the standard “delta” hedging strategy which is commonly studied in the literature.

Thursday, December 7, 2017

4:00 pm in 245 Altgeld Hall,Thursday, December 7, 2017

An Introduction to Dynamic Materials

Suzanne Weekes (Worcester Polytechnic Institute)

Abstract: I will give an overview of work on wave propagation through dynamic materials (DM). DM are spatio-temporal composites - materials whose properties vary in space and in time. Mathematically, we formulate the problem as linear, hyperbolic partial differential equations with spatio-temporally varying coefficients. The variability in the material constituents leads to effects that are unachievable through static (spatial-only) design. For example, with dynamic laminates we are able to screen portions of the material from the effects of a wave disturbance. With checkerboard geometry in space-time, we create pulse compression and energy accumulation, and recent work shows that these effects are structurally stable.

Tuesday, December 12, 2017

4:00 pm in 245 Altgeld Hall,Tuesday, December 12, 2017

Weighted Insurance Pricing Model: Gini Shortfall, economic pricing and capital adequacy

Edward Furman (York University)

Abstract: The Capital Asset Pricing Model (CAPM) fetched the Nobel Prize in Economics to Professor William Sharpe back in 1990. Speaking briefly, the CAPM offers a very simple expression to price an asset in a portfolio of assets, and the just-mentioned expression is magically free of the preferences of the involved investors. As the preferences are generally unobserved in reality, and due to the appealing simplicity of the end-result, the CAPM has become one of the most influential pricing models in finance today. In this talk, among other things, I will discuss the reasons for the CAPM's low penetration into the theory of actuarial pricing, and I will then offer an insurance variant of the model. I will show how a fairly general initial set-up can yield—akin to the Sharpe's CAPM case—simple pricing expressions for a large class of risks that are symmetric or non-symmetric, light-tailed or heavy-tailed, independent or dependent. This talk hinges on a number of joint works with Professors Alexey Kuznetsov (York), Ruodu Wang (Waterloo) and Ricardas Zitikis (Western).