### Spring 2019 Mathematics Colloquium

###### Time and Location:

Minard 212 at 3:00 PM (refreshments at 2:30 PM in the Math Conference Room (Minard 404))

Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.

#### Thursday, January 24 Thomas McConville (MIT)

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Canonical bijections among combinatorial lattices

Abstract: Many significant combinatorial objects such as partitions, permutations, and trees come equipped with natural partial orderings. When such a partial ordering has the structure of a congruence-uniform lattice, there are two accompanying bijections. One bijection takes the lattice to its set of canonical join representations, and the other bijection goes to its core label order. These bijections underlie many interesting enumerative and structural relationships found in Coxeter groups, cluster algebras, and the representations of associative algebras. In this talk, I will demonstrate these bijections using partial triangulations, colored trees, and noncrossing tree partitions. Then I will show how these objects and correspondences arise in the representation theory of a family of gentle algebras. These examples were developed in joint work with Alexander Garver.

#### Tuesday, January 29 Rebecca Patrias (LaCIM)

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A tour of symmetric functions and tableau combinatorics

Abstract: The Schur function basis of the ring of symmetric functions is a central object in algebraic combinatorics, in part due to their deep connection to representation theory of the general linear group and to the cohomology ring of the Grassmannian. Their ties to representation theory make Schur positivity of symmetric functions a sought-after property. After introducing symmetric functions and the Schur basis, I will present a result that gives the probability that a symmetric function is Schur positive. Next, I present analogues of the Schur functions called stable Grothendieck polynomials, which play the role of Schur functions in the K-theory ring of the Grassmannian. I end by discussing several of my results with various coauthors that aim to understand the combinatorics associated to this ring via the stable Grothendieck polynomials and their analogues.

#### Thursday, January 31 Torin Greenwood (Rose-Hulman)

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RNA and Combinatorics

Abstract: Research on RNA has exploded in the last few decades as biologists realized that RNA plays more diverse and nuanced roles than presumed in the 90s. One way to glean information about RNA functions is by understanding the complicated 3-dimensional structural conformations into which sequences of RNA nucleotides fold. In this talk, we will explore the tools from combinatorics and probability used both to predict RNA structures, and to analyze these prediction methods. In particular, we will see how Boltzmann distributions and stochastic context-free grammars can model RNA structural distributions, and how generating functions can help analyze biological features of these distributions.

#### Tuesday, February 19 Cătălin Ciupercă (NDSU)

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Derivations and integral closure

Abstract: We present an overview of derivations in commutative algebra and their behavior with respect to integral closure of rings and ideals. We then apply these results to prove several new properties of a large class of Golod ideals in a polynomial ring over a field of characteristic zero.

#### Tuesday, March 5 Michelle Doyle (Chatham U.)

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Uniqueness results for bodies of constant width in the hyperbolic plane

Abstract: An important difference between the geometry in the Euclidean and the hyperbolic plane is that, while the concepts of width of a body and its projections lengths on geodesics are equivalent in the Euclidean setting, they are unrelated in the hyperbolic one, and there is no natural definition of support function of a convex body. For example, any Euclidean planar body of constant width has constant projections on every line, but this no longer holds in the hyperbolic plane. We discuss the difference between the hyperbolic width and projection lengths, and prove three unique reconstruction results for a hyperbolic disc.

#### Tuesday, March 19 Tri-College Colloquium

Hee Jung Kim at MSUM

Corks, exotic embeddings of 4-manifolds and stabilization

TIME and ROOM : MacLean 274 at 3:00 Pm. Refreshments at 2:30 IN MacLean 268 (the President's Conference Room)

Abstract: The existence of exotic smooth structures of 4-manifolds was highlighted as the h-cobordism theorem applied for the smooth classification of higher dimensional manifolds failed in dimension 4. The study of this phenomenon had led the notion of a `cork' which is a compact, contractible, smooth 4-manifold and S. Akbulut discovered the first example of exotic 4-manifolds using the cork. In this talk, we will discuss corks and their applications.

#### Tuesday, March 26 Alejandro Aceves (SMU)

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*Recent trends in nonlinear optics and photonics: A mathematical modeling perspective*

Abstract: Nonlinear optics and photonics research remains an active field with new phenomena and applications to be explored in the immediate future. It also remains a platform to study similar processes in other fields including nonlinear waves. In this talk, I will point to some of the mathematical challenges and opportunities that one can identify to advance on the theoretical front.

#### Thursday, April 11 Sujit Ghosh (NCSU)

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Dynamic correlation multivariate stochastic volatility with latent factors

Abstract: Modeling the correlation structure of returns is essential in many financial applications. Considerable evidence from empirical studies has shown that the correlation among asset returns is not stable over time. A recent development in the multivariate stochastic volatility literature is the application of inverse Wishart processes to characterize the evolution of return correlation matrices. Within the inverse Wishart multivariate stochastic volatility framework, we propose a flexible correlated latent factor model to achieve dimension reduction and capture the stylized fact of ‘correlation breakdown’ simultaneously. The parameter estimation is based on existing Markov chain Monte Carlo methods. We illustrate the proposed model with several empirical studies. In particular, we use high‐dimensional stock return data to compare our model with competing models based on multiple performance metrics and tests. The results show that the proposed model not only describes historic stylized facts reasonably but also provides the best overall performance.

The talk is primarily based on the joint work with Sheng‐Jhih Wu, Yu‐Cheng Ku and Peter Bloomfield and the publication: https://doi.org/10.1111/stan.12115

#### Thursday, April 18 Qin (Tim) Sheng (Baylor)

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Exponential, Non-Exponential Splitting and Their Applications for Solving Singular PDEs

Abstract: This talk consists of two interactive parts. First, we will pay an attention to optimized operator splitting strategies, such as the non-exponential ADI and exponential LOD methods, and explore their modernizations. Then we will focus at interesting issues involving the constructions and analysis of highly-effective and highly-efficient finite difference approximations of singular partial differential equations which are crucial to rocket engine designs. The quenching phenomenon will be revealed. Adaptive splitting approaches will then be introduced. Mathematical analysis on solution positivity, monotonicity and stability will be discussed. We will also present the latest ideas on moving mesh strategies which can be extended for solving many multiphysics equations including those from laser-materials interactions, oil pipeline decay detections and computational finance. Potentials of collaborative investigations will be explored.

#### Tuesday, April 23 Tri-College Colloquium

Hongyan Hou at NDSU

Convergence Rate for Gauss and Radau Methods Applied to Optimal Control

Abstract: A local convergence rate is established for an orthogonal collocation method based on Gauss and Radau quadrature applied to an unconstrained optimal control problem. If the continuous problem has a smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as the number of collocation points increases, the discrete solution converges to the continuous solution at the collocation points, exponentially fast in the sup-norm.

#### Tuesday, April 30 Sam Hopkins (UMN)

The coincidental down-degree expectations (CDE) phenomenon

Abstract: I will survey some recent developments in algebraic combinatorics, which are based on two observations:

1) there are unexpected product formulas for the number of certain ``barely set-valued'' tableaux;

2) certain finite posets have unexpectedly simple formulas for their edge densities.

Moreover, these two observations are in fact linked via the ``coincidental down-degree expectations (CDE) property'' for posets. I will explain how the study of CDE posets arose independently in the algebraic geometry of curves, and in combinatorial Schubert calculus. I will also highlight the close connections between the study of CDE posets and the emerging field of ``dynamical algebraic combinatorics,'' which studies the dynamical properties of invertible operators acting on objects from algebraic combinatorics.