During the 2021-22 academic year, the CMSA will be hosting an Interdisciplinary Science Seminar, organized by Yingying Wu. This seminar will take place on **Thursdays **at** 9:00am – 10:00am (Boston time)**. The meetings will take place virtually on Zoom. To learn how to attend, please fill out this form or contact Yingying Wu (ywu@cmsa.fas.harvard.edu).

This seminar encompasses theoretical mathematics and applications of geometric analysis to algorithms and machine learning. We hope the seminar will serve the role of facilitating collaborations between mathematicians, physicists, biologists, and medical practitioners to exchange ideas and participate in collaboration.

The schedule below will be updated as talks are confirmed.

Date | Speaker | Title/Abstract |

09/23/2021 | Michael Simkin Harvard CMSA | Title: The number of n-queens configurationsAbstract: The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. The problem has a storied history and was studied by such eminent mathematicians as Gauss and Polya. The problem has also found applications in fields such as algorithm design and circuit development.Despite much study, until recently very little was known regarding the asymptotics of Q(n). We apply modern methods from probabilistic combinatorics to reduce understanding Q(n) to the study of a particular infinite-dimensional convex optimization problem. The chief implication is that (in an appropriate sense) for a~1.94, Q(n) is approximately (ne^(-a))^n. Furthermore, our methods allow us to study the typical “shape” of n-queens configurations. |

09/30/2021 | Itamar Shamir, CMSA | Title: Gravity, geometry and entanglementAbstract: I will review some of the problems and central ideas regarding the quantum nature of gravity. The talk is intended for non-specialists and as much as possible assumes no prier knowledge. |

10/07/2021 | Nicholas J. Magazine Louisiana State University | Title: SiRNA Targeting TCRb: A Proposed Therapy for the Treatment of AutoimmunityAbstract: As of 2018, the United States National Institutes of Health estimate that over half a billion people worldwide are affected by autoimmune disorders. Though these conditions are prevalent, treatment options remain relatively poor, relying primarily on various forms of immunosuppression which carry potentially severe side effects and often lose effectiveness overtime. Given this, new forms of therapy are needed. To this end, we have developed methods for the creation of small-interfering RNA (siRNA) for hypervariable regions of the T-cell receptor β-chain gene (TCRb) as a highly targeted, novel means of therapy for the treatment of autoimmune disorders.This talk will review the general mechanism by which autoimmune diseases occur and discuss the pros and cons of conventional pharmaceutical therapies as they pertain to autoimmune disease treatment. I will then examine the rational and design methodology for the proposed siRNA therapy and how it contrasts with contemporary methods for the treatment of these conditions. Additionally, the talk will compare the efficacy of multiple design strategies for such molecules by comparison over several metrics and discuss how this will be guiding future research. |

10/14/2021 | Ian Gemp DeepMind | Title: D3C: Reducing the Price of Anarchy in Multi-Agent LearningAbstract: In multi-agent systems the complex interaction of fixed incentives can lead agents to outcomes that are poor (inefficient) not only for the group but also for each individual agent. Price of anarchy is a technical game theoretic definition introduced to quantify the inefficiency arising in these scenarios– it compares the welfare that can be achieved through perfect coordination against that achieved by self-interested agents at a Nash equilibrium. We derive a differentiable upper bound on a price of anarchy that agents can cheaply estimate during learning. Equipped with this estimator agents can adjust their incentives in a way that improves the efficiency incurred at a Nash equilibrium. Agents adjust their incentives by learning to mix their reward (equiv. negative loss) with that of other agents by following the gradient of our derived upper bound. We refer to this approach as D3C. In the case where agent incentives are differentiable D3C resembles the celebrated Win-Stay Lose-Shift strategy from behavioral game theory thereby establishing a connection between the global goal of maximum welfare and an established agent-centric learning rule. In the non-differentiable setting as is common in multiagent reinforcement learning we show the upper bound can be reduced via evolutionary strategies until a compromise is reached in a distributed fashion. We demonstrate that D3C improves outcomes for each agent and the group as a whole on several social dilemmas including a traffic network exhibiting Braess’s paradox a prisoner’s dilemma and several reinforcement learning domains. |

10/21/2021 | Guillarmou Colin CNRS/Univ. Paris Saclay | Title: Mathematical resolution of the Liouville conformal field theory.Abstract: The Liouville conformal field theory is a well-known beautiful quantum field theory in physics describing random surfaces. Only recently a mathematical approach based on a well-defined path integral to this theory has been proposed using probability by David, Kupiainen, Rhodes, Vargas. Many works since the 80’s in theoretical physics (starting with Belavin-Polyakov-Zamolodchikov) tell us that conformal field theories in dimension 2 are in general « Integrable », the correlations functions are solutions of PDEs and can in principle be computed explicitely by using algebraic tools (vertex operator algebras, representations of Virasoro algebras, the theory of conformal blocks). However, for Liouville Theory this was not done at the mathematical level by algebraic methods. I’ll explain how to combine probabilistic, analytic and geometric tools to give explicit (although complicated) expressions for all the correlation functions on all Riemann surfaces in terms of certain holomorphic functions of the moduli parameters called conformal blocks, and of the structure constant (3-point function on the sphere). This gives a concrete mathematical proof of the so-called conformal bootstrap and of Segal’s gluing axioms for this CFT. The idea is to break the path integral on a closed surface into path integrals on pairs of pants and reduce all correlation functions to the 3-point correlation function on the Riemann sphere $S^2$. This amounts in particular to prove a spectral resolution of a certain operator acting on $L^2(H^{-s}(S^1))$ where $H^{-s}(S^1)$ is the Sobolev space of order -s<0 equipped with a Gaussian measure, which is viewed as the space of fields, and to construct a certain representation of the Virasoro algebra into unbounded operators acting on this Hilbert space. This is joint work with A. Kupiainen, R. Rhodes and V. Vargas. |

10/28/2021 | Jie Yang Delft University of Technology | Title: ARCH: Know What Your Machine Doesn’t KnowAbstract: Despite their impressive performance, machine learning systems remain prohibitively unreliable in safety-, trust-, and ethically sensitive domains. Recent discussions in different sub-fields of AI have reached the consensus of knowledge need in machine learning; few discussions have touched upon the diagnosis of what knowledge is needed. In this talk, I will present our ongoing work on ARCH, a knowledge-driven, human-centered, and reasoning-based tool, for diagnosing the unknowns of a machine learning system. ARCH leverages human intelligence to create domain knowledge required for a given task and to describe the internal behavior of a machine learning system; it infers the missing or incorrect knowledge of the system with the built-in probabilistic, abductive reasoning engine. ARCH is a generic tool that can be applied to machine learning in different contexts. In the talk, I will present several applications in which ARCH is currently being developed and tested, including health, finance, and smart buildings. |

11/04/2021 | Qifeng Chen The Hong Kong University of Science and Technology | Title: Exploring Invertibility in Image Processing and RestorationAbstract: Today’s smartphones have enabled numerous stunning visual effects from denoising to beautification, and we can share high-quality JPEG images easily on the internet, but it is still valuable for photographers and researchers to keep the original raw camera data for further post-processing (e.g., retouching) and analysis. However, the huge size of raw data hinders its popularity in practice, so can we almost perfectly restore the raw data from a compressed RGB image and thus avoid storing any raw data? This question leads us to design an invertible image signal processing pipeline. Then we further explore invertibility in other image processing and restoration tasks, including image compression, reversible image conversion (e.g., image-to-video conversion), and embedding novel views in a single JPEG image. We demonstrate that customized invertible neural networks are highly effective in these inherently non-invertible tasks. |

11/11/2021 | Lvzhou Chen University of Texas at Austin | Title: The Kervaire conjecture and the minimal complexity of surfaces Abstract: We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture. The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<<w>> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surfaces in HNN extensions. This gives a conceptually simple proof of Klyachko’s theorem that confirms the Kervaire conjecture for any G torsion-free. I will also explain new results obtained using this approach. |

11/18/2021 | Matteo Parisi CMSA/IAS | |

12/02/2021 | Ryosuke Takahashi National Cheng Kung University | |

12/09/2021 | Mike Douglas Stony Brook University | |

12/16/2021 | Boyu Zhang Princeton University | Title: The smooth closing lemma for area-preserving surface diffeomorphisms Abstract: In this talk, I will introduce the smooth closing lemma for area-preserving diffeomorphisms on surfaces. The proof is based on a Weyl formula for PFH spectral invariants and a non-vanishing result of twisted Seiberg-Witten Floer homology. This is joint work with Dan Cristofaro-Gardiner and Rohil Prasad. |

01/06/2022 | An Huang Brandeis University | |

01/13/2022 | Francis Lazarus CNRS / Grenoble University | Title: A universal triangulation for flat toriAbstract: A celebrated theorem of Nash completed by Kuiper implies that every smooth Riemannian surface has a C¹ isometric embedding in the Euclidean 3-space E³. An analogous result, due to Burago and Zalgaller, states that every polyhedral surface, obtained by gluing Euclidean triangles, has an isometric PL embedding in E³. In particular, this provides PL isometric embeddings for every flat torus (a quotient of E² by a rank 2 lattice). However, the proof of Burago and Zalgaller is partially constructive, relying on the Nash-Kuiper theorem. In practice, it produces PL embeddings with a huge number of vertices, moreover distinct for every flat torus. Based on a construction of Zalgaller and on recent works by Arnoux et al. we exhibit a universal triangulation with less than 10.000 vertices, admitting for any flat torus an isometric embedding that is linear on each triangle. Based on joint work with Florent Tallerie. |

02/03/2022 | Fabian Gundlach Harvard University | |

03/17/2022 | Kenji Kawaguchi National University of Singapore |

The CMSA Interdisciplinary Science Seminar, organized by Yingying Wu, will take place on **Thursdays from 9:00 – 10:00am ET.** This seminar concentrates on geometric analysis, algorithms, and mathematical biology with an emphasis on genetics. The seminar is dedicated to applications of mathematics and computer science to life science and medicine. We hope the seminar will serve the role of facilitating collaborations between mathematicians, physicists, and computer scientists with domain experts in biology and medicine.

Date | Speaker | Title/Abstract |
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3/18/2021 | Omri Ben-Eliezer (CMSA) | Title: Adversarially robust streaming algorithmsAbstract: Streaming algorithms are an important class of algorithms designed for analyzing and summarizing large-scale datasets. In this context, the goal is usually to obtain algorithms whose space complexity (or memory consumption) is as small as possible, making them convenient to use on a single machine.Traditionally, streaming algorithms have been analyzed in the static setting, where the stream of incoming data is fixed in advance and does not depend on the algorithm’s outputs. This, however, is unrealistic in many situations. In this talk, I will present and discuss adversarially robust streaming algorithms, whose output is correct with high probability even when the stream updates are adaptively chosen as a function of previous outputs. This regime has received surprisingly little attention until recently, and many intriguing problems are still open. I will mention some of the recent results, discussing algorithms that are well-suited for the adversarially robust regime (random sampling), algorithms that are not robust (linear sketching), and efficient techniques to turn algorithms that work in the standard static setting into robust streaming algorithms. The results demonstrate strong connections between the streaming context and various other areas in computer science, combinatorics and statistics. Based on joint works with Noga Alon, Yuval Dagan, Rajesh Jayaram, Shay Moran, Moni Naor, David Woodruff, and Eylon Yogev. |

3/25/2021 | Cliff Taubes (Department of Mathematics, Harvard University) | Title: Introduction to 4-dimensional differential topology.Abstract: Differential topology is the study of smooth manifolds. I hope to tell you where the frontier lies between knowledge and ignorance with regards to smooth 4-dimensional manifolds (which is by far the hardest dimension to understand). |

4/1/2021 | Canceled – Frontiers in Applied Mathematics and Computation | |

4/8/202112:00pm ET | Enno Keßler | Title: Supergeometry and Super Riemann Surfaces of Genus ZeroAbstract: Supergeometry is a mathematical theory of geometric spaces with anti-commuting coordinates and functions which is motivated by the concept of supersymmetry from theoretical physics. I will explain the functorial approach to supermanifolds by Molotkov and Sachse. Super Riemann surfaces are an interesting supergeometric generalization of Riemann surfaces. I will present a differential geometric approach to their classification in the case of genus zero and with Neveu-Schwarz punctures. |

4/15/2021 | Cheng Yu (Department of Mathematics, University of Florida) | Title: Weak solutions to the isentropic system of gas dynamics Abstract: In this talk, I will discuss the global weak solutions to the isentropic system of gas dynamics: existence and non-uniqueness. In the first part, we generalized the renormalized techniques introduced by DiPerna-Lions to build up the global weak solutions to the compressible Navier-Stokes equations with degenerate viscosities. This existence result holds for any $\gamma>1$ in any dimensional spaces for the large initial data. In the second part, we proved that for any initial data belonging to a dense subset of the energy space, there exists infinitely many global weak solutions to the isentropic Euler equations for any $1<\gamma\leq 1+2/n$. Our result is based on a generalization of convex integration techniques by De Lellis-Szekelyhidi and weak vanishing viscosity limit of the Navier-Stokes equations. The first part is based on the joint works with D. Bresch and A. Vasseur, and the second one is based on our recent joint work with R. M Chen and A. Vasseur. |

4/22/2021 | Matt Novack (New York University) | Title: Convex Integration and Fluid TurbulenceAbstract: The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy even in the vanishing viscosity limit, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. These methods originated in the works of Nash and Gromov and were adapted to the context of fluid equations by De Lellis and Szekelyhidi Jr. In this talk, we will survey the history of both phenomenological theories of turbulence and convex integration. Finally, we discuss recent joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from Kolmogorov’s predictions. |

4/29/2021 | Yijing Wu (Department of Mathematics, University of Maryland, College Park) | Title: An isoperimetric problem with a competing nonlocal singular termAbstract: We are interested in the minimization problem of a functional in which the perimeter is competing with a nonlocal singular term comparable to a fractional perimeter, with volume constraint. We prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or develop fingers of a prescribed width. We connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality. |

5/6/2021 | Aaron Fenyes (Institut des Hautes Études Scientifiques) | Title: Visualizing neutral theoryAbstract: In this expository talk, I’ll use 1d voter models to illustrate basic features of neutral theory—a vision of how genetic and ecological diversity can emerge even without selective pressure. We’ll see how questions about the persistence and spatial organization of lineages can be rephrased, in these models, as questions about random walks. |

5/13/2021 | Jialin Zhang (Institute of Computing Technology, Chinese Academy of Science) | Title: A Tight Deterministic Algorithm for the Submodular Multiple Knapsack ProblemAbstract: Submodular function maximization has been a central topic in the theoretical computer science community over the last decade. Plenty of well-performing approximation algorithms have been designed for the maximization of (monotone or non-monotone) submodular functions over a variety of constraints. In this talk, we consider the submodular multiple knapsack problem (SMKP), which is the submodular version of the well-studied multiple knapsack problem (MKP). Roughly speaking, the problem asks to maximize a monotone submodular function over multiple bins (knapsacks). Recently, Fairstein et al. (ESA20) presented a tight (1−1/e−ϵ)-approximation randomized algorithm for SMKP. Their algorithm is based on the continuous greedy technique which inherently involves randomness. However, the deterministic algorithm of this problem has not been understood very well previously. In this paper, we present a tight (1−1/e−ϵ) deterministic algorithm for SMKP. Our algorithm is based on reducing SMKP to an exponential-size submodular maximizaion problem over a special partition matroid which enjoys a tight deterministic algorithm. We develop several techniques to mimic the algorithm, leading to a tight deterministic approximation for SMKP. |

5/20/2021 | Shang Su (Department of Cancer Biology, The University of Toledo) | Title: In silico design and evaluation of PROTAC-based protein degrader–Introductory case studiesAbstract: Proteolysis-targeting chimeras (PROTACs) are heterobifunctional small molecules consisting of two chemical moieties connected by a linker. The simultaneous binding of a PROTAC to both a target protein and an E3 ligase facilitates ubiquitination and degradation of the target protein. Since its proof-of-concept research in 2001, PROTAC has been vigorously developed by both research community and pharma industry, to act against therapeutically significant proteins, such as BRD4, BTK, and STAT3. However, despite the enthusiasm, designing PROTACs is challenging. Till now, no case of de novo rational design of PROTACs has been reported and the successful PROTACs usually came from the functional screen from a limitedly scaled library. As formation of a ternary complex between the protein target, the PROTAC, and the recruited E3 ligase is considered paramount for successful degradation, several computational algorithms (PRosettaC as the example), have been developed to model this ternary complex, which have got partial agreement with the experimental data and in principle inform future rational PROTAC design. Here I will introduce some of these computational methods and share how they model the ternary complexes. |

5/27/2021 | Ying Hsang Liu & Moritz Spiller (University of Southern Denmark & Otto von Guericke University Magdeburg) | Title: Predicting Visual Search Task Success from Eye Gaze Data for User-Adaptive Information Visualization Systems Abstract: Information visualizations are an efficient means to support the users in understanding large amounts of complex, interconnected data; user comprehension. Previous research suggests that user-adaptive information visualizations positively impact the users’ performance in visualization tasks. This study aims to develop a computational model to predict the users’ success in visual search tasks from eye gaze data and thereby drive such user-adaptive systems. State-of-the-art deep learning models for time series classification have been trained on sequential eye gaze data obtained from 40 study participants’ interaction with a circular and an organizational graph. The results suggest that such models yield higher accuracy than a baseline classifier and previously used models for this purpose. In particular, a Multivariate Long Short Term Memory Fully Convolutional Network (MLSTM-FCN) shows encouraging performance for its use in on-line user-adaptive systems. Given this finding, such a computational model can infer the users’ need for support during interaction with a graph and trigger appropriate interventions in user-adaptive information visualization systems |

6/3/2021 | Joaquim I. Goes (Lamont Doherty Earth Observatory at Columbia University) | Title: Navigating Seas of Change – the Role and Significance of Cross-Disciplinary ResearchAbstract: As atmospheric CO2 levels continue to rise and global and coastal ocean become warmer and more eutrophic as a result of human activities, we need better ways to detect and understand how marine ecosystems are responding to these changes. Until recently, most biological oceanographers relied on shipboard measurements that were limited in their coverage and inadequate to investigate changes at large spatial and temporal scales. With the advent of satellites, autonomous platforms and numerical methods, biological oceanographers are turning to empirical and semi-analytical algorithms to scale limited shipboard measurements from local scales to regional, basin and global scales. While progress has been interdisciplinary research involving collaborations between biological, physical and methodical scientists could help us make rapid advances and mitigate impacts on the livelihoods of coastal communities which are at greatest risk. This presentation will cover a case study from the Arabian Sea in the Indian Ocean and describe the promise and potential of inter-disciplinary research in advancing climate change and ecosystem research for societal benefit. |

6/10/2021 | Tianqi Wu (CMSA) | Title: The deformation space of geodesic triangulations and Tutte’s embeddingAbstract: In 1984, Bloch, Connelly, and Henderson proved that the space of geodesic triangulations of a convex polygon is contractible. It was found that Tutte’s embedding theorem could give a very simple proof to Bloch-Connelly-Henderson’s theorem, and provides an elegant algorithm for image morphing on convex polygons. We recently generalize Tutte’s embedding theorem, and prove that the deformation space of geodesic triangulations of a closed Riemannian surface of negative curvature is contractible. This confirms a conjecture by Connelly, Henderson, Ho, Starbird in 1983, and also indicates a method for image morphing on closed surfaces. |

6/17/2021 | Fang Xie (BIDMC and HMS, Harvard University) | Title: Molecular mechanisms of Taxane resistance in prostate cancerAbstract: Taxanes act by stabilizing microtubules (MTs) and prolong survival in men with prostate cancer (PCa), but biomarkers predictive of responses and clinically actionable mechanism(s) of resistance have yet to be identified. We recently reported that a decrease or absence of MT bundling, despite high levels of intratumoral taxanes, is a basis and a potential pharmacodynamic biomarker of taxane resistance. To determine the molecular basis for this impaired MT bundling, we treated docetaxel sensitive PCa models in vivo for multiple cycles until resistance, and found upregulation of FOXJ1 (a master transcription factor regulating tubulin associated proteins), as well as one of its downstream effector protein, TPPP3, in the resistant tumors. Moreover, mining of patient databases showed that amplification of the FOXJ1 gene is also associated with taxane exposure. Together these data implicate the FOXJ1-TPPP3 regulatory network in taxane resistance. In parallel with these in vivo studies, we have carried in vitro drug screens for agents that enhance responses to docetaxel in 3D/organoid culture. A prominent agent that emerged is a histone methyltransferase inhibitor. Our overall goals are to identify clinically meaningful mechanisms of taxane resistance, to develop therapeutic combinations that enhance efficacy and/or target these resistance mechanisms, and to identify biomarkers indicative of specific mechanisms. Our hypotheses, which are based on our published and preliminary data, are that one major mechanism for taxane resistance is upregulation of the FOXJ1-TPPP3 pathway, and that combination therapies can be developed that enhance taxane efficacy and delay or prevent the emergence of resistance. The specific aims are 1) Determine the effect of FOXJ1-TPPP3 regulatory network on microtubule dynamics, stability and target-engagement by taxanes in vitro and in vivo and 2) Identify effective combination therapies to enhance docetaxel responses and overcome taxane resistance. |

6/24/2021 | Mike Novack (University of Texas at Austin) | Title: 3D Smectic Liquid CrystalsAbstract: Liquid crystals are an intermediate state of matter which flow like liquids but retain molecular ordering similar to that of crystals. Their physical properties make them ideal for a wide range of technological applications. The molecules in smectic liquid crystals form well-defined layers which slide across one another. In this talk we will discuss a model for smectics from the physics literature based on the minimization of a suitable energy and present recent results obtained jointly with Xiaodong Yan. No prior knowledge of liquid crystals or the relevant mathematics will be assumed. |

7/01/2021 | Zheng Shi (Rutgers University) | Title: Mechanics of biomolecular assemblies Abstract: The mechanical properties of biomolecular assemblies play pivotal roles in many biological and pathological processes. In this talk, I’ll focus on two different self-assembled structures in cells: 1) the plasma membrane, which defines the boundary of a cell; and 2) protein condensates, which arise from liquid-liquid phase separation (LLPS) inside cells. In the first part, I’ll discuss recent findings on how cell membranes respond to local mechanical perturbations. In most non-motile cells, local perturbations to membrane tension remain highly localized, leading to subcellular Ca2+ influx and vesicle fusion events. Membrane-cortex attachments are responsible for impeding the propagation of membrane tension. Exception to this rule can be found in the axon of neurons, where a rapid propagation of membrane tension coordinates the growth and branching of the axon. In the second part, I’ll discuss the development of quantitative techniques to measure the surface tension and viscosity of liquid protein condensates. Our results highlight a common misconception about LLPS in biology: ‘oil droplets in water’ is often used to give an intuition about protein condensates in cells. However, oil droplets and protein condensates represent two extremes in the realm of liquid properties. The unique properties of protein condensates have important implications in achieving molecular and functional understanding of LLPS. |

7/08/2021 | Arun Debray (University of Texas at Austin) | Title: Modeling invertible topological phases of matter using homotopy theoryAbstract: Condensed-matter theorists have discovered examples of physical systems with unusual behavior, such as pointlike excitations that behave neither as bosons nor as fermions, leading to the idea of topological phases of matter. Classifying the possible topological phases has been the focus of a lot of research in the last decade in condensed-matter theory and nearby areas of mathematics. In this talk, I’ll focus primarily on the special case of invertible phases, also called symmetry-protected topological (SPT) phases, whose classification uses techniques from homotopy theory. I will discuss two different approaches to this, due to Kitaev and Freed-Hopkins, followed by details of the homotopy-theoretic classifications. The latter includes work of Freed-Hopkins and of myself. |

7/15/2021 | Daniel Kaplan (University of Birmingham) | Title: Representations of quivers and the Deligne-Simpson problemAbstract: A quiver is a directed graph and a representation of a quiver is an assignment of a vector space to each vertex and a linear transformation to each arrow. Many problems in linear algebra can be rephrased in terms of representation theory of quivers. I will highlight one such instance: Crawley-Boevey’s solution to the (additive) Deligne-Simpson problem. Following work of Nakajima, geometers have gained milage by realizing spaces as the space of certain representations of quivers (with relations). For instance, this gives a candidate resolution of singularities using the theory of variation of GIT. Time permitting, I will explain these developments. |

7/22/2021 | Yinbang Lin (Tongji University) | Title: Moduli spaces of stable pairs on algebraic surfacesAbstract: As a variant of Grothendieck’s Quot schemes, we introduce the moduli space of limit stable pairs. We show an example over a smooth projective algebraic surface where there is a virtual fundamental class. We are able to describe this class explicitly. We will also show an application towards moduli of sheaves. |

7/29/2021 | Jeffrey Kuan (Texas A&M University) | Title: Joint moments of multi–species $q$–Boson.Abstract: The Airy_2 process is a universal distribution which describes fluctuations in models in the Kardar–Parisi–Zhang (KPZ) universality class, such as the asymmetric simple exclusion process (ASEP) and the Gaussian Unitary Ensemble (GUE). Despite its ubiquity, there are no proven results for analogous fluctuations of multi–species models. Here, we will discuss one model in the KPZ universality class, the $q$–Boson. We will show that the joint multi–point fluctuations of the single–species $q$–Boson match the single–point fluctuations of the multi–species $q$–Boson. Therefore the single–point fluctuations of multi–species models in the KPZ class ought to be the Airy_2 process. The proof utilizes the underlying algebraic structure of the multi–species $q$–Boson, namely the quantum group symmetry and Coxeter group actions. |

8/5/2021 | Weishan Huang (Louisiana State University and Cornell University) | Title: Designer DNA-based nanoconstructs in viral detection and blockingAbstract: SARS-CoV-2, etiological pathogen of COVID-19, has resulted in a pandemic. There remains an urgent need of innovative technology of developing rapid diagnosis of active infections and affordable antiviral precise medicine for therapeutics. Most viruses have repetitive surface antigen units laid out on the virions following specific patterns, forming the viral capsid or envelop. To develop precise instant diagnosis of active SARS-CoV-2 infections and novel antiviral candidates against SARS-CoV-2 infection and transmission, we exploited the structural characteristics of viral surface proteins that can be matched at nanoscale precision by engineered DNA nanostructure platforms. Our preliminary data demonstrated that these pattern-matching DNA nanostructures can enable specific and sensitive sensing of SARS-CoV-2 viruses and have sufficient antiviral activities against SARS-CoV-2 pseudoviral and live viral infections. Our method can be transferrable to develop rapid diagnosis and precise inhibition of other enveloped viruses such as influenza and HIV. We are seeking expert advice from the mathematical and computational community to help with optimization of the DNA-based nanostructures. |

8/12/2021 | Qingtao Chen (New York University Abu Dhabi) | Title: Recent Progress on Volume Conjectures of links as well as 3-manifoldsAbstract: The original Volume Conjecture of Kashaev-Murakami-Murakami predicts a precise relation between the asymptotics of the colored Jones polynomials of a knot in S^3 and the hyperbolic volume of its complement. I will discuss two different directions that lead to generalizations of this conjecture. The first direction concerns different quantum invariants of knots, arising from the colored SU(n) (with the colored Jones polynomial corresponding to the case n=2). I will first display subtle relations between congruence relations, cyclotomic expansions and the original Volume Conjecture for the colored Jones polynomials of knots. I will then generalize this point of view to the colored SU(n) invariant of knots. Certain congruence relations for the colored SU(n) invariants, discovered in joint work with K. Liu, P. Peng and S. Zhu, lead us to formulate cyclotomic expansions and a Volume Conjecture for these colored SU(n) invariants. If time permits, I will briefly discuss similar ideas for the Superpolynomials that arise in HOMFLY-PT homology. Another direction for generalization involves the Witten-Reshetikhin-Turaev and the (modified) Turaev-Viro quantum invariants of 3-manifolds. In a joint work with T. Yang, I formulated a Volume Conjecture for the asymptotics of these 3-manifolds invariants evaluated at certain roots of unity, and numerically checked it for many examples. Interestingly, this conjecture uses roots of unity that are different from the one usually considered in literature. These 3-manifolds invariants are only polynomially large at the usual root of unity as the level of the representation approaches infinity, which is predicted by Witten’s Asymptotic Expansion Conjecture. True understanding of this new phenomenon requires new physical and geometric interpretations that go beyond the usual quantum Chern-Simons theory. Currently these new Volume Conjectures have been proved for many examples by various groups. However, like the original Volume Conjecture, a complete proof for general cases is still an open problem in this area. In a recent joint work with J. Murakami, I proved the asymptotic behavior of the quantum 6j-symbol evaluated at the unusual root of unity, which could explain the Volume Conjectures for the asymptotics of the Turaev-Viro invariants in general. |

8/19/2021 | Sean Carney (UCLA) | Title: Fluctuating hydrodynamics model for ionic liquidsAbstract: An ionic liquid is a liquid salt with dissociated cations and anions such as molten NaCl. Of particular interest are ionic liquids composed of complex hydrocarbons that are high-viscosity liquids at room temperature. These room temperature ionic liquids (RTILs) can exhibit intriguing physical properties useful in applications such as super-capacitors, batteries, photoelectrochemical cells, and microelectromechanical lubricants. RTIL dynamics are governed by the thermodynamic competition between electrostatic forces, interfacial tension, and short range cation-anion repulsion due to the ions’ complex molecular structures. This competition is quantified by a Gibbs free energy functional from which we derive a PDE model of RTILs. The inviscid hydrodynamics and electrostatics can be derived from the calculus of variations, while the evolution equation for ion concentration and the viscous hydrodynamic force density are derived from the Onsager reciprocal relations of nonequilibrium thermodynamics. The PDEs are further augmented with white noise flux terms modeling thermal fluctuations intrinsic to fluid mixtures at mesoscopic length scales, and a low Mach number formulation is utilized to analytically remove sound waves from the equations of motion based on the assumption that they are relatively unimportant to the system dynamics.After calibrating the strength of the short range repulsive term based on a stability analysis of the concentration equation, we numerically demonstrate the formation of mesoscopic structuring at equilibrium in two and three dimensions. In simulations with electrode boundaries the measured double layer capacitance decreases with voltage, in agreement with theoretical predictions and experimental measurements for RTILs. Finally, we present a shear electroosmosis example to demonstrate that the methodology can be used to model electrokinetic flows. |