Past Year Events of the Week

Week of March 30 - March 26, 2018

*Friday, March 30*

**Campus Closed**

*Thursday, March 29*

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*Wednesday, March 28*

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*Tuesday, March 27*

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*Monday, March 26*

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Week of March 23 - March 19, 2018

Week of March 30 - March 26, 2018

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Week of March 23 - March 19, 2018

**Pramod Achar - LSU
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This talk is about the "Hecke category," a monoidal category that appears in various incarnations in geometric representation theory. I will explain some of these incarnations and their roles in solving classical problems, such as the celebrated Kazhdan-Lusztig conjectures on Lie algebra representations. These conjectures (proved in 1981) hinge on the fact that the derived category of constructible sheaves on a flag variety is equipped with an obvious monoidal action of the Hecke category on the right.

It turns out that there is also a second, "hidden" action of the Hecke category on the left. The symmetry between the "hidden" left action and the "obvious" right action leads to the phenomenon known as Koszul duality. In the last part of the talk, I will discuss new results on Koszul duality with coefficients in a field of positive characteristic, with applications to characters of tilting modules for algebraic groups. This is joint work with S. Makisumi, S. Riche, and G. Williamson.

**Location: Gibson Hall 126
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**Time: 3:00 PM**

**Richard Stanley - Massachusetts Institute of Technology
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The sequence 1, 1, 2, 5, 14, 42, 132, ... of Catalan numbers is perhaps the most ubiquitous integer sequence in mathematics. We will give a survey of these numbers for a general mathematical audience. Topics will include the history of Catalan numbers, some combinatorial interpretations (taken from the 214 interpretations in my monograph on Catalan numbers), some algebraic interpretations, one of the many known generalizations of Catalan numbers, and some connections with number theory and analysis.

**Location:** Gibson Hall 126

**Time: 3:30**

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**Cecilia Mondaini - Institution
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The goal of data assimilation is to obtain an accurate prediction of the future state of a physical system by suitably combining a theoretical model with observational data. The challenge consists in how to use the finite-dimensional information given by the data in order to recover the true state of a complex physical system - possessing a large number of degrees of freedom. In this talk, I will consider the nudging method for data assimilation within an infinite-dimensional framework. Further, I will present about error estimates concerning finite-dimensional numerical approximations of this nudging algorithm.

**Location: Gibson Hall 310
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**Time: 12:00 PM**

**Sergio Villamarin -ITulane University
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In order to make a finite interpretation to the second law of thermodynamics using Boltzmann entropy, we propose a particular finite mathematical model, a Micro-Macro-Dynamical-System (MMD-system), in which we show a characterization of a perfect entropy MMD-system, showing that in a mathematical context the second law of thermodynamics almost never applies. We also find the average number of MM-systems that have a perfect entropy state by fixing the dynamics, the macro-states or both proving a combinatorial identity. After this we show how to bound the error set of an MM-system and characterize the worst-case scenario for the second law.

**Location: Stanley Thomas 316
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**Time: 4:30 PM**

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Week of March 16 - March 12, 2018

**Rafal Komendarczyk - Tulane University
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I will survey the theory of finite type invariants of knots and links, originally developed by Vassiliev. Will also discuss applications to geometric knots theory such as ropelength, embedding thickness and estimates for energies of fluid knots.

**Location:** Gibson Hall 310

**Time: 12:00 pm**

**Sergey Grigorian - Sergey Grigorian
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The group G2 is the automorphism group of the octonion algebra. Given a G2-structure on a 7-dimensional Riemannian manifold we define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonionic covariant derivative on this bundle that is also compatible with the octonion product. The torsion of the G2-structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the Riemann curvature that lies in the 7-dimensional representation of G2. The choice of a particular G2-structure within the same metric class is then interpreted as a choice of gauge and we show that under a change of this gauge, the torsion transforms as an octonion-valued connection 1-form. We will then discuss further properties of this non-associative structure on 7-manifolds.

**Location:** Gibson Hall 126

**Time:** 3:00

**Geng Chen - University of Kansas
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Compressible Euler equations (introduced by Euler in 1757) model the motion of compressible inviscid fluids such as gases. It is well-known that solutions of compressible Euler equations often develop discontinuities, i.e. shock waves. Successful theories have been established in the past 150 years for small solutions in one space dimension. The theory on large solutions is widely open for a long time, even in one space dimension. In this talk, I will discuss some recent exciting progresses in this direction. The talk is based on my joint works with A. Bressan, H.K. Jenssen, R. Pan, R. Young, Q. Zhang, and S. Zhu.

**Location: Gibson Hall 310
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**Time: 3:00**

**Ron Harris - Brigham Young University
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Historical records we have compiled demonstrate that over the past 400 years there have been 105 tsunamis throughout Indonesia, which is an average of at least1 tsunami every 4 years. Just in the past 22 years 8 tsunamis have struck Indonesia, 7 of which caused numerous fatalities (>200,000 total deaths). Women and children account for the majority of these deaths. However, most of these lives could have been saved if those in harm’s way would have known they were at risk, the natural signs that a tsunami was approaching and how to respond. These three fundamental aspects of tsunami disaster preparedness require an integrated, multidisciplinary approach to resolve as demonstrated in Japan during the 2011 tsunami that struck there. The tsunamis in Indonesia and Japan were similar in size and impacted around the same numbers of people; yet, in Japan there were less than 1 death for every 10 in Indonesia. The difference? Resilience!

Building resilience to natural hazards in areas most at risk is the focus of the WAVES consortium, which is a multi-disciplinary partnership of academic and government research institutions dedicated to reversing the increasing losses to nature in developing countries vulnerable to natural hazards. We identify those communities most at risk and help implement community-based disaster mitigation strategies that will save lives from future natural hazards. The project has three integrated goals: 1) ‘Listen to Earth’ to determine who’s most at risk. This task involves applying novel statistical inversion techniques to determine who’s most at risk based on historical, archeological and geological records of past hazardous events. 2) ‘Listen to the People’ through conducting questionnaire surveys to determine levels of awareness and readiness, and town hall meetings. This information is vital in designing presentations and public service videos to communicate risk and effective disaster mitigation strategies in a cultural context. 3) ‘Empower People to Listen to Earth’ by assisting local communities to implement and sustain their own risk reduction strategies connected to natural warning signs.

One of the most important technical aspects of the research is the construction of tsunami flooding maps for at risk communities. These maps provided a way to communicate who is most at risk and the details for each site about expected tsunami escape times and run up heights. A team that integrates the expertise of geoscientists, mathematicians and statisticians at BYU, Virgina Tech and Tulane University conducts the numerical modeling.

We used the tsunami inundation maps as a starting point to assist local communities in tsunami disaster mitigation planning and implementation of risk reduction strategies. These maps include the areas likely to flood during a tsunami, predicted waves heights in these areas, the number of people inhabiting these areas and the time after the earthquake of the arrival of the tsunami. Essentially, they communicate who is most at risk, safe evacuate sites and the time available to evacutate. In most cases, those at risk can know a tsunami is approaching if they feel an earthquake that shakes for > 20 seconds. At that point they may have only 20 minutes to escape to an elevation of 20 m.

**Location: Gibson Hall 310
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**Time: 2:00**

**Jianhong Wu - York University
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**Location: Gibson Hall 126
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**Time: 3:30**

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Week of March 9 - March 5, 2018

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**Sebastian Schreiber - UC Davis (HOST Scott McKinley)
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Demographic stochasticity stems from populations consisting of a finite number of interacting individuals. These dynamics can be represented by Markov chains on a countable state space. For closed populations in a bounded world, extinction in these models occurs in finite time, but may be preceded by long-term transients. Quasi-stationary distributions (QSDs) of these Markov chains characterize this meta-stable behavior. These QSDs correspond to an eigenvector of the transition operator restricted to non-extinction states, and the associated eigenvalue determines the mean time to extinction when the Markov chain is in the quasi-stationary state. I will discuss under what conditions (i) this mean time to extinction increases exponentially with "habitat size" and (ii) the QSDs concentrate on attractors of the mean field model of the Markov chain. These results will be illustrated with models of competing Californian annual plants and chaotic beetles.

On the other hand, environmental stochasticity stems from fluctuations in environmental conditions which influence survival, growth, and reproduction. These effects on population and community dynamics can be modeled by stochastic difference or differential equations. For these models, "stochastic persistence" corresponds to the weak* limit points of the empirical measures of the process placing arbitrarily little weight on arbitrarily low population densities. I will discuss sufficient and necessary conditions for stochastic persistence. These conditions involve Lyapunov exponents corresponding to the "realized" per-capita growth rates of species with respect to stationary distributions supporting subsets of species. These results will be illustrated with models of Bay checkerspot butterflies and eco-evolutionary rock-paper-scissor dynamics.

**Location:** Gibson Hall 126

**Time:** 3:30

**Arkady Leiderman - Department of Mathematics, Ben-Gurion University of the Negev, Israel (joint work with Sidney Morris, and Mikhail Tkachenko)
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**Location:** Gibson Hall 308

**Time: ** 1:00 PM

**Yan gu - Tulane University
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Over the last decade or so, commutative algebraists have become interested in studying the properties of finite simple graphs through monomial ideals. The connection between graph theory and commutative algebra help people import results from graph theory to algebraic results, and at the same time, export algebraic results to graph theory results.

**Location:** Gibson Hall 310

**Time: 12:00 pm**

**Hayden Houser - Tulane University
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Location:** Stanley Thomas 316

**Time:** 4:30pm

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**Karen Zaya - University of Michigan
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The regularity of solutions to equations of fluid motion remains a significant open problem. A vast amount of literature has been devoted to studying regularity criteria, such as the classical Beale-Kato-Majda and Ladyzhenskaya-Prodi-Serrin regularity conditions. We will review some of this literature for the three-dimensional Euler, Navier-Stokes, Boussinesq, and magnetohydrodynamics equations. Then, in the framework of Kolmogorov’s theory of turbulence, we will discuss how work with the dissipation wavenumber and determining modes has produced new, weaker regularity criteria for these equations.

**Location:** Gibson Hall 310

**Time:** 3:00 PM

**Karl Hofmann - Institution
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**Location: Gibson Hall 126
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**Time: 3:00 PM**

**Alejandro Morales - UMass-Amherstn (Host: Tewodros Amdeberhan)
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Flow polytopes of graphs is a rich family of polytopes that include the Pitman-Stanley polytope and a face of the polytope of doubly stochastic matrices called the Chan-Robbins-Yuen polytope. Lattice points of polytopes are counted by Kostant's vector partition function from Lie theory. In the early 2000s, Postnikov-Stanley and Baldoni-Vergne gave remarkable formulas for their volume and lattice points using the Elliott-MacMahon algorithm and residue computations respectively.

In this talk we will describe these polytopes, how to subdivide them to obtain these formulas, and a model for the formulas using certain well-known combinatorial objects called parking functions. We will illustrate the subdivision and the model with known and new examples of flow polytopes with surprising volumes.

This is based on joint work with Karola Meszaros and joint work with Carolina Benedetti, Rafael Gonzalez D'Leon, Chris Hanusa, Pamela Harris, Apoorva Khare and Martha Yip.

**Location: Gibson Hall 126
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**Time: 3:30 PM**

**Simon Mak - Georgia Tech University
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This talk presents a new method for compacting large datasets (or in the infinite-dimensional setting, distributions) into a smaller, representative point set called support points (SPs). In an era where data is plentiful but analysis is oftentimes expensive, the proposed data reduction technique can be used to efficiently tackle many challenging big data problems in engineering, statistics and machine-learning. Using a popular distance-based statistical energy measure introduced in Székely and Rizzo (2004), SPs can be viewed as minimum-energy points under the potential field induced by big data. As such, these point sets enjoy several nice theoretical properties on distributional convergence, integration performance and functional approximation. One key advantage of SPs is that it allows for an efficient and parallelizable reduction of big data via difference-of-convex programming. This talk concludes with several real-world applications of SPs, for (a) compacting Markov chain Monte Carlo (MCMC) sample chains in Bayesian computation, (b) propagating uncertainty in expensive simulations, and (c) efficient kernel learning with big data.

**Location:** Stanley Thomas 316

**Time:** 2:00 PM

**Robyn Brooks - Tulane University
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Directed Topology is a relatively new field of topology that arose in the `90s as a result of the abstraction of homotopy theory. The general aim of this theory is to model non-reversible phenomena. In this talk I will introduce the basics of directed topology and dihomotopy theory, and provide several illustrative examples. Finally, I will discuss a few of the potential tools that may be used to further research in this area.

**Location: Stanley Thomas 316
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**Time: 4:30**

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**Siran Li - Rice University
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On the Geometric Regularity Criteria for Incompressible Navier--Stokes EquationsSiran Li - Rice UniversityAbstract: We present some recent results on the regularity criteria for weak solutions to the incompressible Navier--Stokes equations (NSE) in 3 dimensions. By the work of Constantin--Fefferman, it is known that the alignment of vorticity directions is crucial to the regularity of NSE in $\R^3$. In this talk we show a boundary regularity theorem for NSE on curvilinear domains with oblique derivative boundary conditions. As an application, the boundary regularity of incompressible flows on balls, cylinders and half-spaces with Navier boundary condition is established, provided that the vorticity is coherently aligned up to the boundary. The effects of vorticity alignment on the $L^q$, infty$ norm of the vorticity will also be discussed.

**Location: Gibson Hall 310
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**Time: 3:00**

**Laszlo Fuchs - Tulane University
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**Location: Gibson Hall 126
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**Time: 3:00**

**Aihua Li - Montclair State University
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Let R be a commutative ring with identity 1 and T be the non-commutative ring of all n by n upper triangular matrices over R. In this talk, I will introduce the zero divisor graph of T. Some basic graph theory properties of the graph are given, including determination of the girth and diameter. The structure of such a graph is discussed and bounds for the number of edges are given. In the case that T is a 2 by 2 upper triangular matrix ring over a finite integral domain, the structure of the graph is fully determined. In this case an explicit formula for the number of edges is given.

**Location: Gibson 308
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**Time: 12:00**

**Jinchi Lv - University of Southern California
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Power and reproducibility are key to enabling refined scientific discoveries in contemporary big data applications with general high-dimensional nonlinear models. In this paper, we provide theoretical foundations on the power and robustness for the model-X knockoffs procedure introduced recently in Candes, Fan, Janson and Lv (2017) in high-dimensional setting when the covariate distribution is characterized by Gaussian graphical model. We establish that under mild regularity conditions, the power of the oracle knockoffs procedure with known covariate distribution in high-dimensional linear models is asymptotically one as sample size goes to infinity. When moving away from the ideal case, we suggest the modified model-free knockoffs method called graphical nonlinear knockoffs (RANK) to accommodate the unknown covariate distribution. We provide theoretical justifications on the robustness of our modified procedure by showing that the false discovery rate (FDR) is asymptotically controlled at the target level and the power is asymptotically one with the estimated covariate distribution. To the best of our knowledge, this is the first formal theoretical result on the power for the knockoffs procedure. Simulation results demonstrate that compared to existing approaches, our method performs competitively in both FDR control and power. A real data set is analyzed to further assess the performance of the suggested knockoffs procedure. This is a joint work with Emre Demirkaya, Yingying Fan and Gaorong Li.

**Location: Gibson Hall 126
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**Time: 3:30**

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**Hung Nguyen - Tulane University
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**Location: Gibson Hall 126
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**Time: 3:00 PM**

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**Michael Kelly - Loyola University
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Given a self-map of a compact, connected topological space we consider the problem of determining upper and lower bounds for the fixed point indices of the map. To obtain bounds one needs to restrict attention to the class of spaces considered and also the class of self-maps. Motivated by an elementary result in the case of a 1-dimensional complex this talk will focus attention to the setting of 2-complexes. Some past results and related examples will be presented, leading to some current joint work with D. L. Goncalves (U. Sao Paulo, Brasil).

**Location: Gibson Hall 308
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**Time: 12:30 PM**

**Tewodros Amdeberhan - Tulane University
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How to delve into your research? One might say, just do it! This is by no means to profess to you on "how to". Instead, I will illustrate (in detail) certain mathematical tools with which I approached concrete projects.

**Location: Gibson Hall 310
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**Time: 12:00 pm**

**Vasileios Maroulas - University of Tennessee
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This talk navigates us through the landscape of stochastic filtering, its computational implementations and their applications in science, engineering and national defense. We start by exploring properties of the optimal filtering distribution. Under general conditions, the filtering distribution does not enjoy a closed form solution. Employing several methods, e.g. particle filters, we approximate it and we explore properties of the underlying process and its engaging parameters. The parameter estimation leads us to a research path which involves a novel algorithm of particle filters blended with a Markov Chain Monte Carlo scheme, a sequential Empirical Bayes method and related sufficient estimators. Last, this talk adopts this research path and sheds light on the estimation of the spatiotemporal evolution of radioactive material caused by the disastrous accident at the Fukushima power plant station in 2011.

**Location: Stanley Thomas 316
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**Time: 3:30 PM**

**Joseph Skelton - Tulane University
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This talk will present the foundation for the case k=2 as a base case for showing the equality of symbolic and ordinary powers of edge ideals of cycles. The proof itself works off of basic properties of ideals and rings. I will introduce basic definitions and theorems as needed.

**Location: Stanley Thomas 316
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**Time: 4:30 PM**

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**Daniel Ramras - Institution: Indiana University – Purdue University Indianapolis
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**Location: Gibson Hall 308
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**Time: 9:00 am**

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**Jenifer Hom - Georgia Tech
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The knot concordance group C consists of knots in S^3 modulo knots that bound smooth disks in B^4, with the operation induced by connected sum. We consider various generalizations of the knot concordance group, and compare these to the classical case. This is joint work with Adam Levine and Tye Lidman.

**Location: Gibson Hall 308
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**Time: 12:30**

**Marisa C. Eisenberg - University of Michigan (Hosts: Ricardo Cortez, Lisa J Fauci and James Hyman)
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Connecting dynamic models with data to yield predictive results often requires a variety of parameter estimation, identifiability, and uncertainty quantification techniques. These approaches can help to determine what is possible to estimate from a given model and data set, and help guide new data collection. Here, we will discuss differential algebraic and simulation-based approaches to identifiability analysis, and examine how parameter estimation and disease forecasting are affected when examining disease transmission via multiple types or pathways of transmission. Using examples taken from cholera and polio outbreaks in several settings, we illustrate some of the potential difficulties in estimating the relative contributions of different transmission pathways, and show how alternative data collection may help resolve this unidentifiability.

**Location: Gibson Hall 126
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**Time: 3:30**

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Week of January 26 - January 22, 2017

**Reuven Hodges - Institution
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Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process, due to Alfred Young, that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. It has been a long-standing open problem to give a non-iterative, closed formula for this straightening process.

In this talk I will give such a formula, as well as a simple combinatorial description of the coefficients that arise. Moreover, an interpretation of these coefficients in terms of paths in a directed graph will be explored. I will end by discussing a surprising application of this formula towards finding multiplicities of irreducible representations in certain plethysms and how this relates to Foulkes' conjecture.

**Location: Gibson Hall 126
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**Time: 3:00**

**Pramita Bagchi - Ruhr-Universitat Bochum
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We propose a new measure for stationarity of a functional time series, which is based on an explicit representation of the L^2-distance between the spectral density operator of a non-stationary process and its best (L^2-)approximation by a spectral density operator corresponding to a stationary process. This distance can easily be estimated by sums of Hilbert-Schmidt inner products of periodogram operators (evaluated at diﬀerent frequencies), and asymptotic normality of an appropriately standardised version of the estimator can be established for the corresponding estimate under the null hypothesis and alternative. As a result we obtain conﬁdence intervals for the discrepancy of the underlying process from a functional stationary process and a simple asymptotic frequency domain level α test (using the quantiles of the normal distribution) for the hypothesis of stationarity of functional time series. Moreover, the new methodology allows also to test precise hypotheses of the form “the functional time series is approximately stationarity”, which means that the new measure of stationarity is smaller than a given threshold. Thus in contrast to methods proposed in the literature our approach also allows to test for “relevant” deviations from stationarity.

We demonstrate in a small simulation study that the new method has very good ﬁnite sample properties and compare it with the currently available alternative procedures. Moreover, we apply our test to annual temperature curves.

**Location: Stanley Thomas 316
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**Time: 3:30**

**Mentor Stafa - Tulane University
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**Organizational Meeting
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**Location: TBA
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**Time: 12:30**

**Gideon Simpson - Drexel University (Host Nathan Glatt-Holtz)
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One of the outstanding challenges in atomistic simulations of materials is how to reach physically meaningful time scales. While the fundamental time scale of the atomistic models is that of the femtosecond, physically meaningful phenomenon may take microseconds or longer to occur. This precludes a direct numerical simulation with, for instance, a Langevin model of the material from reaching physical time scales. The time scale separation challenge has motivated the development of a variety of multiscale methods, including accelerated molecular dynamics, kinetic Monte Carlo, phase field models, and diffusive molecular dynamics. In this talk, I will survey some of these approaches and discuss common mathematical assumptions that underlie them while also highlighting where approximations have been made. Rigorous results will be presented, where available, along with outstanding mathematical challenges.

**Location: Stanley Thomas 316
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**Time: 3:00**

**Aram Bingham - Tulane University
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In the words of Scott Aaronson, Geometric Complexity Theory is ``a staggeringly ambitious program for proving P is not equal to NP that throws almost the entire arsenal of modern mathematics at the problem, including geometric invariant theory, plethysms, quantum groups, and Langlands-type correspondences―and that relates the P = NP problem, at least conjecturally, to other questions that mathematicians have been trying to answer for a century.'' We will say as much as we can about this area in 40 minutes or so.

**Location: Stanley Thomas 316
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**Time: 4:30PM**

**Dan Cheng - Texas Tech
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**Abstract: **The height distributions of critical points of random fields arise from p-value computations when performing hypotheses tests at critical points such as local maxima. In this talk, we will show the formulae for the height distributions of critical points of smooth isotropic Gaussian random fields. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. We then apply the results to a topological multiple testing scheme for detecting peaks in images under stationary ergodic Gaussian noise in Euclidean space, where tests are performed at local maxima of the smoothed observed signals. The developed STEM algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency, with specific rates, as the search space and signal strength get large. Simulations show that FDR levels are maintained in non-asymptotic conditions. The methods are illustrated in the analysis of functional magnetic resonance images of the brain. The method of multiple testing of local maxima are also extend to Gaussian random fields on the sphere, providing a powerful tool to detect point sources in CMB data in astronomy. Another important application is detecting change points by performing multiple testing of critical points of the smoothed observed signal. We will also discuss some open problems and future research.

**Location: Stanley Thomas 316
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**Time: 3:30**

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Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu