Mathematics
http://events.berkeley.edu/index.php/calendar/sn/math.html
Upcoming EventsHarmonic Analysis Seminar, May 1
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125533&date=2019-05-01
A short proof of the multilinear Kakeya inequality of Bennett-Carbery-Tao will be presented. This proof (due to Guth, 2015) is based on induction on scales and the Loomis-Whitney inequality, without the nonlinear heat flow of the original proof. In a future lecture, this result will serve as an ingredient in the proof of the multilinear restriction and decoupling inequalities.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125533&date=2019-05-01Topology Seminar (Introductory Talk), May 1
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125566&date=2019-05-01
I will start by motivating cobordism categories by recalling the notion of topological field theories. Then I will explain why “higher” categories appear naturally in this context (and what they are).http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125566&date=2019-05-01Rapidly mixing random walks on matroids and related objectsidly mixing random walks on matroids and related objects, May 1
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125543&date=2019-05-01
A central question in randomized algorithm design is what kind of distributions can we sample from efficiently? On the continuous side, uniform distributions over convex sets and more generally log-concave distributions constitute the main tractable class. We will build a parallel theory on the discrete side, that yields tractability for a large class of discrete distributions. We will use this theory to resolve a 30-year-old conjecture of Mihail and Vazirani that matroid polytopes have edge expansion at least 1. We will also obtain simple nearly-linear time algorithms for sampling from spanning trees of a graph, and easy-to-implement algorithms for volume-based sampling.<br />
The hammer enabling these algorithmic advances is the introduction and the study of a class of polynomials, that we call completely log-concave. We can use very simple and easy-to-implement random walks to perform the task of sampling, and we will use completely log-concave polynomials to analyze the random walk. Based on joint work with Kuikui Liu, Shayan Oveis Gharan, Cynthia Vinzant.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125543&date=2019-05-01Number Theory Seminar, May 1
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125473&date=2019-05-01
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125473&date=2019-05-01Topology Seminar (Main Talk), May 1
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125544&date=2019-05-01
Lurie’s approach to the Cobordism Hypothesis builds upon a suitable higher category of cobordisms. The model of \((\infty,1)\)-categories given by complete Segal spaces (and their higher analogs) are a very natural choice for constructing cobordism categories. A drawback is that the first natural definitions only give Segal spaces, which, for high dimensions, are not complete. This follows directly from the \(s\)-cobordism theorem. In this talk, after explaining and defining the necessary notions in detail, I will explain a very simple model of cobordisms, which is a completion of the usual one. In particular it indeed is complete. This is joint work with Ulrike Tillmann.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125544&date=2019-05-01Center for Computational Biology Seminar, May 1
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120953&date=2019-05-01
Leveraging linkage disequilibrium to identify adaptive and disease-causing mutations<br />
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Abstract: <br />
Correlation among genotypes in human population-genetic datasets complicates the localization of both adaptive mutations and disease-causing mutations. I will describe our latest efforts to develop new methods for localizing adaptive and disease-causing mutations, motivated by (1) incorporating summary statistics at various genomic scales into selection scans, (2) bridging the gap between polygenic and omnigenic complex traits, and (3) testing for differential genetic architecture for the same trait across ancestries.<br />
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Bio: <br />
Sohini Ramachandran is Associate Professor of Ecology and Evolutionary Biology and Director of Brown University's Center for Computational Molecular Biology. During Spring 2019, she is also a Fellow in the Natural Sciences Programme at the Swedish Collegium for Advanced Study in Uppsala, Sweden. Prior to beginning her faculty appointment at Brown University in 2010, Sohini spent 3 years as a Junior Fellow at the Harvard Society of Fellows and postdoctoral fellow in Professor John Wakeley’s group at the Harvard University Department of Organismic and Evolutionary Biology. She completed her PhD in 2007 with Marcus Feldman at Stanford University’s Department of Biological Sciences. Sohini's research has been funded by the National Science Foundation and National Institutes of Health, and she was named a Pew Scholar in the Biomedical Sciences and an Alfred P. Sloan Research Fellow.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120953&date=2019-05-01Mathematics Department Colloquium, May 2
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125569&date=2019-05-02
I shall cover some well-known facts about hydrodynamic turbulence, and present a physically coherent view of intermittency in the energy cascade as a cascade of eddies governed by ideas of statistical mechanics. The approach presented is close to the ideas of Kolmogorov but gives a satisfactory estimate of the intermittency exponents and of the Reynolds number at the onset of turbulence. I shall also indicate why the Kolmogorov-Obukhov does not work.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125569&date=2019-05-02Gammage Seminar, May 3
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125624&date=2019-05-03
This reports on joint work with Umut Varolgunes. In 2007, Entov and Polterovich introduced the notion of heavy and superheavy subsets of symplectic manifolds. We define and study a Floer-theoretic analogue of this notion.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125624&date=2019-05-03'Information and Uncertainty in Data Science' Discussion Forum, May 3
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=124356&date=2019-05-03
Full details about this meeting will be posted here: http://compdatascience.org/entropy.<br />
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The 'Information and Uncertainty in Data Science' Discussion Forum is a forum for open inquiry and discussion about a wide range of recurring data science fundamentals, including information, uncertainty, entropy, bits, probability, machine learning, generalization, and others. The group facilitates academic discourse on the practical use of the fundamental concepts across a wide variety of research disciplines, and strives for clarity and understanding using real-world scenarios, visual examples, cutting edge questions and unique perspectives. This group focusses on understanding and sharing concepts that are often buried in mathematical language, especially entropy, reduction of uncertainty and connections between physical systems and information systems. All interested members of the UC Berkeley, UCSF, LBL and LLNL communities are welcome and encouraged to attend. More details available at http://compdatascience.org/entropy. Contact: BIDS Senior Fellow Gerald Friedland.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=124356&date=2019-05-03Dissertation Talk: Approximate counting, phase transitions and geometry of polynomials, May 3
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125497&date=2019-05-03
In classical statistical physics, a phase transition is understood by studying the geometry (the zero-set) of an associated polynomial (the partition function). In this talk I will show that one can exploit this notion of phase transitions algorithmically, and conversely exploit the analysis of algorithms to understand phase transitions. As applications, I will give efficient deterministic approximation algorithms (FPTAS) for counting q-colorings, and for computing the partition function of the Ising model.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125497&date=2019-05-03Arithmetic Geometry and Number Theory RTG Seminar, May 3
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125574&date=2019-05-03
Given a curve of genus at least $2$ over a number field, Faltings' theorem tells us that its set of rational points is finite. Provably computing the set of rational points remains a major open problem, as does the question of whether the number of rational points can be uniformly bounded. We will survey some recent progress and ongoing work using the Chabauty–Kim method, which uses the fundamental group to construct $p$-adic analytic functions that vanish on the set of rational points. In particular, we present a new proof of Faltings' theorem for superelliptic curves over the rational numbers (due to joint work with Jordan Ellenberg), and a conditional generalization of the Chabauty–Kim method to number fields and higher dimensions.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125574&date=2019-05-03Student 3-Manifold Seminar, May 3
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125626&date=2019-05-03
In this talk, I will try to go through all the ways I am aware of that one can define the Alexander polynomial. To name a few: Alexander's original definition, the Fox calculus, the Conway potential function, polynomial extrapolation of $U_q(\mathfrak {sl}(n))$ quantum invariants, Reidemeister torsion, the Burau representation of braid groups, Alexander representations of the knot quandle, and knot Floer homology.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125626&date=2019-05-03Combinatorics Seminar, May 6
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125468&date=2019-05-06
Both LLT polynomials and k-Schur functions were derived from the study of Macdonald polynomials, and have proved to be fruitful areas of study. A conjecture due to Haglund and Haiman states that k-bandwidth LLT polynomials expand positively into k-Schur functions. This is trivial in the case k=1 and has been recently proved for k=2. In this talk, I will present a proof for the case k=3. To this end, I will introduce a new computational method for establishing linear relations among LLT polynomials.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125468&date=2019-05-06String-Math Seminar, May 6
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125623&date=2019-05-06
It has been understood for some time now that many highlights of Lie theory, such as the representation-theoretic theory of special functions, or the Kazhdan–Lusztig theory, have a natural extension to a much broader setting, the boundaries of which are yet to be explored. In this extension, the focus is shifting from a group \(G\) to various classes of algebraic varieties that possess the key features of \(T^*G/B\). While there are some proposal about what should replace a Lie algebra, root systems, etc., it is less clear what should be the group, or multiplicative analog of these structures. Reflecting the nature of the field, the talk will combine a review of established partial results with unsubstantiated speculations.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125623&date=2019-05-06Northern California Symplectic Geometry Seminar, May 6
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125572&date=2019-05-06
Let $(M,\omega )$ be a closed symplectic manifold. Consider a closed symplectic submanifold $D$ whose homology class is a positive multiple of the Poincare dual of $[\omega ]$. The complement of $D$ can be given the structure of a Liouville manifold, with skeleton $S$. We prove that $S$ cannot be displaced from itself inside $M$ by a Hamiltonian isotopy if we assume that $c_1(M)=0$. Under the same assumption, we also prove that Floer theoretically essential Lagrangians in $M$ have to intersect $S$. These results are related to an understanding of the notions heavy and superheavy in terms of relative symplectic cohomology. Ongoing joint work with Dmitry Tonkonog.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125572&date=2019-05-06Differential Geometry Seminar, May 6
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125422&date=2019-05-06
A spherical surface with $n$ conical singularities is a surface $S$ with cone points $x_1, \dots ,x_n$ and a metric $g$, such that $g$ has curvature 1 on the complement $S \setminus (x_1,...,x_n)$ and has a conical singularity of angle $2\pi (\theta _i)$ at each $x_i$. Moduli spaces of spherical metrics with fixed angles are intriguing objects. Up to very recently the most basic questions about these spaces were open, in particular it was not known for which angles such spaces are non-empty, whether they can be disconnected, whether they project surjectively to the moduli space of curves with $n$ marked points. I'll speak about solutions of such questions, the talk is based on a joint work with Gabirele Mondello.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125422&date=2019-05-06Northern California Symplectic Geometry Seminar, May 6
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125573&date=2019-05-06
The discovery of the Jones polynomial in the early 80s was the beginning of "quantum topology": the introduction of various invariants which, in one sense or another, arise from quantum mechanics and quantum field theory. There are many mathematical constructions of these invariants, but they all share the defect of being first defined in terms of a knot diagram, and only subsequently shown by calculation to be independent of the presentation. As a consequence, the geometric meaning has been somewhat opaque.<br />
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By contrast, in the physics literature, there is a geometric story: Witten showed that the invariants can be extracted from a 3d quantum field theory, and he later showed that this quantum field theory can be found as a boundary condition in string theory. However, it has been difficult to translate these ideas into mathematics, because they a priori depend on infinite dimensional integrals which have no mathematically rigorous definition.<br />
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In the talk I will explain how just enough of the open topological string theory can be made mathematically precise so as to give a manifestly geometric interpretation of the skein relation: it is a boundary term which must be set to zero in order to invariantly count holomorphic curves with boundary. As a consequence one finds that the HOMFLY polynomial (a generalization of the Jones polynomial) is a count of holomorphic curves in a certain 6-dimensional setting which is invariantly and geometrically constructed from the three-dimensional topology.<br />
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This talk draws from the paper “Skeins on Branes” written with Tobias Ekholm.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125573&date=2019-05-06Harmonic Analysis Seminar, May 8
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125611&date=2019-05-08
Multilinear restriction estimates are an important tool in the proof of the decoupling inequality for the paraboloid. This talk will introduce and provide a heuristic proof of a multilinear restriction estimate, relying on the multilinear Kakeya inequality discussed in last week's talk. If time permits, attention will be given to applications to decoupling and multilinear decoupling (the latter being an ingredient in a proof of the former).http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125611&date=2019-05-08Topology Seminar (Introductory Talk), May 8
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125648&date=2019-05-08
Scissor congruence theory of polytopes is an old subject going back to 19th century. One of its first major achievements was appearance of so-called Dehn invariant. This mysterious invariant could be properly understood and generalized in the context of the theory of mixed Hodge structures of mixed Tate type. I will explain this relation and show some applications to hyperbolic geometry.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125648&date=2019-05-08BIDS Data Science Lecture: Hate speech, algorithms, and digital connectivity, May 8
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125241&date=2019-05-08
The Online Hate Index (OHI) is a research partnership between UC Berkeley’s D-Lab and Google Jigsaw that seeks to improve society's understanding of online hate speech (from sources such as YouTube, Reddit, Twitter and other social media sites), including its prevalence over time, variation across regions and demographics, our ability to measure it through crowdsourcing and algorithms, and how to influence it through historical or future interventions. Through a combination of citizen science and machine learning, the team is developing a nuanced measurement methodology that decomposes hate speech into various constituent components, enabling it to be transformed into a continuous “hate speech scale,” making it easier to rate, evaluate and understand than a single omnibus question (i.e. "is this comment hate speech?").<br />
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The project is setting new standards for the data science of hate speech, with goals to 1) establish a theoretically-grounded definition of hate speech inclusive of research/policies/practice, 2) develop and apply a multi-component labeling instrument, 3) create a new crowdsourcing tool to scalably label comments, 4) curate an open, reliable multi-platform labeled hate speech corpus, 5) grow existing data and tool repositories within principles of replicable and reproducible research, enabling greater transparency and collaboration, 6) create new knowledge through ethical online experimentation (and citizen science), and 7) refine AI models. The research team includes Geoff Bacon (Linguistics Ph.D. candidate); Nora Broege (Postdoc at Rutgers University); Chris Kennedy (Biostatistics Ph.D. student, BIDS Fellow); and Alexander Sahn (Political Science Ph.D. candidate).<br />
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Ultimately, we seek to understand the causal mechanisms for intervention and evaluation, while defending free speech. A new open-source platform - to be used by the Anti-Defamation League and other advocacy organizations - will make these resources (along with policy recommendations) available to educate the public and grow the larger data science / citizen science community.<br />
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BIDS Data Science Lectures are open to the entire campus community.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125241&date=2019-05-08Last Passage percolation: modulus of continuity and the slow bond problem, May 8
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125652&date=2019-05-08
The talk has two parts. In the first part we will speak on the modulus of <br />
continuity in Poissonian last passage percolation, a model lying in the <br />
KPZ universality class. In the second part we speak on the “slow bond” <br />
model, where Totally Asymmetric Simple Exclusion Process (TASEP) on <br />
$\mathbb{Z}$ (a model which can be thought to simulate a one-way traffic <br />
movement) is modified by adding a slow bond at the origin, that is, particles <br />
at the origin wait longer before making jumps. A conjectural description <br />
of properties of invariant measures of TASEP with a slow bond at the <br />
origin was provided in Liggett's 1999 book . We establish Liggett’s <br />
conjectures and in particular show that TASEP with a slow bond at the <br />
origin, starting from step initial condition, converges in law to an <br />
invariant measure that is asymptotically close to product measures with <br />
different densities far away from the origin towards left and right. Joint work with Alan Hammond, Allan Sly and Riddhipratim Basu.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125652&date=2019-05-08Arithmetic Geometry and Number Theory RTG Seminar, May 8
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125588&date=2019-05-08
Deligne's "Weil II" paper includes a far-reaching conjecture to the effect that for a smooth variety on a finite field of characteristic $p$, for any prime $\ell $ distinct from $p$, $\ell $-adic representations of the etale fundamental group do not occur in isolation: they always exist in compatible families that vary across $\ell $, including a somewhat more mysterious counterpart for $\ell =p$ (the "petit camarade cristallin"). We explain what such an object is; indicate the role of the Langlands correspondence for function fields in the approach to Deligne's conjecture; and report on prior and ongoing work towards the conjecture (including results of Deligne, Drinfeld, Abe-Esnault, and the speaker).http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125588&date=2019-05-08Topology Seminar (Main Talk), May 8
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125649&date=2019-05-08
I will explain how to construct a rational elliptic surface out of every non-Euclidean tetrahedra. This surface "remembers" the trigonometry of the tetrahedron: the length of edges, dihedral angles and the volume can be naturally computed in terms of the surface. The main property of this construction is self-duality: the surfaces obtained from the tetrahedron and its dual coincide. This leads to some unexpected relations between angles and edges of the tetrahedron. For instance, the cross-ratio of the exponents of the spherical angles coincides with the cross-ratio of the exponents of the perimeters of its faces. The construction is based on relating mixed Hodge structures, associated to the tetrahedron and the corresponding surface.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125649&date=2019-05-08Paris/Berkeley/Bonn/Zürich Analysis Seminar, May 9
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125625&date=2019-05-09
Multiphase mean curvature flow has, due to its importance in materials science, received a lot of attention over the last decades. In this talk, I will show how the gradient-flow structure allows to prove convergence results for several numerically relevant schemes, including phase-field models and thresholding schemes in codimensions one and two. The methods combine basic geometric measure theory, the theory of gradient flows in metric spaces, and multiscale analysis.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125625&date=2019-05-09Applied Math Seminar, May 9
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125570&date=2019-05-09
The facetious and self-serving title refers to four approaches for Navier-Stokes simulations. The first involves the analysis, numerical analysis, and an efficient implementation strategy for a recently proposed fractional Laplacian closure model that accounts for Richardson pair dispersion observed in turbulent flows. The second is the exploitation of accurate and widely applicable ensemble methods in settings in which multiple inputs need to be processed, as may be the case for uncertainty quantification, reduced-order modeling, and control and optimization. The third addresses the lack of regularity of solutions and the resultant loss of accuracy of approximations in the case of white or weakly correlated additive noise forcing. The fourth involves filtered spectral viscosity and hierarchical finite element methods for regularized Navier-Stokes equations.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125570&date=2019-05-09Special Seminar, May 9
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125671&date=2019-05-09
Carmine Emanuele Cella, assistant professor in music and technology at CNMAT, will present work done in the last years in searching good signal representations that permit high-level manipulation of musical concepts. After the definition of a geometric approach to signal representation, the theory of sound-types and its application to music will be presented. Finally, recent research on assisted orchestration will be shown and some possible musical applications will be proposed, with connections to deep learning methods.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125671&date=2019-05-09Applied Math Seminar, May 10
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125571&date=2019-05-10
We use the canonical examples of fractional Laplacian and peridynamics equations to discuss their use as models for nonlocal diffusion and mechanics, respectively, via integral equations with singular kernels. We then proceed to discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. In particular, we discuss the recently developed asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125571&date=2019-05-10Solving Hard Computational Problems using Oscillator Networks, May 10
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125647&date=2019-05-10
Over the last few years, there has been considerable interest in Ising machines, ie, analog hardware for solving difficult (NP hard/complete) computational problems effectively. <br />
We present a new way to make Ising machines using networks of coupled self-sustaining nonlinear oscillators. <br />
Our scheme is theoretically rooted in a novel result that connects the phase dynamics of coupled oscillator systems with the Ising Hamiltonian.<br />
We show that oscillators can be designed to take on a binary phase, and a network of such binary oscillators has phase dynamics evolving naturally towards local minima of the Ising Hamiltonian. <br />
Two simple additional steps (ie, adding noise, and tuning the binarization strength up and down) enable the network to find excellent solutions of Ising problems. <br />
We evaluate our method on Ising versions of the MAX-CUT problems, showing that it improves on previously published results on several benchmark problems. <br />
Our scheme, which is amenable to realization using many kinds of oscillators from different physical domains, is particularly well suited for CMOS, in which it offers significant practical advantages over previous techniques for making Ising machines. <br />
We have demonstrated several working hardware prototypes using CMOS electronic oscillators, built on breadboards and PCBs, implementing Ising machines consisting of 4, 8, 32, 64 and 240 spins.<br />
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In this talk, we will also go over my other Ph.D. work that has led to the development of oscillator-based Ising machines. <br />
In particular, we show that binary oscillators are not only useful for Ising, but can also be used to devise Finite State Machines for general-purpose Boolean computation with phase-based logic encoding, extending a scheme originally proposed by John von Neumann.<br />
We also briefly show my other research topics that have enabled the above work on oscillators, particularly those on the modelling of multi-domain nonlinear devices/systems, and those on advanced simulation analyses (eg, oscillator-specific analyses based on linear periodically time-varying system theory).<br />
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At the end of this talk, there will be a lab demonstration of a prototype oscillator-based Ising machine of 240 spins with programmable couplings.<br />
The prototype is built using off-the-shelf components on PCBs, roughly 10"x6"x4" in size, and interfaces with a laptop through USB.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125647&date=2019-05-10String-Math Seminar, May 13
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125419&date=2019-05-13
Wilson loops are important observables in gauge theory. In this talk, we study half-BPS Wilson loops of a large class of five dimensional supersymmetric quiver gauge theories with 8 supercharges, in a nontrivial instanton background. The Wilson loops are codimension 4 defects of the quiver gauge theory, and their interaction with self-dual instantons is captured by a 1d ADHM quantum mechanics. We compute the partition function as its Witten index. It turns out that we can understand the 5d physics in 3d gauge theory terms. This comes about from so-called gauge/vortex duality; namely, we study the vortices on the Higgs branch of the 5d theory, and reinterpret its partition function from the point of view of the vortices. This perspective has an advantage: it has a dual description in terms of "deformed" Toda Theory on a cylinder, in the Coulomb gas formalism. We show that the gauge theory partition function is equal to a (chiral) correlator of the deformed Toda Theory, with stress tensor and higher spin operator insertions. We derive all the above results from type IIB string theory, compactified on a resolved \(ADE\) singularity \(X\) times a cylinder with punctures. The 5d quiver gauge theory arises as the low energy limit of a system of D5 branes wrapping various two-cycles of \(X\), the Wilson loops are D1 branes, and the duality to Toda theory emerges after introducing additional D3 branes.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125419&date=2019-05-13Representation Theory and Mathematical Physics Seminar, May 16
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125416&date=2019-05-16
The reduced phase space of the Poisson Sigma Model (PSM) comes equipped with a symplectic groupoid structure, when the worldsheet is a disk and the target Poisson structure is integrable. In this talk we describe an extension of this construction when we consider surfaces with arbitrary genus, obtaining the abelianization of the original groupoid. We will also describe the obstructions for smoothness of such abelianization, in terms of the extended monodromy groups. This can be seen as a generalization of the Hurewicz theorem to Lie groupoids and Lie algebroids. Joint work with Rui Fernandes.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125416&date=2019-05-16Analysis and PDE Seminar, May 20
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125769&date=2019-05-20
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^2$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration including Weyl laws; in each case obtaining quantitative improvements over the known bounds.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=125769&date=2019-05-20