Mathematics
http://events.berkeley.edu/index.php/calendar/sn/math.html
Upcoming EventsCombinatorics Seminar, Oct 15
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120269&date=2018-10-15
Many questions in combinatorics, probability and statistical mechanics can be reduced to counting lattice paths (walks) in regions of the plane. A standard approach to counting problems is to consider properties of the associated generating function. These functions have long been well understood for walks in the full plane and in a half plane. Recently much attention has focused on walks in the first quadrant of the plane and has now resulted in a complete characterization of those walks whose generating functions are algebraic, holonomic (solutions of linear differential equations) or at least differentially algebraic (solutions of algebraic differential equations). I will give an introduction to this topic, discuss previous work of Bousquet-Melou, Kauers, Mishna, and others and then present recent work by Dreyfus, Hardouin, Roques and myself applying the theory of QRT maps and Galois theory of difference equations to determine which generating functions satisfy differential equations and which do not.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120269&date=2018-10-15String-Math Seminar, Oct 15
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120725&date=2018-10-15
A conjecture of Gorsky-Negut-Rasmussen asserts the existence of a pair of adjoint functors relating the Hecke category for symmetric groups and the Hilbert scheme of points in the plane. One topological consequence of this conjecture is the prediction of a deformation of the triply graded Khovanov-Rozansky link homology which restores the missing \(q\rightarrow tq^{-1}\) symmetry of KR homology for links. In this talk I will discuss a candidate for such a deformation, constructed in joint work with Eugene Gorsky, which indeed facilitates connections with Hilbert schemes. For instance our main result explicitly computes the homologies (both deformed and undeformed) of the \((n,nk)\) torus links, summed over all \(n\geq 0\), as a graded algebra. Combining with work of Haiman this gives a functor from the Hecke category to sheaves on the relevant Hilbert scheme.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120725&date=2018-10-15Differential Geometry Seminar, Oct 15
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120505&date=2018-10-15
This is joint work with M. Eichmair and V. Moraru. We prove that if a 3-manifold with non-negative scalar curvature contains an absolutely area-minimizing cylinder then the ambient manifold is flat. This can be seen as a scalar curvature analogue of the Cheeger–Gromoll splitting theorem for Ricci curvature.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120505&date=2018-10-15Arithmetic Geometry and Number Theory RTG Seminar, Oct 15
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120840&date=2018-10-15
The local (and global) Langlands conjectures attempt to bridge the major areas of harmonic analysis and number theory by forming a correspondence between representations which naturally appear in both areas. A key insight due to Langlands and Kottwitz is that one could attempt to understand such a conjectural correspondence by comparing the traces of natural operators on both sides of the bridge. Moreover, it was realized that Shimura varieties present a natural means of doing this. For global applications, questions of reduction type (at a particular prime $p$) for these Shimura varieties can often be avoided, and for this reason the methods of Langlands and Kottwitz focused largely on the setting of good reduction. But, for local applications dealing with the case of bad reduction is key. The setting of bad reduction was first dealt with, for some simple Shimura varieties. Harris and Taylor then used this, together with the work of many other mathematicians, to prove the local Langlands conjecture for $GL_n$. A decade later Scholze gave an alternative, more geometric, way to understand the case of bad reduction for certain Shimura varieties and was able to reprove the local Langlands conjecture for $GL_n$ in simpler terms. In this talk we will discuss an extension of the ideas of Scholze to a wider class of Shimura varieties, as well as the intended application of these ideas to the local Langlands conjectures for more general groups.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120840&date=2018-10-15Analysis and PDE Seminar, Oct 15
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120836&date=2018-10-15
In this talk, I will discuss the differential equation $iu_t = Hu, H := H_0 + V$ , where $V$ is a decaying potential and $H_0$ is a Laplacian related operator. In particular, I will focus on when $H_0$ is Laplacian, Bilaplacian and Dirac operators. I will discuss how the threshold energy obstructions, eigenvalues and resonances, effect the $L^1 \to L^\infty $ behavior of $e^{itH} P_{ac} (H)$. The threshold obstructions are known as the distributional solutions of $H\psi = 0$ in certain dimension dependent spaces. Due to its unwanted effects on the dispersive estimates, its absence have been assumed in many work. I will mention our previous results on Dirac operator and recent results on Bilaplacian operator under different assumptions on threshold energy obstructions.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120836&date=2018-10-15Deformation Theory Seminar, Oct 15
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120824&date=2018-10-15
Let $A$ be an algebra. The Koszul duality is a type of derived equivalence between modules over $A$ and modules over its Koszul dual $A^!$. In this talk, we will talk about the general framework and then focus on the classical cases as well as examples.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120824&date=2018-10-15Representation theory seminar, Oct 16
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120838&date=2018-10-16
Khovanov and Rozansky defined a link invariant called triply graded homology. It is conjectured by Gorsky, Negut and Rasmussen that this invariant can be expressed geometrically by a functor from complexes of Soergel bimodules to the derived category of coherent sheaves on the dg flag Hilbert scheme followed by taking cohomology. A functor with similar properties has been constructed by Oblomkov and Rozansky using matrix factorizations and it is believed that this functor solves the conjecture. The aim of this joint work in progress with Roman Bezrukavnikov is to relate the two constructions using previous work of Arkhipov and Kanstrup.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120838&date=2018-10-16Seminar 217, Risk Management: Asymptotic Spectral Analysis of Markov Chains with Rare Transitions: A Graph-Algorithmic Approach, Oct 16
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=118748&date=2018-10-16
Parameter-dependent Markov chains with exponentially small transition rates arise in modeling complex systems in physics, chemistry, and biology. Such processes often manifest metastability, and the spectral properties of the generators largely govern their long-term dynamics. In this work, we propose a constructive graph-algorithmic approach to computing the asymptotic estimates of eigenvalues and eigenvectors of the generator. In particular, we introduce the concepts of the hierarchy of Typical Transition Graphs and the associated sequence of Characteristic Timescales. Typical Transition Graphs can be viewed as a unification of Wentzell’s hierarchy of optimal W-graphs and Friedlin’s hierarchy of Markov chains, and they are capable of describing typical escapes from metastable classes as well as cyclic behaviors within metastable classes, for both reversible and irreversible processes. We apply the proposed approach to conduct zero-temperature asymptotic analysis of the stochastic network representing the energy landscape of the Lennard-Jones cluster of 75 atoms.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=118748&date=2018-10-163-Manifold Seminar, Oct 16
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120839&date=2018-10-16
Bass-Serre theory studies groups acting on trees. The action of a group on a tree determines a quotient "graph of groups", from which one can reconstruct the original group via amalgamated free products and HNN extensions. We will study the tree associated to $PSL(2,{\mathbb Q}_p)$ in detail, discuss various higher-dimensional generalizations, and describe some applications to the study of incompressible surfaces in 3-manifolds.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120839&date=2018-10-16Student Harmonic Analysis and PDE Seminar (HADES), Oct 16
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120842&date=2018-10-16
I will present the method introduced by András Vasy to prove meromorphic continuations of resolvents of Laplacians on asymptotically hyperbolic spaces in a simple model case. In particular, I will show the proof of Melrose's radial estimates indicating the idea behind the general case.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120842&date=2018-10-16Probabilistic Operator Algebra Seminar, Oct 16
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120579&date=2018-10-16
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 than that of classical independence. 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, scenarios in which it appears naturally, together with some results that show the similarities and differences it has with the classical theory of probability.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120579&date=2018-10-16Topology Seminar (Introductory Talk), Oct 17
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120679&date=2018-10-17
Random curves in space and how they are knotted give an insight into the behavior of "typical" knots and links. They have been studied by biologists and physicists in the context of the structure of random polymers. Several randomized models have been suggested and investigated both by theoretical methods and computational experiments. We will review some known and new models of random knotting, and will discuss their nature and the typical properties of the knots they produce.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120679&date=2018-10-17The Lovász theta function for random regular graphs, Oct 17
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120791&date=2018-10-17
The Lovász theta function is a classic semidefinite relaxation of graph coloring. In this talk I'll discuss the power of this relaxation for refuting colorability of uniformly random degree-regular graphs, as well as for distinguishing this distribution from one with a `planted' disassoratative community structure. We will see that the behavior of this refutation scheme is consistent with the conjecture that coloring and community detection exhibit a `computationally hard but information-theoretically feasible' regime typical of random constraint satisfaction and statistical inference problems. The proofs will make use of orthogonal polynomials, nonbacktracking walks, and results on the spectra of random regular graphs.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120791&date=2018-10-17Number Theory Seminar, Oct 17
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120727&date=2018-10-17
We will discuss étale localization in the theory of the de Rham Witt complex.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120727&date=2018-10-17Topology Seminar (Main Talk), Oct 17
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120680&date=2018-10-17
The study of knots and links from a probabilistic viewpoint provides insight into the behavior of "typical" knots, and arises also in applications to the natural sciences. We will discuss knots that arise from random permutations using petal projections (Adams et al. 2012). We will explain why the probability of obtaining any given knot type in this model is positive if the number of petals is at least linear in the knot's crossing number, and why it decays to zero as this number grows to infinity. Our approach uses different knot invariants and arguments than those that have been used in other random models.<br />
<br />
Joint work with Joel Hass, Nati Linial, and Tahl Nowik.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120680&date=2018-10-17Learning in Google Ads, Machines and People, Oct 17
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120789&date=2018-10-17
This talk is in two parts, both of which discuss interesting uses of experiments in Google search ads. In part 1 I discuss how we can inject randomness into our system to get causal inference in a machine learning setting. In part 2. I talk about experiment designs to measure how users learn in response to ads on Google.com.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120789&date=2018-10-17Applied Math Seminar, Oct 18
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120675&date=2018-10-18
Electronic correlation effects play an import role in emergent phenomena such as Mott-insulator-metal transition and unconventional superconductivity. Understanding these effects present a theoretical challenge. In this talk, we will give an overview of dynamical mean-field theory (DMFT) and its combination with the local density approximation in density functional theory. Representative quantum impurity solvers including continuous-time quantum Monte Carlo method will also be discussed, together with a few measurable quantities. Finally, I will present applications of the theoretical approach to strongly correlated f-electron systems.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120675&date=2018-10-18Mathematics Department Colloquium, Oct 18
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120841&date=2018-10-18
This is a joint work with Piermarco Cannarsa and Wei Cheng. <br />
<br />
If A is a closed subset of the Euclidean space $R^k$, the Euclidean distance function $d_A : R^k \to [0, + \infty[$ is defined by<br />
<br />
$$d_A(x) = \mathrm{min}_{a \in A} ||x − a||.$$<br />
<br />
This function is Lipschitz, therefore differentiable almost everywhere. We will give topological properties of the set Sing(F) of points in $R^k \setminus M$ where F is not differentiable. For example it is locally connected. We will also discuss the homotopy type of Sing(F).<br />
<br />
Although, we will concentrate on $d_A$, we will explain that it is a particular case of a more general result on the singularities of a viscosity solution $F:R^k × ]0, +\infty[ \to R$ of the evolution Hamilton-Jacobi equation<br />
<br />
$$ \partial_t F + H(x, \partial_x F) = 0,$$<br />
<br />
where $H : R^k × R^k \to R$, $(x, p) \mapsto H(x, p)$ is a $C^2$ Tonelli Hamiltonian, i.e. convex and superlinear in the momentum p. If time permits we will explain the methods of proof for the case of $d_A$.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120841&date=2018-10-184th Annual CDAR Symposium 2018, Oct 19
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=119719&date=2018-10-19
The fourth annual CDAR Symposium, presented in partnership with State Street, will convene on Friday, October 19, 2018, from 8:30 am to 6:30 pm at UC Berkeley’s Memorial Stadium. Our conference will feature new developments in data science, highlighting applications to finance and risk management. Confirmed speakers include Jeff Bohn, Olivier Ledoit, Ulrike Malmendier, Steven Kou, Ezra Nahum, Roy Henriksson, and Ken Kroner.<br />
<br />
The Consortium for Data Analytics in Risk (CDAR) supports research into innovation in data science and its applications to portfolio management and investment risk. Based in the Economics and Statistics Departments at UC Berkeley, CDAR was co-founded with State Street, Stanford, Berkeley Institute for Data Science (BIDS), and Southwestern University of Finance and Economics (SWUFE). This year, CDAR welcomes a new founding member, Swiss Re based in Switzerland, and a new industry partner, AXA Rosenberg. CDAR organizes conferences, workshops, and research programs, bringing together academic researchers from the physical and social sciences, and industry researchers from financial management firms and technology development companies large and small.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=119719&date=2018-10-19Student Probability/PDE Seminar, Oct 19
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120837&date=2018-10-19
We shall first recall how to obtain macroscopic PDEs by taking limits of Hamiltonian dynamics as the number of molecules increases to infinity. We shall then construct along these lines explicit examples of spontaneous energy generation (and therefore establish non-uniqueness) for the compressible Euler system, with and without pressure. The examples come from rescalings of well-posed deterministic systems of molecules that either collide elastically or interact via singular pair potentials. They live in space dimension 1 for the Euler with pressure and in higher dimensions, but have singular support, for the pressureless Euler. (Work with Jianfei Xue.)http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=120837&date=2018-10-19