### Colloquia

Colloquia for the Department of Mathematics and Statistics are normally
held in **University Hall 4010 on Fridays at 4:00pm**.
Any departures from this are indicated below.

Light refreshments are served after the colloquia in 2040 University Hall.

Driving directions, parking information, and maps are available on the university website.

# 2016-2017 Colloquia

What follows is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.

## Next Colloquium

- April 27, 2017 (Thursday, 4:00pm, University Hall 4010)
Ekaterina Shemyakova (SUNY at Newpaltz)

*Differential operators on the superline, Berezinians and Darboux transformations*Abstract: In this talk we consider Darboux transformations (DTs), which are certain non-group symmetries of linear partial differential equations, in the setting of supergeometry, an area on the borderline between quantum physics and differential geometry.

We consider the superline, by which we mean a supermanifold of dimension 1|1, i.e. the case of one even and one odd variable. Among all linear partial differential operators on the superline, we have isolated a class of "nondegenerate operators". We show that they are close by their properties to ordinary differential operators. In particular, we show how every such operator can be expressed via certain "super Wronskians" (particular Berezinians, i.e. superdeterminants). This is applied to classification of Darboux transformations (DTs). Namely, we show that every DT on the superline corresponds to an invariant subspace of the source operator and upon a choice of basis is given by a super-Wronskian formula. These formulas are non-trivial because Berezinians are rational functions, not polynomials. On the way we establish important properties of Berezinians which are non-trivial analogs of textbook formulas for ordinary determinants, but seem to be not known in the super case.

We also discuss a generalization of the above to differential operators acting on the algebra of densities (instead of the algebra of functions).

See Differential operators on the superline, Berezinians, and Darboux transformations and Darboux transformations for differential operators on the superline

## Spring Semester

- April 28, 2017
Antonio Moro (Northumbria University Newcastle)

*Dressing networks: towards an integrability approach to collective and complex phenomena*Abstract: A large variety of real world systems can be naturally modelled by networks, i.e. graphs whose nodes represent the components of a system linked (interacting) according to specific statistical rules. A network is realized by a graph typically constituted by a large number of nodes/links. Fluid and magnetic models in physics are just two among many classical examples of systems which can be modelled by simple or complex networks. In particular "extreme" conditions (thermodynamic regime), networks, just like fluids and magnets, exhibit a critical collective behavior intended as a drastic change of state due to a continuous change of the model parameters. Using an approach to thermodynamics, recently introduced to describe a general class of van der Waals type models and magnetic systems in mean field approximation, we analyze the integrable structure of corresponding networks and use the theory of integrable conservation laws combined with a suitable "dressing" procedure to calculate order parameters outside and inside the critical region.

- April 27, 2017 (Thursday, 4:00pm, University Hall 4010)
Ekaterina Shemyakova (SUNY at Newpaltz)

*Differential operators on the superline, Berezinians and Darboux transformations*Abstract: In this talk we consider Darboux transformations (DTs), which are certain non-group symmetries of linear partial differential equations, in the setting of supergeometry, an area on the borderline between quantum physics and differential geometry.

We consider the superline, by which we mean a supermanifold of dimension 1|1, i.e. the case of one even and one odd variable. Among all linear partial differential operators on the superline, we have isolated a class of "nondegenerate operators". We show that they are close by their properties to ordinary differential operators. In particular, we show how every such operator can be expressed via certain "super Wronskians" (particular Berezinians, i.e. superdeterminants). This is applied to classification of Darboux transformations (DTs). Namely, we show that every DT on the superline corresponds to an invariant subspace of the source operator and upon a choice of basis is given by a super-Wronskian formula. These formulas are non-trivial because Berezinians are rational functions, not polynomials. On the way we establish important properties of Berezinians which are non-trivial analogs of textbook formulas for ordinary determinants, but seem to be not known in the super case.

We also discuss a generalization of the above to differential operators acting on the algebra of densities (instead of the algebra of functions).

See Differential operators on the superline, Berezinians, and Darboux transformations and Darboux transformations for differential operators on the superline

- April 25, 2017 (Tuesday, 4:00pm, University Hall 4010)
David Kaspar (Brown Univeristy)

*Kinetic Theory: Old and New*Abstract: Following the time from Bernoulli to Boltzmann, kinetic theory has grown from an attempt to describe gases to a subject with an enormous diversity of applications and considerable mathematical challenges. After a brief orientation to the origins, themes, and scope of kinetic theory, we focus on a surprising connection between scalar conservation laws with Markov process initial data and mean-field coagulation phenomena. A rigorous validity result appearing in [Kaspar and Rezakhanlou, PTRF 2016] will be presented, and ongoing and future efforts at generalization (including the multidimensional setting) will be indicated.

- April 24, 2017 (Monday, 4:00pm, University Hall 4010)
Funda Gultepe (University of Illinois Urbana-Champaign)

*A geometric look at free group and group of outer automorphisms of the free group*- April 21, 2017
Alden Waters (University College London)

*An introduction to the Modern Calderon Problem*Abstract: Ever wonder how X-ray machines work? This is your chance! In this talk we give an introduction to the Calderon Problem and it's modernisations. We discuss the use of the Gaussian beam Ansatz to develop accurate solutions to the problem and highlight the connection to control theory.

- Shoemaker Lecture Series Apr 10-12, 2017, 4:00-5:00pm, UH 4010
Gigliola Staffilani (Massachusetts Institute of Technology)

**Lecture 1: The many faces of dispersive equations as infinite dimensional Hamiltonian systems.**Abstract: In this lecture, I will give an overview of several results obtained for dispersive and wave equations that are Hamiltonian systems. I will talk about conservation laws, Strichartz estimates, energy transfer, Gibbs measures and non-squeezing theorems.

**Lecture 2: Energy transfer for certain nonlinear Schrodinger (NLS) initial value problems.**Abstract: In this lecture, I will concentrate on the question of energy transfer and weak turbulence. I will first show how bounds in time of higher Sobolev norms of solutions to certain NLS are related to energy transfer, then I will show some recent results on polynomial bounds for these norms.

**Lecture 3: Almost sure well-posedness and randomization of initial data.**Abstract: In this lecture, I will go back to the concept of Gibbs measure, outline the work of Bourgain for the 2D cubic nonlinear periodic NLS and I will describe further results on almost sure well-posedness obtained by randomizing the initial data.

- April 7, 2017
Junfeng Shang (Bowling Green State University)

*A modeling diagnostic and a model selection criterion in random effects models*Abstract: This talk consists of two parts. Part I presents a modeling diagnostic for detecting influential cases based on the information complexity (ICOMP) criteria for detecting influential cases which substantially affect the model selection criterion ICOMP in generalized linear mixed models (GLMM). The diagnostic compares the ICOMP criterion between the full data set and a case-deleted data set. The computational formula of the ICOMP criterion is evaluated using the Fisher information matrix. A simulation study is accomplished and a real data set of cancer cells is analyzed using the logistic linear mixed model for illustrating the effectiveness of the proposed diagnostic in detecting the influential cases. Part II presents a model selection criterion, Conceptual Predictive Statistic (CCp) for mixed model selection based on a conditional Gauss discrepancy. We define the conditional Gauss discrepancy for measuring the distance between the true model and the candidate model under the conditional mean of response variables. The best linear unbiased predictor (BLUP) is employed for the estimation of the random effects. The simulation results demonstrate that when the true model includes significant fixed effects, the conditional criteria perform effectively in selecting the most appropriate model. The penalty term in the CCp computed by the estimated effective degrees of freedom yields a very good approximation to the penalty term between the target discrepancy and the goodness-of-fit term.

- March 31, 2017
Farzad Fathizadeh (California Institute of Technology)

*The term $a_4$ in the heat kernel expansion of noncommutative tori*Abstract: The analog of the Riemann curvature tensor for noncommutative tori manifests itself in the term $a_4$ appearing in the heat kernel expansion of the Laplacian of curved metrics. This talk presents a joint work with Alain Connes in which we obtain an explicit formula for the $a_4$ associated with a general metric in the canonical conformal structure on noncommutative two-tori. Our final formula has a complicated dependence on the modular automorphism of the state or volume form of the metric, namely in terms of several variable functions with lengthy expressions. We verify the accuracy of the functions by checking that they satisfy a family of conceptually predicted functional relations. By studying the latter abstractly we find a partial differential system which involves a natural flow and action of cyclic groups of order two, three and four, and we discover symmetries of the calculated expressions with respect to the action of these groups. At the end, I will illustrate the application of our results to certain noncommutative four-tori equipped with non-conformally flat metrics and higher dimensional modular structures.

- March 24, 2017
Paramasamy Karuppuchamy (University of Toledo)

*Representations of algebraic groups.*Abstract: One of the main problems in representations of algebraic groups is finding a character formula for irreducible modules. There are many character formulas in characteristic zero case but not in characteristic p. Lusztig's conjectured character formula in characteristic p is recently proved to be wrong, more precisely, Andersen, Jantzen, Soergel proved the conjecture is true for large primes (1994) and recently Williamson proved it is not true for small primes. A partial survey will be presented in this talk. Also I will state a couple of theorems from recent papers with Leonard Scott and Terrell Hodge on "Induction Theorem".

It will be accessible to our graduate students.

- March 17, 2017
Vladimir Dragovic (The University of Texas at Dallas)

*Algebro-Geometric approach to the Schlesinger systems: from Poncelet to Painleve' VI and beyond*Abstract: A new method of construction of algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve represented as a ramified double covering of $\mathbb{CP}^1$, a meromorphic differential is constructed with the following property: the common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of a Painleve' VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve VI equations. A generalization of this differential to hyperelliptic curves is also constructed. The corresponding solutions of the rank two Schlesinger systems associated with elliptic and hyperelliptic curves are constructed in terms of these differentials. The initial data for the construction of the meromorphic differentials include a point in the Jacobian of the curve, under the assumption that this point has non-variable coordinates with respect to the lattice of the Jacobian while the branch points vary.

The research has been partially supported by the NSF grant 1444147. This is joint work with Vasilisa Shramchenko.

- March 14, 2017 (Tuesday, 4:00pm, University Hall 4010)
Nate Iverson (University of Toledo)

*Circumference over diameter; the different universes of pi*Abstract: Pi is the ratio of circumference to diameter in a circle. We define a circle to be a set of points equidistant from a common point. When the method of measuring distance is changed different ratios are possible. This talk will discuss the ratio of circumference to diameter in all p-norms including p=1, the taxicab norm, and p=$\infty$, infinity the supremum norm. Results dating to 1932 using the Minkowski functional norms will also be discussed along with further generalizations.

- February 24, 2017
Zeljko Cuckovic (University of Toledo)

*$L^p$ Regularity of Bergman Projections on Domains in $\mathbb{C}^n$*Abstract. Bergman projections and Bergman kernels are among the central objects in complex analysis. In this talk we will discuss the $L^p$ regularity of weighted Bergman projec- tions on various domains in $\mathbb{C}^n$. Then we will show an $L^p$ irregularity of weighted Bergman projections on complete Reinhardt domains with exponentially decaying weights (joint work with Yunus Zeytuncu). Finally we establish estimates of the $L^p$ norms of Bergman projec- tions on strongly pseudoconvex domains. The talk should be accessible to graduate students.

- February 17, 2017
Dean A. Carlson (Mathematical Reviews, American Mathematical Society, Ann Arbor, MI)

*Minimizers for Nonconvex Variational problems in the plane via Convex/Concave Rearrangements*Abstract: Recently, A. Greco utilized convex rearrangements to present some new and interesting existence results for noncoercive functionals in the calculus of variations. Moreover, the integrands were not necessarily convex. In particular, using convex rearrangements permitted him to establish the existence of convex minimizers essentially considering the uniform convergence of the minimizing sequence of trajectories and the pointwise convergence of their derivatives. The desired lower semicontinuity property is now a consequence of Fatou's lemma.

In this paper we point out that such an approach was considered in the late 1930's in a series of papers by E. J. McShane for problems satisfying the usual coercivity condition. Our goal is to survey some of McShane‘s results and compare them with Greco's work. In addition, we will update some hypotheses that McShane made by making use of a result due to T. S. Angell on the avoidance of the Lavrentiev phenomenon.

- February 10, 2017
Jim Albert (Bowling Green State University)

*Situation Statistics in Baseball*Abstract: Baseball fans are fascinated with splits, where player offensive or defensive statistics are broken down by different situations, such as home/away, date, same side/opposite arm pitchers, etc. We provide a general overview of the use of random effects models to fit these data. It is well-known that hitters have different talents, but it is not clear that hitters have talents to take advantage of particular situations. For example, fans like to think that specific players have clutch ability, but there is little statistical evidence to indicate that clutch ability exists.

- February 3, 2017
Chunhua Shan (University of Toledo)

*Turning points and relaxation oscillations in epidemic models*Abstract: We study the interplay between effects of disease burden on the host population and the effects of population growth on the disease incidence, in an epidemic model of SIR type with demography and disease-caused death. Under the assumption that the host population has a small intrinsic growth rate, using singular perturbation techniques and the phenomenon of the delay of stability loss due to turning points, we prove the existence of large-amplitude relaxation oscillation cycles, which contrast sharply to oscillations via Hopf bifurcation. Simulations are provided to support the theoretical results. Our results offer new insight into the classical periodicity problem in epidemiology.

- January 27, 2017
Alimjon Eshmatov (University of Toledo)

*Homotopy braid closure and knot contact homology*Abstract: In this talk, I will present joint work with Yu. Berest and W. Yeung, where we give a universal construction, called homotopy braid closure, that produces invariants of links in $\mathbb{R}^3$. Applying this construction to the Gelfand-MacPherson-Vilonen (GMV) braid action, we obtain a differential graded (DG) category that gives knot contact homology in the sense of L. Ng. As an application, we show that the category of finite-dimensional modules over the 0-th homology of this DG category is equivalent to the category of perverse sheaves on $\mathbb{R}^3$ with singularities at most along the link.

## Fall Semester

- December 2, 2016
Timothy Clos (University of Toledo)

*Compactness of Hankel Operators with Continuous Symbols*Abstract: Hankel operators are an area of research in operator theory. I will begin this talk by surveying some background material concerning Hankel operators and compactness of operators. I will then give some previous results on compactness of Hankel operators on the Bergman spaces of domains in $\mathbb{C}^n$ for $n\ge 1$. I will also outline the proof of our main result, which concerns compactness of Hankel operators with symbols which are continuous up to the closure of convex Reinhardt domains in $\mathbb{C}^2$.

This is a joint work with Sönmez Şahutoğlu.

- November 18, 2016
Adrian Lam (Ohio State University)

*Evolutionarily Stable Strategies in the Evolution of Condition Dispersal*Abstract: Dispersal, which refers to the movement of an organism between two successive areas impacting survival and reproduction, is one of the most studied concepts in ecology and evolutionary biology. How do organisms adopt their dispersal patterns? Is there an "optimal", or evolutionarily stable, dispersal strategy that emerges from the underlying ecology? In this talk, we consider a reaction-diffusion model of two competing species for the evolution of conditional dispersal in a spatially varying but temporally constant environment. Two species are different only in their dispersal strategies, which are a combination of random dispersal and biased movement upward along the resource gradient. In the absence of biased movement or advection, A. Hastings (1983) showed that dispersal is selected against in spatially varying environments. When there is a small amount of biased movement or advection, we show that there is a positive random dispersal rate that is both locally evolutionarily stable and convergent stable. Our analysis of the model suggests that a balanced combination of random and biased movement might be a better habitat selection strategy for populations.

This is joint work with Y. Lou of Ohio State University.

- October 7, 2016
Vani Cheruvu (University of Toledo)

*Numerical methods, grids in atmospheric and oceanic sciences*High-order numerical methods offer the promise of accurately capturing many physical processes and have been shown to efficiently scale to large number of processors. There is a considerable effort in using high-order methods to solve partial differential equations that model physical phenomena in atmospheric and oceanic sciences. In this talk, I would present three different high-order methods and discuss their advantages and disadvantages. These methods are compared by applying to a PDE. Several issues are involved when these methods are applied to a PDE on a sphere for instance, suitability of a grid. I will conclude the talk with two suitable grids on a sphere.