Tensor amplification and decomposition and their applications in biomedicine
N. Tokcan, H. Derksen , in preparation
I am a Postdoctoral Assistant Professor at the Math Department, at the University of Michigan, Ann Arbor (UMICH). I am currently holding a double appointment also as a Postdoctoral Researcher at the Biomedical and Clinical Informatics Lab, Department of Computational Medicine and Bioinformatics at UMICH.
I received my Ph.D.from the University of Illinois at Urbana-Champaign (UIUC) in May 2017 working under the supervision of Prof. Bruce Reznick.
My research is largely focused on high-order tensor analysis. More specifically, I am working on formulating a novel, mathematically sound theoretical framework to perform analysis of data structured as n-dimensional tensors while preserving the integrity of their structure -- i.e. without slicing. My research interests also include real algebraic geometry, field theory and polynomials, semidefinite optimization, combinatorics and graph theory. I am also interested in techniques for symmetric tensor decomposition – also known as the Waring problem for forms, sum of squares optimization, Schubert calculus and graph matching.
Please feel free to get in touch with me at tokcan@umich.edu. You can also find additional ways to contact me in the Contact section.
N. Tokcan, H. Derksen , in preparation
N. Tokcan, H. Derksen, G. Jonathan, L. Hernandez, K. Najarian, in preparation
N. Tokcan, B. Reznick, in preparation
B. Reznick and N. Tokcan, accepted for publication in Proceedings of the American Mathematical Society
N. Tokcan, acctepted for publication in Linear Algebra and Its Applications
C. Monical, N. Tokcan and A. Yong, available on arXiv
R. Mancuso, R. Pellizzoni, N. Tokcan and M. Caccamo, in In Proceedings of the 29th Euromicro Conference on Real-Time Systems (ECRTS 2017), Dubrovnik, Croatia. To Appear
Postdoctoral Assistant Professor in Mathematics, and Computational Medicine and Bioinformatics
University of Michigan, Ann Arbor
Ph.D. in Mathematics
University of Illinois at Urbana-Champaign Department of Mathematics
Master of Science in Mathematics
University of Illinois at Urbana-Champaign Department of Mathematics
Bachelor of Science in Mathematics
Cukurova University, Turkey
I am a Postdoctoral Assistant Professor at the Math Department, at the University of Michigan, Ann Arbor (UMICH). I am currently holding a double appointment also as a Postdoctoral Researcher at the Biomedical and Clinical Informatics Lab, Department of Computational Medicine and Bioinformatics at UMICH.
My research is largely focused on high-order tensor analysis. More specifically, I am working on formulating a novel, mathematically sound theoretical framework to perform analysis of data structured as n-dimensional tensors while preserving the integrity of their structure -- i.e. without slicing. My research interests also include the study of symmetric tensor decomposition – also known as the Waring problem for binary forms, computational (real) algebraic geometry, and field theory and polynomials. The topics explored in my thesis find applications in different areas such as signal processing, machine learning, data mining, and neuroscience. I am also currently conducting research in the areas of semidefinite optimization, combinatorics and graph theory. I am interested in interdisciplinary projects and applications in many engineering areas. Furthermore, I have identified a number of promising directions to investigate as a follow up of my thesis research. I would summarize the main directions of my thesis research as follows:
Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \subseteq \mathbb C$. The $K$-rank of $f$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d \ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fields. The $K$-rank of a form $f$ (such as $x^3y^2$) may depend on whether -1 is a sum of two squares in $K$.
The $K$-rank of a binary form $f$ in $K[x,y],~K\subseteq \mathbb{C},$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We completely classify binary forms of Waring rank 3.
A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others. Our principal construction is the Schubitope. For any subset of $[n]$ x $[n]$, we describe it by linear inequalities. This generalized permutahedron conjecturally has positive Ehrhart polynomial. We conjecture it describes the Newton polytope of Schubert and key polynomials. We also define dominance order on permutations and study its poset-theoretic properties.
In the last decade there has been a steady uptrend in the popularity of embedded multi-core platforms. This represents a turning point in the theory and implementation of real-time systems. From a real-time standpoint, however, the extensive sharing of hardware resources (e.g. caches, DRAM subsystem, I/O channels) represents a major source of unpredictability. Budget-based memory regulation (throttling) has been extensively studied to enforce a strict partitioning of the DRAM subsystem’s bandwidth. The common approach to analyze a task under memory bandwidth regulation is to consider the budget of the core where the task is executing, and assume the worst-case about the remaining cores’ budgets.
In this work, we propose a novel analysis strategy to derive the WCET of a task under memory bandwidth regulation that takes into account the exact distribution of memory budgets to cores. In this sense, the proposed analysis represents a generalization of approaches that consider (i) even budget distribution across cores; and (ii) uneven but unknown (except for the core under analysis) budget assignment. By exploiting the additional piece of information, we show that it is possible to derive a more accurate WCET estimation. Our evaluations highlight that the proposed technique can reduce overestimation by 30% in average, and up to 60%, compared to the state of the art.
E. M. Forster once said: “Spoon feeding, in the long run teaches us nothing but the shape of the spoon”. I believe that this quote embeds a fundamental guideline for effective teaching: long-term learning for a student cannot be achieved through passive listening, but only through active participation. This is because there is usually no way to convey an abstract idea unaltered from the teacher’s mind to the student. Instead, the teacher should provide the basic knowledge/material, allow students to reason about it thoroughly and develop their own way of thinking. The main advantage of this approach is that students are encouraged to be creative. Moreover, it generates a sense of self-accomplishment that will be a push for further learning.
My teaching methodology revolves around active learning, I adopt two main strategies. First, in discussion sections I put students into groups of 3 or 4 people, so that they have the opportunity to discuss problems and course material with their peers. At the same time, they develop collaboration and teamwork skills. In fact, when students are given the opportunity to interact with each other, they are more likely to feel like part of a community and become engaged with the course. Second, I believe that thinking is not driven by answers but by questions and that only students who have questions are really thinking and learning. For this reason, I try to create an environment where each student feels comfortable about asking questions.
My full teaching statement is available here.
Workshop for undergraduate students with high academic potential who are members of groups, such as ethnic minorities and women, who tend to be underrepresented in STEM. The workshop involves challenging problems to encourage critical thinking and is designed around in-class activities to promote class discussion and active participation. An example of a class activity performed in the workshop is available below. Sample worksheets, solutions and lecture notes are also available/
First course in calculus and analytic geometry for students with some calculus background; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.
First course in calculus and analytic geometry; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.
Class material is available here.
A calculus course intended for those studying business, economics, or other related business majors. The following topics are presented with applications in the business world: functions, graphs, limits, exponential and logarithmic functions, differentiation, integration, techniques and applications of integration, partial derivatives, optimization, and the calculus of several variables.
First course in calculus and analytic geometry for students with some calculus background; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.
A calculus course intended for those studying business, economics, or other related business majors. The following topics are presented with applications in the business world: functions, graphs, limits, exponential and logarithmic functions, differentiation, integration, techniques and applications of integration, partial derivatives, optimization, and the calculus of several variables.
First course in calculus and analytic geometry; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.
The best way to get in contact with me is either by email or by arranging an appointment. Please use the following information to get in touch with me.
You can find me at my office located at University of Michigan, Ann Arbor. The address is: 2074 East Hall 530 Church Street Ann Arbor, MI 48109-1043 - Room 4863
You may consider sending an email to arrange an appointment.
First course in calculus and analytic geometry; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions.
Lectures: 243 Atgeld Hall
Time: Mon, Tue, Wed, Thurs, Fri - 10 AM to 11.40 AM
Instructor: Neriman Tokcan
Email: tokcan2@illinois.edu