From Ecstasy to Agony and Back: Mathematics, Motherhood, and Postpartum Depression

In this essay I address some challenges I had to overcome as a non-tenured mathematician mother married to another mathematician. While some of these challenges, such as lack of sleep, were predictable, many were not. For example, I was not aware that just getting pregnant would be a difficulty. Also, I did not know how paralyzing postpartum depression could be. Looking back on these experiences after a few years, I have a new perspective. I share my experiences, feelings about them, and some of the ways I figured out to overcome the hardships I experienced

Straightening Identities in the Onsager Algebra

The purpose of this work is to formulate and prove some straightening identities in the Onsager algebra. Our identities allow one to rewrite specific products of basis elements as linear combinations of products which are in a different order. Such identities could be helpful in understanding the representation theory of the Onsager algebra

Hemangioblastoma of the Central Nervous System: A Case Series of Patients Surgically Treated at

Objective: Hemangioblastoma refers to a benign vascular neoplasm that comprises stromal and capillary cells. Based on the classification of nervous system tumors proposed by the World Health Organization, hemangioblastomas are classified as Grade I meningeal tumors of uncertain origin. These tumors are found almost exclusively in the central nervous system (CNS) and account for 0.9 to 2.1% of all primary CNS tumors.Methods: In this descriptive retrospective study, the archives of pathology reports were reviewed in the department of pathology of Shohadaye Tajrish Hospital and patients with definite diagnosis of hemangioblastoma made through histpopathological examinations during 2004-2014 were identified. Age, gender and the location of tumor were extracted from the medical records and entered into SPSS statistical software v.22 for analysis.Results: A total of 30 patients including 16 males (53.3%) and 14 females (46.7%) were identified. The mean age of the patients was calculated to be 41.2±13.47 years, ranging from 19 to 62 years old. The majority of lesions had been found in the cerebellum of the patients (93.3%); only one had occurred in the cerebrum (3.3%) and another in the fourth ventricle (3.3%). Conclusion: Cerebellum is the most commonly affected location in patients with CNS hemangioblastomas, and a male preponderance is observed in these cases.

Using Twisted Alexander Polynomials to Detect Fiberability

The purpose of this dissertation is to discuss how certain algebraic invariants of 3-manifolds, the twisted Alexander polynomials, can be effectively used in the study of fiberability and the Thurston norm of links. The links to which we have applied this technique belong to the class of graph links. For graph links, D. Eisenbud and W. Neumann introduced splice diagrams and developed a method to use the combinatorial informationincluded in splice diagrams to determine berability and the Thurston norm]. We use twisted Alexander polynomials to prove that the exterior of a certain graph knot, whose splice diagram is given, is not fibered. Then we consider three 2-component graph links built out of this knot. For these links we use the same technique, involving twisted Alexanderpolynomials, to discuss their berability and Thurston norm. This allows us to demonstrate the effectiveness of twisted Alexander polynomials in this context (links in homology spheres different from S3), where no calculations exist in the literature