Mathematics & Statistics Seminar Talks

Seminar on Analysis and Applications
Lee Jones (UMass Lowell): Some Improvements of Hoeffding and Chernoff Bounds with Applications to Machine Learning and Confidence Analysis

Organizer: Alexander Kheifets, email: Alexander_Kheifets@uml.edu
February 25, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: We first show how to improve known bounds on Pr ( S - m is greater than or equal t ) where S is the sum of k independent but not identically distributed (i. nid.)Bernoulli random variables, t is positive and m = E ( S ) . No further distributional assumptions are made on S. These improvements are best possible among those based on the application of Markov’s inequality to the above probability using the moment generating function of S. The results yield significant improvements of Hoeffding's inequality when t/k is not too small ( 28% better for t = 5, k = 10) and significant improvements for smaller t/k (like halving type 2 error) that can be applied to the k- nearest neighbor estimate s/k of class probability for a query in a machine learned feature space, obtaining a reproduceable or compatible confidence statement (95.4% vs. a frequentist 91.1% upper confidence interval ((s - t)/100 , 1] if t = 11, s = 80, k = 100). Or they can be used by a pollster who accepts that responders j have varying probabilities p(j ) of favoring a candidate, especially early in a campaign.
The more general problem with S being a sum of k i.nid. random variables with finite support is then considered. A (best) improvement of Hoeffding’s lemma is derived both with no assumptions on the value of the mean and when the value of the mean is specified. These cases have attracted recent interest as well as cases where higher moment information is available. The improvement of the lemma yields a significant (but not best) improvement of Hoeffding’s inequality for the finite support case. Obtaining best ( via Markov inequality) bounds in the Bernoulli case for weighted k- nearest neighbor machine learning turns out to be computationally challenging. This weighted sum Bernoulli problem is equivalent to the best bound determination for the Hoeffding inequality in the finite support case.

Seminar on Analysis and Applications
Clement Tayie (UMass Lowell): An Accurate and Efficient Numerical Method for Singular Integrals on a Sphere

Organizer: Bobbie Wu
March 25, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: Many application problems, such as fluid simulation, require the accurate and efficient numerical evaluation of singular integrals on the sphere. In this talk, we will introduce the "Spherical Zeta Quadrature", a fast and high-order numerical method for singular surface integration on spherical domains. The method combines the theory of generalized Euler-Maclaurin summation formulas for singular quadrature and a fast spherical rotation algorithm based on the Non-Uniform Fast Fourier Transform to handle singularities at all locations on the spherical mesh efficiently. Numerical examples will be presented to demonstrate the accuracy and computational efficiency of the method.

Student Seminar
Gabriel Ong (University of Bonn): P-adic Geometry

Organizers: Stephanie Atherton, Billy Wehring
Part I: March 25, 3 p.m. - 4 p.m., Room: Southwick Hall 350W, Zoom: https://uml.zoom.us/j/8051858440
Part II: April 1, 3 p.m. - 4 p.m., Room: Southwick Hall 350W, Zoom: https://uml.zoom.us/j/8051858440

Abstract: In this two part talk, we parallel non-Archimedean geometry with classical settings. In Part I, we discuss rigid geometry from Tate's perspective in comparison with complex algebraic geometry. One of the most important tools in geometry is cohomology as an invariant of a space. In Part II, we give a broad overview of p-adic cohomology theories comparing the cohomological framework of classical Hodge theory.

Seminar: Working on What (WOW)
Billy Wehring (UMass Lowell): Quite Remarkable Quiver Representations

Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
October 1, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: In this beginner-friendly talk, we will learn from the ground up what quiver representations are. In particular, we will learn what a quiver is and how to construct two types of representations of quivers, known as projective and injective representations. We will also learn about morphisms between representations. Finally, we will learn how to construct what are known as 'Auslander–Reiten Quivers' using polygons.

Seminar on Analysis and Applications
Nguyen Lam (Memorial University of Newfoundland): Uncertainty Principles: Best constants, Optimizers and the stability
Organizer: Hung Phan, email: hung_phan@uml.edu
February 24, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: The Heisenberg uncertainty principle, which is a fundamental result in quantum mechanics, and related inequalities such as the hydrogen and Hardy uncertainty principles, belong to the family of geometric inequalities known as the Caffarelli-Kohn-Nirenberg inequalities. In this talk, we discuss some recent results about the optimal uncertainty principles, Caffarelli-Kohn-Nirenberg inequalities, and their quantitative stability.

Seminar: Working on What (WOW)
Julio Enrique Castrillon Candas (Boston University): Statistical computational mathematics for complex models: Applications to machine learning, protein interactions and Alzheimer’s disease
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
March 19, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: With the advent of massive data sets and high dimensional complex stochastic models appearing in many scientific and engineering fields, novel approaches to prediction, data and statistical analysis, machine learning, and optimization under uncertainty are needed. The merging of statistics, stochastics, computational applied math, and high performance computing has opened the door to solving these and many related complex problems. Computational prediction, inference, optimization and machine learning form the basis for critical decisions in areas of engineering and science such as infrastructure/disaster management, drug discovery, remote sensing and medical diagnosis, among many others. By adapting to the nature of their uncertainty these problems can be solved in a more optimal and accurate manner. My specific interests lie in mathematically rigorous functional analysis for random fields, Machine Learning (ML), Uncertainty Quantification (UQ), stochastic Partial Differential Equations, non-linear stochastic networks and large scale computational statistics spanning pure and applied mathematics with high performance computing. Complementing the research are several funded high impact applications such as power systems, protein interactions, deforestation detection and Alzheimer's disease subtyping.
The speaker's website

Seminar: Working on What (WOW)
Jianping Pan (Arizona State University): Polynomials from Schubert calculus via diagrams
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
April 7, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: Polynomials are powerful tools in many fields, for example, representation theory, geometry, and topology. Understanding the combinatorics arising from polynomials may reveal important information in problems from these fields. I will talk about several families of polynomials arising from Schubert calculus, which originated in enumerative geometry. I will discuss some combinatorial models related to these polynomials, including (K)-Kohnert diagrams and snow diagrams. These discrete objects, along with operations defined on them, exhibit rich combinatorial properties. They can be used to simplify computations and extract algebraic and geometric information within these polynomials.

Seminar: Working on What (WOW)
Lee K. Jones (UMass Lowell): On local statistical machine learning
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
September 18, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: We first review some optimal finite sample accuracy bounds in Loader [1999] and J. [2009] for weighted k-nearest neighbor rules for estimating a function f at a fixed point x under “smoothness” assumptions and conditions of approximate linearity in the spherical neighborhood about x containing the k nearest neighbors. Such bounds are ancillary i.e.depend only on the design points and not on the responses (unlike many obtained in logistic regression and with variance/bias tradeoff plug-in estimates). We extend these results here to include neighborhoods which consist of the convex hull of x and any sub-collection of the neighbors. These new bounds can be computed using Karmarkar’s linear programming algorithm (with a quadratic constraint ) and minimization of a non-smooth convex function. The convex minimization problem, using optimization algorithms of Nesterov (2004), is slower than that for smooth convex functions but is achievable with today’s computers. We conjecture that an appropriate greedy relaxed approach converges at a faster rate. We can show that speed-ups using proximal functions ( as are used in global machine learning e.g. lasso regression, inverse problems, matrix completions, etc.) are not possible.

We indicate how to implement both the old and new results for binary classification problems or proportions sampling and search over lower dimensional subspaces of the design domain for accurate estimates of conditional success probability given the projection of x.

Seminar: Working on What (WOW)
Jim Propp (UMass Lowell):“Tilings? (Again?)”
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
October 23, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: From the late 1980s to the early 2000s I did a lot of work on tiling. For the next 20 years I did very little work in this area. Suddenly I’m working on tilings again. What’s up with that?

Seminar on Algebraic Combinatorics
Benjamin Dequêne (University of Picardie Jules Verne): A generalized RSK correspondence via the combinatorics of (type A) quiver representations
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
January 24, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: The Robinson-Schensted-Knuth (RSK) correspondence is a bijection from nonnegative integer matrices to pairs of semi-standard Young tableau. A generalized version of RSK gives a bijection from fillings of a tableau of shape lambda to reverse plane partitions of shape lambda.

From the quiver representation point of view, the RSK correspondence provides a transformation between two different invariants of a module X (in a certain subcategory). The entries in the arbitrary filling of shape lambda correspond to multiplicities of indecomposable summands of the representation, while the entries in the reverse plane partition of shape lambda record the generic Jordan form data of X, an invariant introduced by Garver, Patrias and Thomas.

My talk aims to present a version of RSK that works from the most general possible choice of a subcategory of the category of representations of a type A quiver. Note that this talk will not assume that the audience has prior knowledge of quiver representations.

This is a combinatorial extraction (in progress) of my Ph.D. work, supervised by Hugh Thomas.

Seminar: Working on What (WOW)
Daniel Glasscock (UMass Lowell): Difference sets: not Bohring but potentially Bohr
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
February 21, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: It is a basic and often useful fact in analysis that convolutions make things smoother. Less famous is the additive combinatorial analogue: set sums and differences support richer structures (and, hence, are “smoother”). This fact – known almost a century ago – is still being refined and finding new applications in additive number theory today. And there are still many open questions. In this talk, we will discuss the following open question: must the difference set A-A of a syndetic subset of integers A contain a Bohr set?

Seminar: Working on What (WOW)
Amanda Redlich (UMass Lowell): Eeny-Meeny-Miney-Moe: Random approaches to solving hard problems
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
March 20, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: Some problems seem impossibly hard: computing the Ramsey number R(5,5), modeling the internet, picking a grocery store checkout line. It turns out that using random guesses is a great approach to all three of these! A lot of my research uses randomized algorithms and the probabilistic method. In this talk I'll give an overview of some of the classical results in the area.

Seminar on Analysis and Applications
Yuan Liu (Wichita State University): High order structure-preserving numerical methods for convection-diffusion-reaction equations
Organizer: Shiwen Zhang, email: shiwen_zhang@uml.edu
April 1, 11 a.m. - Noon, Room: Southwick Hall 350W

Abstract: Convection-diffusion-reaction (CDR) equation is one of the widely used mathematical models in science and engineering. It describes how one or more substances distributed under the influences of convection, diffusion and reaction processes. In this talk, we will present some recent work on high order numerical methods for solving CDR equation under two cases. (1) When there are only convection terms, the CDR equation is hyperbolic conservation laws. We will talk about the development of high order bound-preserving numerical methods. (2) When there are only diffusion and reaction terms, Krylov implicit integration factor discontinuous Galerkin methods on sparse grids are proposed to solve the equation in high dimensional cases.

Seminar: Working on What (WOW)
Tibor Beke (UMass Lowell): Hex
Organizer: Emily Gunawan, email: emily_gunawan@uml.edu
April 24, 11 a.m. - Noon, Room: Southwick Hall 350W

• a thoroughly enjoyable board game
• a case study in taking the continuum limit of a discrete theorem (the fact that no Hex game can end in a draw is "equivalent" to the Brouwer Fixed Point theorem)
• a case study in game theory (the so-called "strategy stealing" argument proves that the first player has a winning strategy)
• a case study in computational complexity (finding the winning move is PSPACE-complete, probably explaining why no explicit winning strategy has been found beyond 10x10 boards).