Rigidity in enveloping algebras

我的研究在广义上属于代数福尔斯。李代数和包络代数是活跃和重要的研究领域都在代数和其他领域，如数学物理，粒子物理，弦论，几何，超平面Aarrangement等李理论方法所使用的领域奖牌得主叶菲姆Zelmanov导致他解决有限Kourosh问题的群论。 李代数是非交换和非结合的对象。什么是“非交换性”？考虑一下我们把两种化学物质倒进一个烧瓶的动作。在这里，我们先倒入哪个产品并不重要。换句话说，这个作用是可交换的。相比之下，“非结合性”涉及三个对象。例如，考虑通过在烧瓶中混合三种不同的化学品A、B和C获得的化学产品。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，我们可以将这个过程概括为（A+B）+C。相反，如果我们先加上C和A，然后加上B，我们可能会得到不同的产品。换句话说，C+（A+B）可能与（C+A）+B不同。我们可以解释这一经验，说添加化学物质的作用是交换的而不是联想的。 我在代数方面的专业知识使我扩大了我的研究计划。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（如医疗保健系统或税务系统）中的异常访问。我们还在开发人脸检测和识别的新方法。 我的研究项目也有助于癌症诊断和免疫治疗等重要问题。全基因组关联研究（genome-wide association study，GWAS）是对人类基因组的部分或全部进行测序，以发现遗传因素与疾病之间的关联。在GWAS中，考虑的遗传变异是单核苷酸多态性（SNP），这是人类最常见的变异类型。在一个数据集中，SNP的数量通常超过100万;因此，需要一套强大的数据挖掘和机器学习方法来研究遗传数据，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，同时去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被广泛研究。Golub等人是第一个鉴定出50个基因子集的人，这些基因可以区分急性髓细胞白血病和急性淋巴细胞白血病，并随后预测新白血病病例的类别成员。 作为我们的研究计划的一部分，我们研究了一种新的特征选择方法的基础上扰动理论。我们的方法的有效性进行了一系列的比较与传统的和新的特征选择方法在文献中的一些癌症数据集。

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.