Rigidity in enveloping algebras

我的研究从广义上讲属于代数。李代数和包络代数是代数和数学物理、粒子物理、弦理论、几何、超平面排列等领域中活跃而重要的研究领域。菲尔兹奖获得者Efim Zelmanov使用的李理论方法使他解决了群论中的有界Kourosh问题。 李代数是非交换的、非结合的对象。什么是“非对易”？考虑一下我们在烧瓶中倒入两种化学物质的动作。在这里，我们首先将哪种产品倒入烧瓶中并不重要。换句话说，行动是可交换的。相比之下，“非关联性”涉及三个对象。例如，考虑将三种不同的化学品A、B和C混合在烧瓶中而得到的化学产品。所以配方是先把A和B加在一起，然后再加C。在数学公式中，我们可以把这个过程概括为(A+B)+C。相反，如果我们先把C和A相加，然后再加B，我们可能得到不同的乘积。换句话说，C+(A+B)可能不等同于(C+A)+B。我们可以解释这个经验，说加化学品的行为是可交换的，而不是联想的。 我在代数方面的专业知识使我拓宽了我的研究计划。例如，我们使用人工智能和机器学习来检测和防止隐私敏感组织(如医疗系统或税务系统)的异常访问。我们还在开发人脸检测和识别的新方法。 我的研究项目也对癌症诊断和免疫治疗等重要问题做出了贡献。在全基因组关联研究中，对部分或全部人类基因组进行测序，以发现遗传因素与疾病之间的关联。在GWA中，正在考虑的遗传变异是单核苷酸多态(SNPs)，这是人类中最常见的变异类型。在一个数据集中，SNPs的数量通常超过一百万；因此，需要一套强大的数据挖掘和机器学习方法来调查遗传数据，以揭示导致疾病的最重要的遗传变异。 在机器学习和模式识别中，特征选择是选择问题中最重要的特征，同时删除不必要的特征的过程。在微阵列数据集或基因表达中使用特征选择来检测癌症是被广泛研究的。戈卢布等人。他们首先确定了可以区分急性髓系白血病和急性淋巴细胞白血病的50个基因的子集，并随后预测了新白血病病例的类别成员。 作为研究项目的一部分，我们研究了一种新的基于扰动理论的特征选择方法。通过在一些癌症数据集上与传统的和新颖的特征选择方法进行一系列的比较，验证了我们的方法的有效性。

My research in a broad sense falls in algebra. Lie algebras and enveloping algebras are active and important research areas both within algebra and in other fields such as Mathematical Physics, Particle Physics, String Theory, Geometry, Hyperplane Aarrangement, etc. Lie theory methods used by the Fields medallist Efim Zelmanov led him to solve the Bounded Kourosh Problem in Group Theory. Lie algebras are non-commutative and non-associative objects. What is "non-commutativity"? Consider the action where we pour two chemicals in a flask. Here, it doesn't matter which product we pour first into the flask. In other words, the action is commutative. In comparison "non-associativity" involves three objects. For example, consider a chemical product obtained by mixing three different chemicals A, B, and C in a flask. So the recipe is to add together A and B first and then add C. In Mathematics formulation, we can summarize this procedure as (A+B)+C. Instead, if we first add C and A and then add B we could possibly get a different product. In other words, C+(A+B) may not be the same as (C+A)+B. We can interpret this experience and say that the action of adding chemicals is commutative bot not associative. My expertise in Algebra has led me to broaden my research program. For example, we use Artificial Intelligence and machine learning to detect and prevent abnormal access in privacy sensitive organizations like Healthcare System or TAX system. We are also developing new methods in face detection and recognition. My research program also contributes to vital problems such as Cancer diagnoses and immunotherapy. In genome-wide association study (GWAS), partial or all of the human genome is sequenced for discovering the associations between genetic factors and a disease. In GWAS the genetic variants under consideration are single nucleotide polymorphisms (SNPs), the most common type of variation among people. The number of SNPs usually goes over one million in a dataset; therefore, a set of powerful data mining and machine learning methods are needed in order to investigate genetic data to reveal the most significant genetic variants that cause a disease. In machine learning and pattern recognition, feature selection is the process of selecting the most important features of a problem while removing unnecessary ones. The use of feature selection in microarray datasets or gene expression for detecting cancer is widely investigated. Golub et al. were the first to identify a subset of 50 genes that can discriminate acute myeloid leukemia from acute lymphoblastic leukemia, and subsequently predict class membership of new leukemia cases. As part of our research program, we investigate a new feature selection method based on perturbation theory. The effectiveness of our method is verified by performing a series of comparisons with conventional and novel feature selection methods in the literature on some Cancer datasets.