## Thesis Defences

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Thesis defenses in all programs should be announced online 10 days before the oral examination. The details of the thesis presentations are listed here and the calendar located in "Home" page. The students are strongly encouraged to attend the presentations.

## Schedule

** Speaker: **Sharoy Augustine Samuel

** Affiliation: **Ph.D. in Financial Mathematics

**Advisor:** Ali Devin Sezer

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**Date: **July 2, 2021 10:30 Ankara

The **Abstract: **We study a class of nonlinear BSDEs with a superlinear driver process $f$ adapted to a filtration${\mathbb F}$ and over a random time interval $[0,S]$ where $S$ is a stopping time of ${\mathbb F}$. The filtration is assumed to support at least a $d$-dimensional Brownian motion as well as a Poisson random measure. The terminal condition $\xi$ is allowed to take the value $+\infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability $1$. We focus on non-Markovian terminal conditions of the following form:1) $\xi = \infty \cdot {\bm 1}_{\{\tau \le S\}}$ and 2) $\xi_2 = \infty \cdot {\bm 1}_{\{ \tau >S \}}$ where $\tau$ is another stopping time.

We call a stopping time $S$ solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value $\infty$ at terminal time $S$. The concept of solvability plays a key role in many of the arguments. We also use the solvability concept to relax integribility conditions assumed in previous works for continuity results for BSDE with singular terminal conditions for terminal values of the form $\infty \cdot {\bm 1}_{\{\tau \le T \}}$ where $T$ is deterministic.We provide numerical examples in cases where the solution is explicitly computable and a basic application in optimal liquidation.

** Speaker: **Süleyman Yıldız

** Affiliation: **PhD in Scientific Computing

**Advisor:** Bülent Karasözen

** Place: https://zoom.us/j/92728802662?pwd=YkRxSUlwZ0s3MVJGZXNmdkxLRFo1QT09**

**Meeting number:** **:** 927 2880 2662

**Password:** 361150

**Date: **June 18, 2021 13:30 Ankara

**Abstract: **The shallow water equations (SWEs) consist of a set of two-dimensional partial differential equations (PDEs) describing a thin inviscid fluid layer flowing over the topography in a frame rotating about an arbitrary axis. SWEs are widely used in modeling large-scale atmosphere/ocean dynamics and numerical weather prediction. Highresolution simulations of the SWEs requires long time horizons over global scales, which when combined with accurate resolution in time and space makes simulations very time-consuming. While high-resolution ocean-modeling simulations are still feasible on large HPC machines, performing many query applications, such as repeated evaluations of the model over a range of parameter values, at these resolutions, is not feasible. Techniques such as reduced-order modeling produces an efficient reduced model based on existing high-resolution simulation data. In this thesis, ROMs are investigated for the rotating SWE, with constant (RSWE) and non-traditional SWE with full Coriolis force (NTSWE), and for rotating ther- mal SWE (RTSWE) while preserving their non-canonical Hamiltonian-structure, the energy and Casimirs, i.e. mass, enstrophy, vorticity, and buoyancy. Two different approaches are followed for constructing ROMs; the traditional intrusive model order reduction with Galerkin projection and the data-driven, non-intrusive ROMs. The full order models (FOM) of the SWE, which needed to construct the ROMs is obtained by discretizing the SWE in space by finite differences by preserving the skew-symmetric structure of the Poisson matrix. Applying intrusive proper orthogonal decomposition (POD) with the Galerkin projection, energy preserving ROMs are constructed for the NRSWE and RTSWE in skewgradient form. Due to nonlinear terms, the dimension of the reduced-order system scales with the dimension of the FOM. The nonlinearities in the ROM are computed by applying the discrete empirical interpolation (DEIM) method to reduce the computational cost. The computation of the reduced-order solutions is accelerated further by the use of tensor techniques. For the RSWE in linear-quadratic form, the dimension of the reduced solutions is obtained using tensor algebra without necessitating hyper-reduction techniques like the DEIM. Applying POD in a tensorial framework by exploiting matricizations of tensors, the computational cost is further reduced for the rotating SWE in linear-quadratic as well in skew-gradient form. In the data-driven, nonintrusive ROMs are learnt only from the snapshots by solving an appropriate leastsquares optimization problem in a low-dimensional subspace. Data-driven ROMs are constructed for the NTSWE and RTSWE with the operator inference (OpInf) using, (non-Markovian) and with re-projection (Markovian) dynamics, respectively. Computational challenges are discussed that arise from the optimization problem being ill-conditioned. Moreover, the non-intrusive model order reduction framework is extended to a parametric case, whereas we make use of the parameter dependency at the level of the PDE without interpolating between the reduced operators. The overall procedure of the intrusive and non-intrusive ROMs for the rotating SWEs in linear-quadratic and skew-gradient form yields a clear separation of the offline and online computational cost of the reduced solutions. The predictive capabilities of both models outside the range of the training data are shown. Both ROMs behave similarly and can accurately predict in the test and training data and capture system behavior in the prediction phase. The preservation of physical quantites in the ROMs of the SWEs such as energy (Hamiltonian), and other conserved quantities, i.e., mass, buoyancy, and total vorticity, enables that the models fit better to data and stable solutions are obtained in long-term predictions which are robust to parameter changes while exhibiting several orders of magnitude computational speedup over the FOM.

** Speaker: **Umut Gölbaşı

** Affiliation: **MSc in Financial Mathematics

**Advisor:** A. Sevtap Kestel

** Place: https://zoom.us/j/95369171602?pwd=ek1UUm90a3VDR1c5dWtRNGdzQkZMUT09**

**Meeting number:** **:** 953 6917 1602

**Password:** 783645

**Date: **15 March 2021, 10:00

**Abstract: **Electricity generation cost and environmental effects of electricity generation continue to be among central themes in energy planning. The choice of electricity generation technology and energy source affect the environment through released greenhouse gases and other waste. United States is the world’s second-largest CO2 emitter and electricity consumer. This thesis aims to estimate the optimal capacity expansion of electric power sector in the United States for 2022-2050. We develop a fuzzy multi-objective linear program to minimize cost and environmental effects. In sensitivity analyses, we show how different policies and price evolution may alter the mix. Later on, we examine the effects of the new capacity mix and implied generation on the cost of electricity and emissions. We find that direct modeling of capacity factors give meaningful results. According to this thesis, renewable energy is expected to reach more than 1100 GW installed capacity by 2050. This reduces average cost of electricity generation by more than 70 percent and reduces CO2 emissions by more than 80 percent compared to expected end-2021 levels.

** Speaker: **Burcu Aydogan

** Affiliation: **PhD in Financial Mathematics

**Advisor:** Ömür Ugur

** Place: https://zoom.us/j/94118246108?pwd=LzhyL255SkE1eHpQRHA2RW5vSG5aUT09**

**Meeting number:** 941 1824 6108

**Password:** 316831

**Date: **15 March 2021, 13:00

**Abstract: **In this thesis, we intend to develop optimal market making strategies in a limit order book for high-frequency trading using stochastic control approach. Firstly, we address for evolving optimal bid and ask prices where the underlying asset follows the Heston stochastic volatility model including jump components to explore the effect of the arrival of the orders. The goal of the market maker is to maximize her expected return while controlling the inventories where the remaining is charged with a liquidation cost. Two types of utility functions are considered: quadratic and exponential with a risk averse degree, respectively. Then, we take into consideration a model considering an underlying asset with jumps in stochastic volatility. We derive the optimal quotes for both models under the assumptions. For the numerical simulations, we apply finite differences and linear interpolation as well as extrapolation methods to obtain a solution of the nonlinear Hamilton-Jacobi-Bellman (HJB) equation. We discuss the influence of each parameter on the best bid and ask prices in the models and demonstrate the risk metrics including profit and loss distribution (PnL), standard deviation of PnL and Sharpe ratio which play important roles for the trader to make decisions on the strategies in high-frequency trading. Moreover, we provide the comparisons of the strategies with the existing ones. The thesis reveals that our models describe and fit the real market data better since a real data has jumps and the volatility is fluctuating in reality. As a real data application, we conduct our simulations for the developed strategies in this thesis on the high-frequency data of Borsa Istanbul (BIST). For this purpose, we first estimate the parameters of each model and then perform the numerical experiments on the optimal quotes. Our aim is to investigate the qualitative behaviour of an investor who is trading in an emerging market by our strategies in terms of the PnL, standard deviation of PnL and inventory process. Furthermore, we provide the applications on global stocks in order to see that the models are applicable, reasonable and profitable also for the developed markets. Lastly, we take account of the optimal market making models with stochastic latency in the price. The jump components are included on these models, as well. We contribute to this study by providing the numerical experiments with artificial data. Finally, the thesis ends up with a conclusion and a showcase on future research.

** Speaker: **Deniz Kenan Kilic

** Affiliation: **PhD in Financial Mathematics

**Advisor:** Ömür Ugur

** Place: https://zoom.us/j/99252972182?pwd=Tk9Yc0d6MzlvSWhra29yYldLcGVsQT09**

**Meeting number:** 992 5297 2182

**Password:** 711612

**Date: **15 February 2021, 11:00

**Abstract: **The thesis aims to combine wavelet theory with nonlinear models, particularly neural networks, to find an appropriate time series model structure. Data like financial time series are nonstationary, noisy, and chaotic. Therefore using wavelet analysis helps for better modeling in the sense of both frequency and time. Data is divided into several components by using multiresolution analysis (MRA). Subsequently, each part is modeled by using a suitable neural network structure. In this step, the design of the model is formed according to the pattern of subseries. Then predictions of each subseries are combined. The combined prediction result is compared to the original time series’s prediction result using only a nonlinear model. Moreover, wavelets are used as an activation function for LSTM networks to form a hybrid LSTM-Wavenet model. Furthermore, the hybrid LSTM-Wavenet model is fused with MRA as a proposed method. In brief, it is studied whether using MRA and hybrid LSTM-Wavenet model decreases the loss or not for both S&P500 (∧GSPC) and NASDAQ (∧IXIC) data. Four different modeling methods are used: LSTM, LSTM+MRA, hybrid LSTM-Wavenet, hybrid LSTM-Wavenet+MRA (the proposed method). Results show that using MRA and wavelets as an activation function together decreases error values the most.

**Speaker: **Merve Gözde Sayın

**Affiliation: **MSc in Financial Mathematics

**Advisor:** Ceylan Yozgatligil

**Place: https://zoom.us/j/91434127130?pwd=QVB1VmlRNzgrenl6b014c0ZiY3lWdz09**

Meeting number: 914 3412 7130

Password: 205326

**Date: **15 February 2021, 13:00

**Abstract: ** Estimating stock indices that reflect the market has been an essential issue for a long time. Although various models have been studied in this direction, historically, statistical methods and then various machine learning methods have to introduced artificial intelligence into our lives. Related literature shows that neural networks and tree-based models are mostly used. In this direction, in this thesis, four different models are examined. The first one is the most preferred neural network method for financial data called LSTM, and the second one is one of the most preferred tree-based models called XGBoost, and the third and the fourth models are the hybridizations of LSTM and XGBoost. Besides, these models have been applied to eight different stock market indices, and the model that gives the best results is determined according to the Mean Absolute Scaled Error (MASE) evaluation criteria.

** Speaker: **Esra Günsay

** Affiliation: **MSc in Cryptography

**Advisor:** Murat Cenk

** Place: https://zoom.us/j/91560425604?pwd=VXZqK2dOd1NYWVViVTE4UTZLR2NoUT09**

**Meeting number:** 915 6042 5604

**Password:** 052190

** Date: **12 February 2021, 16:30

**Abstract: **Appropriate, effective, and efficient use of cryptographic protocols contributes tomany novel advances in real-world privacy-preserving constructions. One of the most important cryptographic protocols is zero-knowledge proofs. Zero-knowledge proofs have the utmost importance in terms of decentralized systems, especially in context of the privacy lately. In many decentralized systems, such as electronic voting, e-cash, e-auctions, or anonymous credentials, zero-knowledge range proofs are used as building blocks. The main purpose of this thesis is to explain range proofs with detailed primitives and examine their applications in decentralized, so-called blockchain systems such as confidential assets, Monero, zkLedger, and Zether.In this thesis, we have examined, summarised, and compared range proofs based on zero-knowledge proofs.

** Speaker: **Gizem Kara

** Affiliation: **MSc in Cryptography

**Advisor:** Ali Doğanaksoy

**Co-Advisor: **Oğuz Yayla

**Place: https://zoom.us/j/91560425604?pwd=VXZqK2dOd1NYWVViVTE4UTZLR2NoUT09**

**Meeting number:** 915 6042 5604

**Password:** 052190

** Date: **12 February 2021, 17:30

**Abstract: **A number of arithmetization-oriented ciphers emerge for use in advanced cryptographic protocols such as secure multi-party computation (MPC), fully homomorphic encryption (FHE) and zero-knowledge proofs (ZK) in recent years. The standard block ciphers like AES and the hash functions SHA2/SHA3 are proved to be efficient in software and hardware but not optimal to use in this field for this reason, new kind of cryptographic primitives proposed. However, unlike traditional ones, there is no standard approach to design and analyze such block ciphers and the hash functions, therefore their security analysis needs to be done carefully. In 2018, StarkWare launched a public STARK-Friendly Hash (SFH) Challenge to select an efficient and secure hash function to be used within ZK-STARKs, transparent and post-quantum secure proof systems. The block cipher JARVIS is one of the first ciphers designed for STARK applications but, shortly after its publication, the cipher has been shown vulnerable to Gröbner basis attack. This master thesis aims to describe a Gröbner basis attack on new block ciphers, MiMC, GMiMCerf (SFH candidates) and the variants of JARVIS. We present the complexity of Gröbner basis attack on JARVIS-like ciphers, results from our experiments for the attack on reduced-round MiMC and a structure we found in the Gröbner basis for GMiMCerf.

** Speaker: **Pınar Ongan

** Affiliation: **MSc in Cryptography

**Advisor:** Ali Doğanaksoy

**Co-Advisor:** Burcu Gülmez Temür

** Place: https://zoom.us/j/98707521401?pwd=OCtPRnV2a013QnVNUnhiTTFOQjVtUT09**

**Meeting number:** 987 0752 1401

**Password:** 068665

** Date: **11 February 2021, 16:30

**Abstract: **This thesis consists of two main parts: In the first part, a study of several classes of permutation and complete permutation polynomials is given, while in the second part, a method of construction of several new classes of bent functions is described. The first part consists of the study of several classes of binomials and trinomials over finite fields. A complete list of permutation polynomials of the form f(x)=x^{(q^n-1)/(q-1)+ 1} + b x in GF(q^n)[x] is obtained for the case n=5, and a criterion on permutation polynomials of the same type is derived for the general case. Furthermore, it is shown that when q is odd, trinomials of the form f(x)= x^5 h(x^{q-1}) in GF(q^2)[x], where h(x)=x^5+x+1 never permutes GF(q^2). A method of constructing several new classes of bent functions via linear translators and permutation polynomials forms the second part of the thesis. First, a way to lift a permutation over GF(2^t) to a permutation over GF(2^m) is described, where t divides m. Then, via this method, 3-tuples of particular permutations that lead to new classes of bent functions are obtained. As a last step, the fact that none of the bent functions obtained here will be contained in Maiorana-McFarland class is proved.

**Speaker: **Fatih Cingöz

** Affiliation: **MSc in Financial Mathematics

**Advisor:** Adil Oran

** Place: https://metu.webex.com/metu/j.php?MTID=m5385a4be312d961b1c561de397bb22af**

Meeting number: 121 130 773

Password: Fp8vgrbq9g6

** Date: **4 February 2021, 14:00

** Abstract:**