Find Academic Supervisors

Maja J MatarićView profile

Professor

• Socially Assistive Robotics
• Human-Robot Interaction
• Assistive Technology
• +7 more

Maja J Matarić is the Chan Soon-Shiong Chaired and Distinguished Professor of Computer Science at the University of Southern California's Viterbi School of Engineering, with courtesy appointments in Neuroscience and Pediatrics. She serves as founding director of the USC Robotics and Autonomous Systems Center, co-director of the USC Robotics Research Lab, and Principal Scientist at Google DeepMind. Previously, she held leadership roles as USC's interim Vice President of Research (2020-2021) and Vice Dean for Research (2006-2019). Her educational background includes: PhD in Computer Science and Artificial Intelligence from MIT (1994) MS in Computer Science from MIT (1990) BS in Computer Science from University of Kansas (1987) Matarić pioneers Socially Assistive Robotics (SAR) , a field her lab named, focusing on human-robot interaction that provides assistance through social rather than physical support. Her research targets critical health and wellness challenges including post-stroke rehabilitation, autism spectrum disorder therapy, cognitive exercises for Alzheimer's patients, ADHD academic support, and mental health interventions. She develops systems modeling user engagement, personality, and motivation, with extensive real-world deployments in schools, rehabilitation centers, and homes. Analysis of her recent publications reveals dominant themes in cognitive health robotics (2025 CHI paper on LLM-powered elder care), pediatric assistive technology (2025 IDC speech therapy review), and adaptive preference modeling (2025 HRI contrastive learning work). Her research consistently bridges machine learning with human-centered design for vulnerable populations. Major scientific recognition includes: ACM Athena Lecturer Award (2024) ACM Eugene L. Lawler Humanitarian Award (2024) ACM Fellow (2020) Presidential Mentoring Award (2011) Multiple society fellowships (AAAS, IEEE, AAAI) As a dedicated mentor, Matarić has championed underrepresented groups through CRA-W, placing numerous women in faculty positions. She leads USC Viterbi's K-12 STEM Outreach Program serving low-income Los Angeles schools and authored The Robotics Prime for student education. Her research has secured significant funding enabling real-world technology transfer, with documented impact in rehabilitation centers and homes through deployable SAR systems. Her Robotics Research Lab at USC drives innovation in embodied AI, with current projects spanning LLM-integrated elder care robots, ADHD academic companions, and autism therapy systems. The lab emphasizes co-design with end-users and rigorous real-world validation across diverse populations.

Richard M. MurrayView profile

Professor

• Control Theory
• Autonomous Vehicles
• Biological Circuit Design
• +3 more

Richard M. Murray is the Thomas E. and Doris Everhart Professor of Control and Dynamical Systems and Bioengineering at the California Institute of Technology (Caltech). He holds a B.S. from Caltech (1985), M.S. from UC Berkeley (1988), and Ph.D. from UC Berkeley (1990). He has served in academic roles from Assistant Professor (1991–1997) to his current endowed professorship. He chaired the Engineering and Applied Science division (2000–2005) and Biology and Biological Engineering (2020–2024). His research focuses on feedback control in biological and autonomous systems, synthetic cells, and networked control systems. Collaborators include experts in robotics, synthetic biology, and systems biology. Key awards include the IEEE Control Systems Award and election to the National Academy of Engineering. His educational contributions span courses on control systems, robotics, and bioengineering. Current research projects include the Developer Cell initiative (Sloan Foundation), layered testing for autonomous systems (AFOSR), and microbiome-based environmental solutions (CHARMME, ARO). He advises numerous graduate students and postdocs, with notable alumni in academia and industry. Labs include facilities in Keck and Steele laboratories at Caltech. His work bridges control theory, synthetic biology, and autonomous systems to address societal challenges like environmental monitoring and safe autonomy.

Asaf CidonView profile

Associate Professor

• Software Systems
• Storage Systems
• Machine Learning
• +4 more

Asaf Cidon is an Associate Professor at Columbia University, jointly affiliated with the Department of Electrical Engineering and Computer Science, and a member of the Data Science Institute. His research focuses on software systems , storage , large-scale machine learning , and cybersecurity . Stanford University - PhD in Electrical Engineering Stanford University - MS in Electrical Engineering Technion - BS in Computer and Software Engineering His work in distributed storage systems has been commercialized by companies such as Facebook, Tibco, and Rubrik. He has led projects like Sentinel and Forensics during his industry career. Recent publications highlight advancements in software-based radiation protection (ASPLOS'26), PCIe pooling with CXL (HotOS'25), and AI phishing detection (IMC'25). These reflect his expertise in system architecture , security , and networking . Scientific recognitions include best paper awards at OSDI, Usenix Security, CIDR, and ATC, along with NSF CAREER and ARO Young Investigator Awards . His papers Cookie Monster (SOSP'24) and Chablis (CIDR'24) received notable accolades. He has mentored numerous PhD and Master's students , including Edward Guo, Harry Wang, and Teng Jiang, many of whom now hold roles at Google, Meta, Amazon, and academic institutions. His lab at Columbia is actively recruiting CS and EE PhD students. Prior to Columbia, he founded and led the startup Sookasa to acquisition and served as Senior Vice President of Email Protection at Barracuda Networks , managing a $200M business with 100 engineers.

Trevor CampbellView profile

Associate Professor

• Bayesian Inference
• Bayesian Nonparametrics
• Machine Learning
• +5 more

Trevor Campbell is an Associate Professor in the Department of Statistics at the University of British Columbia (UBC), Vancouver. He holds a Ph.D. in Machine Learning and Statistics from MIT and a B.A.Sc. in Aerospace Engineering from the University of Toronto. His research focuses on automated, scalable Bayesian inference algorithms, Bayesian nonparametrics, and streaming data analysis. Campbell is a core contributor to probabilistic programming tools like Pigeons.jl and has developed influential methods for coreset-based Bayesian inference. Education : Ph.D. in Machine Learning and Statistics (2016), MIT M.S. in Aeronautics and Astronautics (2013), MIT B.A.Sc. in Aerospace Engineering (2011), University of Toronto Research Interests : His work emphasizes scalable Bayesian computation, including variational inference, MCMC optimization, and coresets. He develops algorithms that balance statistical accuracy with computational efficiency, particularly for large-scale datasets. His recent work explores adaptive samplers (e.g., AutoStep, autoMALA) and theoretical guarantees for coreset methods. Applications span astrophysics, materials science, and network analysis. Awards & Grants : NSERC Discovery Grant (2025) Blackwell-Rosenbluth Award (2021) PIMS Early Career Award (2023) Google Perception Academic Funding (2020) Advising & Labs : He supervises a team of PhD and M.Sc. students at UBC, focusing on Bayesian methodology and computational tools. His lab collaborates with institutions like SFU and MIT on projects involving distributed sampling and probabilistic modeling. He co-organizes workshops on Bayesian computation and serves on editorial boards for Bayesian Analysis and TMLR.

Hyesoon KimView profile

Professor

• Computer Architecture
• GPU
• Compilers and Runtime Systems
• +3 more

Hyesoon Kim is a Professor at the Georgia Institute of Technology , affiliated with the College of Computing and leading the HPArch research group . She co-directs the Center for Research into Novel Computing Hierarchies (CRNCH) . Her research focuses on Computer Architecture , GPU , Compilers and Runtime Systems , and Hardware Security , particularly for heterogeneous systems. Contact : hyesoon@cc.gatech.edu Location : 266 Ferst Drive, KACB 2344, Atlanta, GA Research Trends Her recent work spans RISC-V extensions for security, CUDA optimization on softcore GPUs, memory safety techniques, and energy-efficient deep learning architectures. Articles emphasize heterogeneous computing , GPU performance scaling, and IoT -oriented neural network methods. Open Source Projects She leads development of Macsim (heterogeneous architecture simulator) and Vortex (open-source GPU platform).

Daniel W. BlissView profile

Professor

• Radar
• Wireless Communications
• Precision Positioning
• +7 more

Daniel W. Bliss is a Professor in the School of Electrical, Computer and Energy Engineering at Arizona State University and Director of ASU's Center for Wireless Information Systems and Computational Architectures (WISCA). With over $50 million in research funding as principal investigator from organizations including DARPA, ONR, Google, and Airbus, his work bridges theoretical foundations with practical implementations across multiple domains of wireless systems. Dr. Bliss received his educational foundation with a B.S.E.E. from Arizona State University (1989), followed by M.S. and Ph.D. degrees in Physics from the University of California-San Diego (1995, 1997). His academic journey includes significant industry experience at General Dynamics (1989-1993) and MIT Lincoln Laboratory (1997-2012) before joining ASU. His research program focuses on advanced wireless systems spanning radar, communications, precision positioning, computational architectures, and medical monitoring applications. Bliss employs information theory, estimation theory, and signal processing to develop novel system concepts with disruptive capabilities. Current research emphasizes RF convergence, integrated sensing and communications, and anticipatory medical analytics using wireless technologies, with particular focus on extracting physiological data from radar signals. Analysis of recent publications reveals a strong trend toward integrated sensing and communications systems, particularly utilizing mmWave and radar technologies for medical monitoring applications. His work increasingly bridges traditional communications and radar domains while expanding into physiological monitoring, demonstrating a clear trajectory toward convergence of wireless technologies for healthcare applications and remote vital sign detection. Dr. Bliss has received significant recognition for his contributions: Fellow of the IEEE (2015) 2021 IEEE Warren D. White Award for Excellence in Radar Engineering 2016-2017 Top 5% Teaching Award at ASU 2017 ASU Fulton Engineering Exemplar Faculty As a dedicated mentor, Dr. Bliss has supervised numerous graduate students through successful dissertation and thesis defenses across both PhD and Master's programs. His research portfolio includes substantial funding from diverse sources with over $50 million secured as principal investigator. Current projects include the $17M DARPA DASH project focused on advanced software-reconfigurable heterogeneous SoCs for next-generation RF systems, and multiple initiatives in contactless vital sign monitoring using radar technologies. Dr. Bliss leads the BLISS Lab and serves as director of WISCA, fostering interdisciplinary research in wireless systems. His team includes researchers working on distributed coherent systems, MIMO radar, RF convergence, and medical monitoring applications, with recent successes including the Making Waves team that tied for first place in the Air Force Spark Tank challenge. He has founded two startup companies: DASH Tech Integrated Circuits Company and the Big Little Sensor Company, focusing on high-performance embedded processing and small-scale radar physiological monitoring, respectively.

Simon KingView profile

Professor

• Speech Synthesis
• Speech Processing
• Acoustic Modeling
• +4 more

Simon King is a Professor of Speech Processing at the University of Edinburgh , affiliated with the School of Philosophy, Psychology and Language Sciences . He serves as Director of the Centre for Speech Technology Research (CSTR) and teaches courses like Speech Processing and Speech Synthesis , while directing the MSc in Speech and Language Processing . Research Interests His research focuses on: Developing new acoustic models (e.g., Linear Dynamical Models, factorial-HMMs) for speech recognition Advancing unit selection and HMM-based speech synthesis Integrating articulatory measurement data for enhanced modeling Exploring perceptual measures in synthesis criteria Building multilingual speech systems to identify universal speech building blocks Publication Trends Simon's recent work emphasizes deep learning (DNNs, LSTMs) in speech synthesis, multilingual frameworks , and articulatory-acoustic feature integration . His studies often bridge grapheme-based modeling , perceptual error reduction , and noise-robust synthesis . Scientific Awards EPSRC Advanced Research Fellowship (2005-2009) Students & Collaborations He has supervised numerous PhD students including Rasmus Dall, Tom Merritt, and Srikanth Ronanki. Current research fellows like Mirjam Wester and Zhizheng Wu contribute to projects such as Natural Speech Technology (NST) and Simple4All .

• Operations Research
• Machine Learning
• Optimization under Uncertainty
• +4 more

Minseok Ryu is an Assistant Professor at the School of Computing and Augmented Intelligence, Arizona State University (ASU). He holds a Ph.D. in Industrial & Operations Engineering from the University of Michigan (2020). Prior to ASU, he was a postdoctoral appointee at Argonne National Laboratory’s Mathematics and Computer Science Division. His research focuses on optimization methodologies for decentralized and stochastic decision-making distributed algorithms for machine learning and operations research applications in healthcare systems and energy grids Teaching responsibilities include courses on applied deterministic operations research (IEE 574), optimization (IEE 622), and research practicums (IEE 792). His work emphasizes computational challenges in decision-making under uncertainty, with recent projects addressing nurse staffing optimization, federated learning frameworks (e.g., APPFL/APPFLX), and resilient power grid systems. He has no recorded academic awards but actively contributes to open-source software and cross-disciplinary research collaborations. Research interests bridge theory and practice, targeting social goods through optimization techniques like distributionally robust optimization, federated learning, and heuristic algorithms for energy and healthcare systems.

Kenneth P. BirmanView profile

Professor

• Distributed Systems
• Cloud Computing
• Machine Learning Systems
• +5 more

Kenneth P. Birman is the N. Rama Rao Professor of Computer Science at Cornell University, where he has had a long and impactful career in distributed systems, cloud computing, and AI/ML infrastructure. He is known for foundational contributions to reliable and scalable distributed systems, and for leading high-impact projects such as Cascade, Vortex, and Derecho. He is also the author of a widely used textbook on reliable distributed systems and has founded multiple companies based on his research. Education: Ph.D. in Computer Science, University of California, Berkeley M.S. in Computer Science, University of California, Berkeley B.A. in Computer Science, Columbia University Research Interests: Professor Birman's research focuses on building reliable, secure, and scalable distributed systems . His current emphasis is on AI and ML infrastructure , particularly in reducing latency and improving performance through hardware acceleration, RDMA-based communication, and edge computing. He explores how to eliminate data movement bottlenecks in AI pipelines and how to support real-time, mission-critical applications in domains like healthcare, smart grids, and industrial IoT. His work spans systems programming, cloud computing, fault tolerance, and formal verification . He has designed systems that have been deployed in high-stakes environments such as the New York Stock Exchange, the Swiss Exchange, and the French Air Traffic Control system. Scientific Awards: ACM Fellow (1999) IEEE Fellow (2014) IEEE Tsutomu Kanai Award for innovations in distributed computing Teaching and Mentorship: Professor Birman teaches two courses in the fall semester: CS4414: Systems Programming and CS5416: Cloud and ML Systems Programming . He has advised numerous Ph.D. and M.S. students, including Alicia Yang, Tiancheng Yuan, Yifan Wang, Weijia Song, Edward Tremel, Sagar Jha, Jonathan Behrens, and Mae Milano. He has announced that Fall 2025 will be his last semester teaching, and he is no longer recruiting new students, though he will continue supervising current ones. Labs and Projects: He leads the Derecho Project and the Cascade/Vortex Project , both focused on high-performance distributed systems. These projects are collaborative efforts with students and industry partners, and the software is released under open-source licenses. He also maintains strong ties with Cornell's systems group and collaborates with faculty across CS, ECE, IS, and the Cornell Tech NYC campus.

Andrea ColadangeloView profile

Assistant Professor

• Quantum Computation
• Quantum Cryptography
• Quantum Learning Theory
• +3 more

Andrea Coladangelo is an Assistant Professor at the Allen School of Computer Science & Engineering , University of Washington , co-leading the Quantum group and contributing to the Theory and Crypto groups. He coordinates the NSF-funded Quantum@UW REU program , fostering undergraduate research in quantum information. Previously, he was a postdoctoral researcher at UC Berkeley and the Simons Institute , advised by Umesh Vazirani , following a PhD in Computer Science at Caltech under Thomas Vidick . His academic journey began with a B.A. in Mathematics from Oxford and a Master in Mathematics from Cambridge . He co-founded qBraid , a platform for quantum computing education. Research Interests : Andrea explores the intersection of quantum computation and cryptography , focusing on foundational questions in entanglement , quantum correlations , and quantum learning theory . His work investigates quantum pseudorandomness , device-independent security , quantum algorithms , and quantum copy-protection , often leveraging computational assumptions to bridge quantum information theory with cryptographic applications. Scientific Awards : 2025 Google Research Scholar Program Award in Quantum Computing 2023 CSE Undergraduate Teaching Award for his course on quantum computation 2019 Best Student Paper Award at QIP Teaching & Outreach : Andrea designed and taught CSE 434: Intro to Quantum Computation (Spring 2023, 2024, 2025), CSE 534: Quantum Information and Computation (Autumn 2023), and CSE 599C: Quantum Learning Theory (Winter 2025). He also delivered lectures at the 22nd Bellairs Crypto Workshop (2024) and led a quantum programming tutorial using qBraid . Labs & Teams : As co-leader of the Quantum group at the Allen School, he collaborates with researchers in Theory and Crypto , advancing quantum computing through interdisciplinary projects and educational initiatives.

April Yi WangView profile

Assistant Professor

• Human-Computer Interaction
• Educational Technology
• Programming Support
• +4 more

April Yi Wang is a tenure-track Assistant Professor in the Department of Computer Science at ETH Zürich, where she directs the Programming, Education, and Computer-Human Interaction Lab (PEACH Lab). She is a core faculty member at the Institute for Intelligent Interactive Systems and associated with the ETH AI Center. Wang is also an active member of ETH HCI and Swiss CHI communities, contributing significantly to human-computer interaction and educational technology research. Dr. Wang's educational background includes: Ph.D. in Information Science from University of Michigan (2023), advised by Steve Oney and Christopher Brooks M.Sc. in Computer Science from Simon Fraser University (2018), advised by Parmit Chilana B.Eng in Computer Science from Zhejiang University (2016) Dr. Wang's research focuses on human-centered approaches to programming and data science. Her work reimagines programming as a form of literature that communicates with both machines and people, exploring creative representations like text, shapes, animations, and everyday objects. She investigates how to make programming more natural and intuitive through literate programming environments, with applications in professional and educational contexts. Her research spans human-computer interaction, educational technology, and AI-assisted programming tools. Analysis of Dr. Wang's recent publications reveals a strong focus on AI-enhanced educational tools, particularly for programming and data literacy. Her work increasingly integrates large language models to scaffold learning while maintaining user agency. There's a clear trajectory toward developing situated learning approaches that connect abstract concepts to real-world contexts through augmented reality and tangible interfaces. Her research bridges HCI, education, and AI to create more accessible and engaging technical learning experiences. Dr. Wang has received numerous prestigious awards including: 2023 Gary M. Olson Award and Honourable Mention Award at ACM CHI 2022 Rising Stars in EECS and Heidelberg Laureate Forum Young Researcher 2020 Best Short Paper Award at IEEE VL/HCC and Honourable Mention at ACM CHI 2019 Best Paper Award at ACM CSCW Dr. Wang actively mentors students through thesis projects at ETH Zürich, supervising numerous bachelor's and master's students on topics ranging from AI-assisted programming to data literacy tools. Her lab, PEACH Lab, has secured funding including the recent innovedum funding for the Coducate project. She serves on program committees for major conferences including CHI and UIST, and regularly reviews for top HCI and education journals. The PEACH Lab, directed by Dr. Wang, focuses on creating expressive, intelligent, and human-centered systems that make technical topics more accessible. The lab explores textual, visual, and embodied representations for programming, with emphasis on enhancing communication, collaboration, and learning. Current research directions include balancing automation with user agency, supporting diverse learning needs, and developing tools for interdisciplinary technical communication.

Priyanka RainaView profile

Assistant Professor

• Domain-Specific Architectures
• Hardware/Software Co-design
• Energy-Efficient Computing
• +3 more

Priyanka Raina is an Assistant Professor of Electrical Engineering at Stanford University, with a courtesy appointment in Computer Science. She leads the Stanford Accelerate research group, focusing on domain-specific hardware architectures and agile hardware-software co-design. Her work emphasizes high-performance, energy-efficient accelerators for emerging technologies. Education: B.Tech. from IIT Delhi (2011), M.S. and Ph.D. from MIT (2013/2018). Postdoctoral experience includes NVIDIA Research (2018) and Amazon Visiting Academic (2023–present). Awards include the Sloan Research Fellowship (2024), NSF CAREER Award (2023), and Terman Faculty Fellowship (2018). Research Interests: Domain-specific architectures, near-memory computing, design productivity, and machine learning acceleration. Her group develops frameworks like AHA (Agile Hardware) for efficient accelerator design and compilers. Awards: Over 10 major awards, including best paper recognitions at VLSI, MICRO, and JSSC. Active roles as Associate Editor for IEEE JSSC and Program Chair for IEEE Hot Chips (2020). Advising & Teaching: Supervises multiple PhD and MS students in hardware design, CGRAs, and ML accelerators. Teaches VLSI design courses (EE271/272/372) and oversees independent studies in embedded systems and chip design. Labs & Collaborations: Affiliated with Stanford PORTAL Center, AHA Center, and SystemX Alliance. Projects include MINOTAUR (edge AI accelerator), Amber (CGRA-based SoC), and EMBER (RRAM macros).

Krishnan MaheshView profile

Professor

• High-fidelity simulation of turbulent flows
• Multi-physics fluid dynamics
• Computational Fluid Dynamics (CFD)
• +10 more

Dr. Krishnan Mahesh is a Professor at the University of Michigan with joint appointments in Mechanical Engineering and Naval Architecture and Marine Engineering. He serves as Director of the Center for Naval Research and Education and leads the Computational Fluids Laboratory, where he develops advanced numerical methods for simulating multi-physics turbulent flows. Education: Ph.D. (1996), M.S. (1990) from Stanford University, B.Tech (1989) from IIT Bombay Leadership: Director, Center for Naval Research and Education (2022-present) His research focuses on high-fidelity simulations of turbulent flows with applications in marine propulsors, multiphase systems, cavitation, hydroacoustics, superhydrophobic surfaces, biofouling, fluid-structure interaction, and flow stability. His group develops the MPCUGLES software for unstructured grid simulations on parallel computing platforms. Recent work examines cavitation dynamics , tip vortex flows , and roughness-induced transition in complex marine and aerospace systems. His 15 most recent articles demonstrate expertise in LES/DNS of multi-physics flows, with emphasis on marine propulsion, bubble collapse, and turbulent noise prediction. Scientific Honors: 2021 AIAA Best Paper Award 2018 Fulbright Scholar 2017 Marine Propulsors Symposium Best Paper 2011 APS Fellow 2010 Taylor Award for Distinguished Research He mentors numerous graduate students and postdoctoral fellows, with collaborative projects spanning jet in crossflow analysis, gas turbine simulations, and shock-turbulence interactions. His research receives funding from ONR, NSF, and international naval programs.

Ajmal MianView profile

Professor

• Artificial Intelligence
• Computer Vision
• Machine Learning
• +8 more

Ajmal Mian is a Professor of Computer Science at the University of Western Australia (UWA), affiliated with the School of Physics, Maths and Computing. He holds an Australian Research Council Future Fellowship (2022) and leads research in Artificial Intelligence, Computer Vision, and Machine Learning. His work focuses on 3D computer vision, adversarial AI defense, and explainable AI. His research interests include 3D point cloud analysis, face recognition, human action recognition, and remote sensing. He has published over 300 papers and secured major grants from ARC, NHMRC, and DARPA, totaling millions in funding. He has supervised 29 PhD students and mentored 12 postdoctoral researchers. Key projects include 3D diffusion models for scene generation, robust 3D vision systems, and defense against AI deception attacks. He serves as a fellow of IAPR, an ACM Distinguished Speaker, and has editorial roles at IEEE Transactions on Neural Networks and Pattern Recognition. Research Awards: HBF Mid-Career Scientist of the Year, West Australian Early Career Scientist of the Year, IAPR Best Scientific Paper Award. Grants: ARC Discovery Projects, National Intelligence & Security Discovery grants, DARPA grants for AI security. His teaching spans computer vision, machine learning, and programming courses. Collaborations include defense, medical, and agricultural applications.

• Software Verification
• Formal Methods
• Programming Languages
• +5 more

Jenna Wise DiVincenzo is an Assistant Professor at the Elmore Family School of Electrical and Computer Engineering at Purdue University. She specializes in research areas such as software verification, formal methods, and programming languages, with a focus on gradual verification techniques that combine static and dynamic analysis. Her work emphasizes usability and scalability in verification tools, and she has contributed to projects like Gradual C0 and gradual null-pointer analysis. Dr. DiVincenzo earned her PhD in Software Engineering from Carnegie Mellon University (2023) and a BS in Mathematics and Computer Science from Youngstown State University (2017). She has interned at IBM Research, MIT Lincoln Laboratory, and the Software Engineering Research and Empirical Studies Lab at YSU. Her awards include the Google PhD Fellowship, NSF GRFP Fellowship, and 2022 Rising Star in EECS. Her research projects span theoretical advancements in gradual verification, empirical studies on usability, and practical tool development. She advises PhD students (e.g., Craig Liu, Conrad Zimmerman) and collaborates on initiatives like gradual verification for Rust and educational tools to teach verification concepts. Her work also explores leveraging large language models for specification generation and enhancing verification tool soundness through formal proofs.