IITG Mathematics Seminar Series

 Lecture Number: 361 Title: Endpoint estimate of rough maximal singular integral operator Speaker: Prof.Parasar Mohanty Affiliation: Professor, Department of Mathematics and Statistics, Indian Institute of Technology Kanpur Date: 7th October, 2021 (Thursday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: See the PDF file for abstract

 Lecture Number: 360 Title: Lower Bounds in Algebraic Circuit Complexity Speaker: Ms. Prerona Chatterjee Affiliation: Final year PhD student, School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai Date: 14th September, 2021 (Tuesday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: Algebraic circuits and formulas are the standard computational models for computing multivariate polynomials. They are similar to boolean circuits and formulas except that in this case, gates are labelled by '+' or 'x'. Another important model is that of an algebraic branching program (ABP), which is an intermediate model between circuits and formulas in terms of computational power. I will begin the talk by defining these models of computation formally, and then survey some of the results that have helped shape our understanding of algebraic circuits.

Along the way, we will see that even though very strong lower bounds are known against various structured models, the best known lower bound against general algebraic circuits is barely super-linear [Str73, BS83]. This was also the best known bound against ABPs. On the other hand, against formulas, the best known lower bound was \Omega(n^2/log n) [Kal85, SY11] for multilinear polynomials (which is the interesting case). In a joint work with Mrinal Kumar, Adrian She and Ben Lee Volk, we improved the bounds for ABPs and formulas by showing a quadratic lower bound against both these models. We will see the proof ideas of this result in some detail.

 Lecture Number: 359 Title: The journey of Voxelgrids - Story of an Indian Magnetic Resonance Imaging (MRI) Scanner startup Speaker: Dr. Arjun Arunachalam Affiliation: Founder and Chief Executive Officer (CEO) of Voxelgrids Innovations Pvt Ltd. Date: 25th August, 2021 (Wednesday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: Voxelgrids is an Indian Medical Devices startup that was formally founded in March 2017. The company's main vision is to build next generation, compact, lightweight, affordable, full body Magnetic Resonance Imaging [MRI] Scanners.

During an MRI scan, the human anatomy is subjected to a strong magnetic field under the influence of which, water from within the human anatomy emits radio frequency (RF) signals that are detected and digitized to form an image. However, this process of detecting and digitizing radio signals has to be repeated many times over during a scan in order to acquire sufficient data to build a complete image. This is therefore a time consuming process that presents a significant clinical limitation as the data acquisition rate during an MRI scan is not fast enough for real time, high resolution imaging of rapidly moving organs such as the heart.

Additionally, there are also serious practical limitations that have limited the widespread adoption of MRI. For instance, the magnetic field that the magnet of an MRI scanner produces should be strong and highly uniform over the anatomical region that is to be imaged. This is a demanding design specification that inevitably results in bulky, heavy and very expensive magnets that also require liquid helium, an expensive and rapidly diminishing non-renewable resource. In summary, a combination of the clinical, practical and financial limitations listed above have currently limited the widespread adoption of MRI.

Voxelgrids has developed a next generation full body MRI scanner that provides the following benefits:

1. World’s first liquid helium free, commercial high field full body human MRI scanner
2. World’s fastest MRI scanner with Imaging speeds exceeding those of existing advanced systems by factors of 3 to 4.
3. An extremely cost effective high field (1.5 Tesla) scanner with an expected 3-4X increase in Return on Investment (ROI).

The societal impact of this development is the following:

(a)The cost of an MRI scan for a patient is expected to become cheaper by a factor of 3-4. This will provide more people access to this Imaging modality, which at this point is prohibitively expensive for patients from lower income backgrounds.

(b)The 3-4X speed up factor can enable greater patient throughput and a higher Return on Investment for a hospital or Imaging center. Alternatively, the 3-4X speed up factor can be employed in real time high resolution Imaging of rapidly moving organs such as the heart. This will greatly enhance MRI’s diagnostic utility beyond its traditional application areas.

This talk will focus on the Voxelgrids journey and shall shed light on the challenges and successes that have contributed towards this successful milestone.

 Lecture Number: 358 Title: Lugsail lag windows for estimating time-average covariance matrices Speaker: Dr. Dootika Vats Affiliation: Assistant Professor, Department of Mathematics and Statistics, Indian Institute of Technology Kanpur Date: 10th August, 2021 (Tuesday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: Lag windows are commonly used in time series, econometrics, steady-state simulation, and Markov chain Monte Carlo to estimate time-average covariance matrices. In the presence of positive correlation of the underlying process, estimators of this matrix almost always exhibit significant negative bias, leading to undesirable finite-sample properties. We propose a new family of lag windows specifically designed to improve finite-sample performance by offsetting this negative bias. Any existing lag window can be adapted into a lugsail equivalent with no additional assumptions. We employ the lugsail lag windows in weighted batch means estimators due to their computational efficiency on large simulation output and arrive at some key theoretical results. Superior finite-sample properties are illustrated in a few examples.

 Lecture Number: 357 Title: (Polynomial) convexity properties of real submanifolds in complex Euclidean spaces Speaker: Dr. Purvi Gupta Affiliation: Assistant Professor, Department of Mathematics, Indian Institute of Science Bangalore Date: 06th August, 2021 (Friday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: A compact subset of $\mathbb{C^n}$ is said to be polynomially convex if it is defined by a family of polynomial inequalities. In this talk, we will elaborate on this definition, and discuss its approximation-theoretic implications. We will focus on the special but natural setting of real submanifolds of $\mathbb{C^n}$, where some of the factors influencing polynomial convexity are: the topology of the submanifold, the (partial) complex structure inherited by the submanifold from the ambient space, and the nature of the singularities of this inherited structure. Within this context, we will discuss some recent results on a related embedding problem.

 Lecture Number: 356 Title: Universal static hedging of contingent claims Speaker: Dr. Shashi Jain Affiliation: Assistant Professor, Department of Management Studies, Indian Institute of Science Bangalore Date: 31st March, 2021 (Wednesday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: There is a growing interest in static hedging of option contracts, as dynamic hedging often breaks down when there are sharp movements in markets or when the market faces liquidity issues. Unfortunately, these are the precise moments where an effective hedge is highly desired. The principle of static replication is to construct a portfolio of instruments that mirrors the value function of a target security in every possible state of the world. We present an algorithm, inspired by neural networks, for a universal approach towards static hedging of a wide variety of contingent claims (including path dependent options).

 Lecture Number: 355 Title: Foundational Aspects of Blockchain Security & Privacy Speaker: Dr. Sushmita Ruj (won Samsung Global Research Outreach 2014 award, NetApp Faculty Fellowship 2016, OCSP grant from IBM Research 2017) Affiliation: Senior Research Scientist, CSIRO Data61 and Associate Professor, Cryptology and Security Research Unit, Indian Statistical Institute, Kolkata. Date: 16th March, 2021 (Tuesday) Time: 10:00 AM (Venue: Microsoft Teams)

Abstract: Blockchain is a verifiable immutable distributed ledger. The talk will introduce and discuss cryptographic primitives and consensus algorithms used in blockchains. We will then discuss various attacks on popular proof of work based blockchains. We will end with some privacy concerns in blockhains and possible mitigations.

 Lecture Number: 354 Title: Efficient generation of ideals Speaker: Dr. Md. Ali Zinna (Young Scientist Award from INSA (2020) and Inspire Faculty Award from DST (2016)) Affiliation: Assistant Professor, Department of Mathematics and Statistics, Indian Institute of Science Education & Research Kolkata Date: 23th February, 2021 (Tuesday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: Let $R$ be a Noetherian ring. An ideal $I$ of $R$ is called efficiently generated if $\mu(I)=\mu(I/I^2)$, where $\mu(I)$ denotes the minimal number of generators of $I$ (as an $R$-module). In this talk we will provide sufficient conditions for an ideal $I$ to be efficiently generated.

 Lecture Number: 353 Title: Digital Transformation enabled by Data Science Speaker: Ritish Menon Affiliation: Senior Manager-II, Data Science, Walmart Global Tech India Date: 10th February, 2021 (Wednesday) Time: 04:00 PM (Venue: Microsoft Teams)

Abstract: In this talk, we will be discussing how Data Science projects are executed to enable Digital Transformation of large scale Enterprises. We will spend some time in demystifying practical implementation of data science and covering a broad set of use cases leveraging Machine Learning and Statistical Models. We will deep dive into one specific problem statement. Finally, we will also spend some time discussing Data Science career path for fresh graduates.

 Lecture Number: 352 Title: Probabilistic Techniques in Nonlocal PDE Speaker: Dr. Anup Biswas (Awarded SwarnaJayanti Fellowships 2020 in Mathematical Science) Affiliation: Associate Professor, Department of Mathematics, Indian Institute of Science Education and Research-Pune Date: 27th January, 2021 (Wednesday) Time: 04:30 PM (Venue: Microsoft Teams)

Abstract: Nonlocal PDE has been an active research area in differential equations for the last two decades. In this talk we see how simple probabilistic tools can contribute in the development of nonlocal PDE. More precisely, I shall illustrate two specific problems and find solutions using some simple estimates in probability.

The talk is intended for a general math audience.

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