## Fall 2017 – Summer 2018 – Abstracts

Wednesday, September 13, 2017. Statistics Colloquium.

Prof. Randy Lai, Department of Mathematics and Statistics, University of Maine.
“Julia, the programming language for 21st century”
3:30 – 4:20 pm, 100 Neville Hall. Snacks at 3:15pm.

Abstract: In this seminar, I will give an introduction to a relatively new programming language called Julia. Before Julia, we had to be satisfied with user-friendly, easy to understand yet decades old and slow scripting languages such as Matlab, R, and Python. To speed up the actual computation, we often had to rely on compiled languages such as C and Fortran. Now, with Julia JIT (just in time) compiler, we are now able to write code in a higher level with less development time while enjoying the fast performance offered by it.

Wednesday, September 27, 2017. Mathematics Colloquium.

Prof. Jerrod Smith, Department of Mathematics and Statistics, University of Maine.
“Distinguished representations and relative discrete series for $p$-adic groups”
3:30 – 4:20 pm, 100 Neville Hall. Snacks at 3:15pm.

Abstract: Let $F$ be a $p$-adic field and $G = \mathbf G(F)$ the $F$-points of a connected reductive group defined over $F$. Given an involution $\theta$ of $G$ we define $H$ to be the subgroup of $\theta$-fixed points. The quotient $H \backslash G$ is a $p$-adic symmetric space.

In this talk, we will discuss the harmonic analysis on $H \backslash G$ and the notion of distinguished representations. In particular, we will consider the problem of constructing the irreducible $G$-representations that occur as subrepresentations of the space of square-integrable functions on $H\backslash G$, the relative discrete series (RDS).

We construct families of non-discrete RDS representations for three symmetric spaces:

1. $\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$,

2. $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, where $E$ is a quadratic Galois extension of $F$, and

3. $\mathbf{U}_{E/F}(F) \backslash \mathbf{GL}_{2n}(E)$, where $\mathbf{U}_{E/F}$ is a quasi-split unitary group over $F$.

We will review Casselman’s Criterion for Square Integrability and its relative (symmetric space) analogue.

Wednesday, October 18, 2017. Mathematics Colloquium.

Prof. Peter Stechlinski, Department of Mathematics and Statistics, University of Maine.
“Dynamic Optimization of Nonsmooth Systems”
3:30 – 4:20 pm,  Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: A variety of engineering problems require dynamic simulation and optimization, but exhibit a mixture of continuous and discrete behavior. Such hybrid behavior can cause failure in conventional methods. Nonsmooth dynamical systems provide a natural modeling framework for a large class of these problems. Moreover, thanks to recent advancements in nonsmooth analysis, nonsmooth dynamical systems now have a suitable foundational theory for use in, for example, dynamic optimization. Motivated by control and optimization problems in process systems engineering, the theory allows for numerical implementations that scale efficiently for large-scale problems. This work is placed in the context of state-of-the-art modeling efforts for systems displaying hybrid behavior.

Wednesday, October 25, 2017. Mathematics Colloquium.

Prof. Sougata Dhar, Department of Mathematics and Statistics, University of Maine.
“Linear and half linear Lyapunov-type inequalities and applications”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: In this talk, we will discuss several Lyapunov-type inequalities for third order linear and half-linear differential equations. These inequalities utilize integrals of both $q_+(t)$ and $q_-(t)$ rather than those of $|q(t)|$ as in most papers in the literature for higher order Lyapunov-type inequalities. In the process, we also obtain a sharper inequality. Furthermore, by combining these inequalities with the “uniqueness implies existence” theorems by several authors, we establish the uniqueness and hence existence-uniqueness for several classes of boundary value problems for third-order linear equations. We believe that this is the first time for Lyapunov-type inequalities to be used to deal with boundary value problems and expect that this approach can be further applied to study general higher-order boundary value problems.

Thursday, November 16, 2017. Statistics Colloquium.

Prof. Erin Conlon, Department of Mathematics and Statistics, University Massachusetts Amherst.
“Parallel Markov Chain Monte Carlo Methods for Bayesian Analysis of Big Data”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: Recently, new parallel Markov chain Monte Carlo (MCMC) methods have been developed for massive data sets that are too large for traditional statistical analysis. These methods partition big data sets into subsets, and implement parallel Bayesian MCMC computation independently on the subsets. The posterior MCMC samples from the subsets are then joined to approximate the full data posterior distributions. Current strategies for combining the subset samples include averaging, weighted averaging and kernel smoothing approaches. Here, I will discuss our new method for combining subset MCMC samples that directly products the subset densities. While our method is applicable for both Gaussian and non-Gaussian posteriors, we show in simulation studies that our method outperforms existing methods when the posteriors are non-Gaussian. I will also discuss computational tools we have developed for carrying out parallel MCMC computing in Bayesian analysis of big data.

Wednesday, November 29, 2017. Mathematics Colloquium.

Prof. Evan Randles, Department of Mathematics and Statistics, Colby College.
“Convolution powers of complex-valued functions on $\mathbb{Z}^d$
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB) Snacks at 3:15pm.

Abstract: The study of convolution powers of a finitely supported probability distribution $\phi$ on the $d$-dimensional square lattice is central to random walk theory. For instance, the $n$th convolution power $\phi^{(n)}$ is the distribution of the $n$th step of the associated random walk and is described by the classical local limit theorem. When such distributions take on complex values, their convolution powers exhibit surprising and disparate behaviors not seen in the probabilistic setting. In this talk, I will discuss new results concerning the asymptotic behavior of convolution powers of complex-valued functions on $\mathbb{Z}^d$, specifically generalized local limit theorems and sup-norm estimates. This joint work with Laurent Saloff-Coste extends previous results by I. J. Shoenberg, T. N. E. Greville, P. Diaconis and L. Saloff-Coste.

Wednesday, December 6, 2017. Mathematics Grad Seminar.

Alice Wise, UMaine Mathematics MA student.
“From Analytic Continuation to the Schwarz Reflection Principle”
3:00 – 3:50 pm, 421 Neville Hall.

Abstract: Every complex function has a largest naturally defined domain, and when such a function is holomorphic and expanded in a Taylor Series, the domain of that series is usually much smaller. In this talk, we introduce the relationship between an analytic function defined as a Taylor Series and the global analytic function it represents by means of analytic continuation. Further, we will display different techniques of analytic continuation which help to determine the domain of a global analytic function much faster and more easily.

Thursday, December 14, 2017. Mathematics Grad Seminar.

Abel Lourenco, UMaine Mathematics MA student.
“Dirichlet’s Theorem on primes in arithmetical progressions”
2:00 – 2:50 pm, 421 Neville Hall.

Abstract: The study of prime numbers has always amazed mathematicians for its beauty and surprisingly hard questions. Euclid was one of the first to work on them and showed that there are infinitely many prime numbers. Euler studied the series of inverses of prime numbers $\sum_{p \in \mathbb{P}} \frac{1}{p}$ and showed it diverges. Dirichlet showed that any arithmetic progession $a, a+m, a+2m, \ldots$, where $\gcd(a,m)=1$, contains infinitely many prime numbers. In this talk we will elaborate on these topics, with a focus on Dirichlet’s theorem.

Wednesday, April 11, 2018. Mathematics Colloquium.

Omer Offen, Department of Mathematics, Brandeis University and Technion-Israel Institute of Technology.
“Models for representations of general linear groups”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB) Snacks at 3:15pm.

Abstract: A part of Gelfand’s philosophy is to study the representations of a group G by finding a natural representation (a model) that is isomorphic to the direct sum of all irreducible representations of G. A beautiful model for GL(n,F) where F is a finite field was introduced by Klyachko in 1983. I will discuss the Klyachko model and to what extent can the construction generalize when F is a local field.