Mathematics
http://events.berkeley.edu/index.php/calendar/sn/math.html
Upcoming EventsCombinatorics Seminar, Jan 22
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114453&date=2018-01-22
In 1944, Selberg evaluated a multivariate integral, which generalizes Euler's beta integral. In 1980, Askey conjectured a $q$-integral version of the the Selberg integral, which was proved independently by Habsieger and Kadell in 1988. In this talk, we focus on the combinatorial aspects of the Selberg integral. First, we review the following fact observed by Igor Pak: evaluating the Selberg integral is essentially the same as counting the linear extensions of a certain poset. Considering $q$-integrals over order polytopes, we give a combinatorial interpretation for Askey's $q$-Selberg integral. We also find a connection between the Selberg integral and Young tableaux. As applications we enumerate Young tableaux of various shapes.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114453&date=2018-01-22Probabilistic Operator Algebra Seminar, Jan 22
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114444&date=2018-01-22
With the introduction of free independence by D.V. Voiculescu, it became clear that in the framework of non-commutative probability there are other notions of independence different than that of (classical) independence. In 1997, R. Speicher defined a notion of universal product for which he showed that there are three types of independence. In the category of unital algebras the tensor and free independence are the only existing ones. On the other hand when algebras are not required to have a unit, the product provided by Boolean independence is also admitted as a universal product. The Boolean convolution between measures was formally introduced by Speicher and R. Woroudi in 1993, although it had previously appeared in the literature in different contexts, for example, as partial cumulants in stochastic differential equations. Later, in 2006, H. Bercovici provided the product for Hilbert spaces that, in the context of operator algebras, corresponds to the Boolean convolution between measures. In this talk we will survey the basics of Boolean probability together with some results that show the similarities and differences that it has with the classical theory of probability.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114444&date=2018-01-22Differential Geometry Seminar, Jan 22
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114399&date=2018-01-22
We establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collection of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove that the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is a isometric to a flat polyhedron.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114399&date=2018-01-22Arithmetic Geometry and Number Theory RTG Seminar, Jan 22
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114780&date=2018-01-22
We will discuss recent developments in the computation of Brauer groups of some algebraic stacks, namely the moduli stack of elliptic curves $\mathscr M_{1,1}$ and torsion $\mathbb G_m$-gerbes.<br />
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Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114780&date=2018-01-22Understanding rare events in models of statistical mechanics, Jan 22
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114509&date=2018-01-22
Statistical mechanics models are ubiquitous at the interface of probability theory, information<br />
theory, and inference problems in high dimensions. In this talk, we will focus on<br />
sparse networks, and polymer models on lattices. The study of rare behavior (large deviations)<br />
is intimately related to the understanding of such models. In particular, we will<br />
consider the rare events that a sparse random network has an atypical number of certain<br />
local structures and that a polymer in random media has atypical weight. Such events can<br />
have different geometric consequences, ranging from local to more global. We will discuss<br />
some recent results concerning such phenomena, and connections to stochastic block models,<br />
exponential random graphs, eigenvalues of random matrices, and fundamental growth<br />
models.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114509&date=2018-01-22Thematic Seminar: Numerical Methods, Jan 22
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114374&date=2018-01-22
Machine intelligence for processing big data sets is big business. A mathematician's point of view has led to (1) effective large-scale principal component analysis and singular value decomposition, and (2) theoretical foundations for convolutional networks (convolutional networks underpin the recent revolution in artificial intelligence).http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114374&date=2018-01-22Understanding rare events in models of statistical mechanics, Jan 22
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114559&date=2018-01-22
Statistical mechanics models are ubiquitous at the interface of probability theory, information<br />
theory, and inference problems in high dimensions. In this talk, we will focus on<br />
sparse networks, and polymer models on lattices. The study of rare behavior (large deviations)<br />
is intimately related to the understanding of such models. In particular, we will<br />
consider the rare events that a sparse random network has an atypical number of certain<br />
local structures and that a polymer in random media has atypical weight. Such events can<br />
have different geometric consequences, ranging from local to more global. We will discuss<br />
some recent results concerning such phenomena, and connections to stochastic block models,<br />
exponential random graphs, eigenvalues of random matrices, and fundamental growth<br />
models.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114559&date=2018-01-223-Manifold Seminar, Jan 23
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114774&date=2018-01-23
We'll discuss a proof that knot recognition is in NP, using a certificate that encodes a sutured manifold hierarchy.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114774&date=2018-01-23Thematic Seminar: Applied Mathematics, Jan 23
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114402&date=2018-01-23
Fiber-reinforced structures arise in many engineering and biological applications. Examples include space inflatable habitats, vascular stents supporting compliant vascular walls, and aortic valve leaflets. In all these examples a metallic mesh, or a collection of fibers, is used to support an elastic structure, and the resulting composite structure has novel mechanical characteristics preferred over the characteristics of each individual component. These structures interact with the surrounding deformable medium, e.g., blood flow or air flow, or another elastic structure, constituting a fluid-structure interaction (FSI) problem. Modeling and computer simulation of this class of FSI problems is important for manufacturing and design of novel materials, space habitats, and novel medical constructs.<br />
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Mathematically, these problems give rise to a class of highly nonlinear, moving-boundary problems for systems of partial differential equations of mixed type. To date, there is no general existence theory for solutions of this class of problems, and numerical methodology relies mostly on monolithic/implicit schemes, which suffer from bad condition numbers associated with the fluid and structure sub-problems.<br />
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In this talk we present a unified mathematical framework to study existence of weak solutions to FSI problems involving incompressible, viscous fluids and elastic structures. The mathematical framework provides a constructive existence proof, and a partitioned, loosely coupled scheme for the numerical solution of this class of FSI problems. The constructive existence proof is based on time-discretization via operator splitting, and on our recent extension of the Aubin-Lions-Simon compactness lemma to problems on moving domains. The resulting numerical scheme has been applied to problems in cardiovascular medicine, showing excellent performance, and providing medically beneficial information.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114402&date=2018-01-23Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring, Jan 23
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114773&date=2018-01-23
The two topics for the student portion this semester are Intersection Theory and Linkage. A detailed outline of the topics for the first half of the seminar can be found at "math.berkeley.edu/~ritvik/Eisenbud-Seminar-Outline.pdf". In the meeting we will describe the main goals of this seminar and sign-up speakers for respective topics.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114773&date=2018-01-23Topology Seminar (Introductory Talk), Jan 24
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114728&date=2018-01-24
We will introduce Thurston's norm on the second homology of a 3-manifold, and some associated constructions including branched surfaces and Agol’s veering triangulation.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114728&date=2018-01-24Topology Seminar (Main Talk), Jan 24
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114727&date=2018-01-24
Let \(M\) be a closed hyperbolic 3-manifold with a fibered face \(\sigma\) of the unit ball of the Thurston norm on \(H_2(M)\). If \(M\) satisfies a certain condition related to Agol’s veering triangulation, we can construct a taut branched surface in \(M\) spanning \(\sigma\). This partially answers a 1986 question of Oertel, and extends an earlier partial answer due to Mosher.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114727&date=2018-01-24Applied Math Seminar, Jan 24
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114775&date=2018-01-24
(Note the special location) A spacetime simulation region can be subdivided into tent-shaped subregions. Tents appear to be natural for solving hyperbolic equations. Indeed, one can ensure causality by constraining the height of the tent pole. More precisely, the domain of dependence of all points within the tent can be guaranteed to be contained within the tent, by constraining the tent pole height. We consider techniques to advance the numerical solution of a hyperbolic problem by progressively meshing a spacetime domain by tent shaped objects. Such tent pitching schemes have the ability to naturally advance in time by different amounts at different spatial locations. One obtains spacetime discontinuous Galerkin (SDG) schemes - extensively studied by many authors - when the hyperbolic system on the tent is discretized in spacetime. We pursue another alternative by mapping each tent to a spacetime cylinder. These maps transform tents into domains where space and time are separated, thus allowing standard methods to be used within tents. Several open mathematical and computational issues surrounding these methods will be touched upon.<br />
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Reference: J. Gopalakrishnan, J. Schlberl, and C. Wintersteiger. "Mapped tent pitching schemes for hyperbolic systems." SIAM J Sci Comp, 39:6, p.B1043-B1063, 2017.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114775&date=2018-01-24Thematic Seminar: Partial Differential Equations, Jan 24
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114375&date=2018-01-24
An intriguing feature of the explicit charged (Reissner-Nordstrom) or spinning (Kerr) black hole spacetimes is the existence of a regular Cauchy horizon, beyond which the Einstein equation loses its predictive power. The strong cosmic censorship conjecture of Penrose is a bold claim that, nevertheless, such a pathological behavior is nongeneric.<br />
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In this lecture, I will give a short introduction to general relativity and the strong cosmic censorship conjecture. Then I will describe my recent joint work with J. Luk, where we rigorously establish a version of this conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry, which has long been studied by physicists and mathematicians as a useful model for this problem.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114375&date=2018-01-24Workshop: Introduction to Deep Learning, Jan 24
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114699&date=2018-01-24
Are you interested in learning what's the machinery behind the Deep Learning hype? Come to ML@B's Intro to Deep Learning Workshop! <br />
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We will teach you how neural networks work and how to use them through hands-on demos. The workshop is structured so that you have enough knowledge to get the most out of our future workshops. <br />
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You can find the full list at ml.berkeley.edu/workshops. Although no strict prerequisites, Python knowledge and knowledge of multivariable calculus will be helpful.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114699&date=2018-01-24Paris/Berkeley/Bonn/Zürich Analysis Seminar, Jan 25
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114308&date=2018-01-25
I show that on a compact hyperbolic surface, the mass of an $L^2$-normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassical version, has many applications including control for the Schrödinger equation and the full support property for semiclassical defect measures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114308&date=2018-01-25Seminar 217, Risk Management: PageRank on directed complex networks, Jan 25
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114314&date=2018-01-25
The talk will center around a set of recent results on the analysis of Google’s PageRank algorithm on directed complex networks. In particular, it will focus on the so-called power-law hypothesis, which states that the distribution of the ranks produced by PageRank on a scale-free graph (whose in-degree distribution follows a power-law) also follows a power-law with the same tail-index as the in-degree. We show that the distribution of PageRank on both the directed configuration model and the inhomogeneous random digraph does indeed follow a power-law whenever the in-degree does, and we provide explicit asymptotic limits for it. Moreover, our asymptotic expressions exhibit qualitatively different behaviors depending on the level of dependence between the in-degree and out-degree of each vertex. On graphs where the in-degree and out-degree are close to independent, our main theorem predicts that PageRank will tend to grant high ranks to vertices with large in-degrees, but also to vertices who have highly-ranked inbound neighbors. However, when the in-degree and out-degree are positively correlated, the latter can potentially disappear, strengthening the impact of high-degree vertices on the ranks produced by the algorithm.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114314&date=2018-01-25DiPerna Lecture, Jan 25
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114730&date=2018-01-25
We will give an overview of some of the developments in recent years dealing with the description of asymptotic states of solutions to semilinear evolution equations ("soliton resolution conjecture"). New results will be presented on damped subcritical Klein-Gordon equations, joint with Nicolas Burq and Genevieve Raugel.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114730&date=2018-01-25Student Probability/PDE Seminar, Jan 26
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114829&date=2018-01-26
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114829&date=2018-01-26Applied Math Seminar, Jan 26
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114776&date=2018-01-26
(Note special date and location) A wide variety of physical and biological systems can be described as continuum limits of interacting particles. Many of these problems are gradient flows and their dynamics are governed by a monotonically decreasing interaction energy that is often non-local in nature. We show how to exploit these energies numerically, analytically, and asymptotically to characterize the observed behavior. We describe three such systems. In the first, a Langmuir layer, line tension (the two-dimensional analog of surface tension) drives the fluid domains to become circular and the rate of relaxation to these circular domains can be used to deduce the magnitude of the line tension forces. In the second, a Hele-Shaw problem, vexing changes in topology are observed. The third system models the formation of the convoluted fingered domains observed experimentally in ferrofluids for which pattern formation is driven by line tension and dipole-dipole repulsion. We show that noise in this system plays an unexpected but essential role and deduce an algorithm for extracting the dipole strength using only a shape's perimeter and morphology.<br />
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Biosketch: Andrew Bernoff is the Kenneth & Diana Jonsson Professor of Mathematics at Harvey Mudd College. His research specializes in bridging the gaps between Mathematics, Physics, Biology and Engineering with a particular emphasis on using dynamical systems methods to understand experiments and natural phenomena. Prof. Bernoff was an undergraduate at MIT where he received BS degrees in Mathematics and Physics. He was awarded a Marshall Scholarship to pursue a PhD at the University of Cambridge in England. His PhD studies were on the application of dynamical systems methods in fluid mechanics in the Department of Applied Mathematics and Theoretical Physics (DAMTP). Prof. Bernoff spent time on the faculty at Northwestern before settling in at Harvey Mudd College. He is passionate about mentoring undergraduate research, coaching the Harvey Mudd College Putnam Team, and supporting Harvey Mudd College’s Clinic Program, a year-long practicum in which teams of undergraduates work for industrial sponsors on real-world problems and applications. His research program centers on understanding the behavior of fluids at small scales and modeling the swarming of organisms, in particular locusts.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=114776&date=2018-01-26