Georgia Institute of Technology College of Sciences

The Bochner Classification Theorem (1929) characterizes the polynomial sequences $\{p_n\}_{n=0}^\infty$, with $\deg p_n=n$ that simultaneously form a complete set of eigenstates for a second order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders: Hermite, Laguerre, Jacobi and Bessel polynomials. In 2009, G\'{o}mez-Ullate, Kamran, and Milson found that for sequences $\{p_n\}_{n=1}^\infty$, with $\deg p_n=n$ (i.e.~without the constant polynomial) the only such sequences are the \emph{exceptional} Laguerre and Jacobi polynomials. They also studied two Types of Laguerre polynomial sequences which omit $m$ polynomials. We show the existence of a new "Type III" family of Laguerre polynomials and focus on its properties.

The symmetric configurations for the equilibrium shape of a fluid interfaceare given by the geometric differential equation mean curvature isproportional to height. The equations are explored numerically tohighlight the differences in classically treated capillary tubes andsessile drops, and what has recently emerged as annular capillary surfaces. Asymptotic results are presented.

A non-trivial group G is called left-orderable if there exists a strict total ordering < on its elements such that g

Bayesian approaches to statistical model selection requires the evaluation of the marginal likelihood integral, which, in general, is difficult to obtain. When the statistical model is regular, it is well-known that the marginal likelihood integral can be approximated using a function of the maximized log-likelihood function and the dimension of the model. When the model is singular, Sumio Watanabe has shown that an approximation of the marginal likelihood integral can be obtained through resolution of singularities, a result that has intimately tied machine learning and Bayesian model selection to computational algebraic geometry. This talk will be an introduction to singular learning theory with the factor analysis model as a running example.

A panel discussion featuring postdocs (Kelly Bickel, Adam Fox, Kit-Ho Mak, David Murrugarra, and Will Perkins) who have been on the job market.

Some mixed-type PDE problems for transonic flow and isometric embedding will be discussed. Recent results on the solutions to the hyperbolic-elliptic mixed-type equations and related systems of PDEs will be presented.

Spatially discrete stochastic models have been implemented to analyze cooperative behavior in a variety of biological, ecological, sociological, physical, and chemical systems. In these models, species of different types, or individuals in different states, reside at the sites of a periodic spatial grid. These sites change or switch state according to specific rules (reflecting birth or death, migration, infection, etc.) In this talk, we consider a spatial epidemic model where a population of sick or healthy individual resides on an infinite square lattice. Sick individuals spontaneously recover at rate *p*, and healthy individual become infected at rate O(1) if they have two or more sick neighbors. As *p* increases, the model exhibits a discontinuous transition from an infected to an all healthy state. Relative stability of the two states is assessed by exploring the propagation of planar interfaces separating them (i.e., planar waves of infection or recovery). We find that the condition for equistability or coexistence of the two states (i.e., stationarity of the interface) depends on orientation of the interface. We also explore the evolution of droplet-like configurations (e.g., an infected region embedded in an all healthy state). We analyze this stochastic model by applying truncation approximations to the exact master equations describing the evolution of spatially non-uniform states. We thereby obtain a set of discrete (or lattice) reaction-diffusion type equations amenable to numerical analysis.

We will discuss a few introductory results in real algebraic geometry concerning semi-algebraic sets. A semi-algebraic subset of R^k is the set of solutions of a boolean combination of finitely many real polynomial equalities and inequalities. These sets arise naturally in many areas of mathematics as well as other scientific disciplines, such as discrete and computational geometry or the configuration spaces in robotic motion planning. After providing some basic definitions and examples, we will outline the proof of a fundamental result, the Oleinik-Petrovsky-Thom-Milnor bound of d(2d-1)^{k-1} on the sum of the Betti numbers of a real algebraic variety, as well as indicate the direction of recent and ongoing research generalizing this result.

In this talk, we will discuss a result due to Gabai which states that a minimal genus Seifert surface for a knot in 3-sphere can be realized as a leaf of a taut foliation of the knot complement. We will give a fairly detailed outline of the proof. In the process, we will learn how to construct taut foliations on knot complements.

In this talk we will discuss applications of a new method of proving vector-valued inequalities discovered by M. Bateman and C. Thiele. We give new proofs of the Fefferman-Stein inequality (without using weighted theory) and vector-valued estimates of the Carleson operator using this method. Also as an application to bi-parameter problems, we give a new proof for bi-parameter multipliers without using product theory. As an application to the bilinear setting, we talk about new vector-valued estimates for the bilinear Hilbert transform, and estimates for the paraproduct tensored with the bilinear Hilbert transform. The first part of this work is joint work with Ciprian Demeter.

I will explain the construction of the essential skeleton of a one-parameter degeneration of algebraic varieties, which is a simplicial space encoding the geometry of the degeneration, and I will prove that it coincides with the skeleton of a good minimal dlt-model of the degeneration if the relative canonical sheaf is semi-ample. These results, contained in joint work with Mircea Mustata and Chenyang Xu, provide some interesting connections between Berkovich geometry and the Minimal Model Program.

In this talk, we discuss our recent progress on the famous directed cycle double cover conjecture of Jaeger. This conjecture asserts that every 2-connected graph admits a collection of cycles such that each edge is in exactly two cycles of the collection. In addition, it must be possible to prescribe an orientation to each cycle so that each edge is traversed in both ways. We plan to define the class of weakly robust trigraphs and prove that a connectivity augmentation conjecture for this class implies general directed cycle double cover conjecture. This is joint work with Martin Loebl.

Rare events, metastability and Monte Carlo methods for stochastic dynamical systems have been of central scientific interest for many years now. In this talk we focus on multiscale systems that can exhibit metastable behavior, such as rough energy landscapes. We discuss quenched large deviations in related random rough environments and design of provably efficient Monte Carlo methods, such as importance sampling, in order to estimate probabilities of rare events. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies.

Robert will be talking and leading the discussion on Chapter 2 Section 7 of Bounded Analytic Functions: Beurling's Theorem.

We say that a cover of surfaces S-> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a cover has this property. We give new explicit examples of irregular branched covers that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.

We introduce concepts of entropy and methods of calculation of entropy and examples. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

We will explain the concept of aproximate well-supported Nash equilibrium and show that one must consider equilibria with large supports to achieve good approximation ratio. Our arguments use tools from probabilistic, extremal and additive combinatorics. Joint work with Y. Anbalagan, R. Savani and A. Vetta.