Title: Topological recursion for matrix models and abstract topological recursion (a course of 3 lectures)

Date: 03/01/2017

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

I will begin with the description of a generating function for numbers of
Grothendieck's dessins d'enfant, or Belyi pairs. This generating function
is given by a random matrix model integral, I describe what is a
topological expansion (AKA genus expansion) of such models. The method
allowing finding corrections in all orders of the genus expansion is
Topological Recursion formulated in its present form by Eynard, Orantin
and the speaker in 2005-2006. This method had already found numerous
applications in mathematics and mathematical physics, so I describe the
general construction underlying the topological recursion and present an
(incomplete) list of its applications. Very recently, this method was
developed into an abstract topological recursion by Kontsevich and
Soibelman. In my last lecture I explain their construction and our
interpretation of it (forthcoming paper by J.Andersen, G.Borot, L.Ch., and
N.Orantin).

Title: The genus of a special cube complex and its applications

Date: 03/02/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Recently, the geometry of non-positively curved (NPC) cube complexes has featured prominently in low-dimensional topology. We introduce an invariant of NPC special cube complexes called the genus, which generalizes the classical notion of genus for a closed orientable surface. We then show that having genus one characterizes special cube complexes with abelian fundamental group and discuss some applications.

In this talk, I will give an introduction to the so called mutation operation of a 3-manifold, which can be described as cutting the 3-manifold along a given surface and then sew it back in a different way. In particular, this induces corresponding mutation operations on knots obtained by mutating the 3-sphere along a genus 2 surface in the knot complement. I will survey a few properties and conjectures regarding mutant knots.

In a classical paper, Cafferelli, Gidas and Spruck discussed positive solutions of the Yamabe equation, corresponding to the positive scalar curvature of the conformal metrics, with a nonremovable isolated singularity. They proved that solutions are asymptotic to radial singular solutions. Korevaar, Mazzeo, Pacard, and Schoen expanded solutions to the next order. In this talk, we discuss how to expand solutions up to arbitrary order. We also discuss positive solutions of the Yamabe equation, corresponding to the negative scalar curvature of the conformal metrics, that become singular in an (n-1)-dimensional set.

Speaker: Julia Aguirre, University of Washington, Tacoma

Title: Math Strong: Amplifying Equity and Justice in Mathematics Education Research and Practice

Date: 03/15/2017

Time: 3:30 PM - 5:00 PM

Place: 252 EH

In this interactive talk, Dr. Aguirre will challenge math education researchers to amplify equity in research and practice. We will discuss fundamental questions about mathematics education and its role in perpetuating systems of power, privilege and oppression. Through exploration of various empirical, instructional, and organizational tools, we will identify concrete actions as math educators to disrupt the negative effects of these systems and re-align our intent and impact to cultivate a more just and equitable mathematics education experience for our nation’s youth.

We will introduce mapping class groups of surfaces, give several specific examples, and then discuss finite generating sets for mapping class groups of surfaces that are oriented, connected, compact, with possibly finitely many punctures.

Title: Using geometry and combinatorics to move robots quickly.

Date: 03/16/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

How do we move a robot quickly from one position to another? To answer this question, we need to understand the 'space of possibilities” containing all possible positions of the robot. Unfortunately, these spaces are tremendously large and high-dimensional, and are very difficult to visualize. Fortunately, geometers and algebraists have encountered and studied these kinds of spaces before. Thanks to the tools they’ve developed, we can build “remote controls” to navigate these complicated spaces, and move (some) robots optimally.
This talk is based on joint work with my students Arlys Asprilla, Tia Baker, Hanner Bastidas, César Ceballos, John Guo, and Rika Yatchak. It will be accessible to undergraduate students, and assume no previous knowledge of the subject.

Title: Composition algebras: from a Dublin bridge to your cellphone

Date: 03/17/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In 1843 Sir William Rowan Hamilton scratched
the quaternions onto a bridge in Dublin,
and in 1998 Siavash M. Alamouti used them as
a coding scheme, now embedded in hundreds
of millions of cellphones. In discussing
this we also touch on combinatorial
designs, sums of squares, and those odd
towers in your local supermarket parking
lot.

Title: When medical challenges meet time series analysis and manifold learning

Date: 03/17/2017

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Adaptive acquisition of correct features from massive datasets
is at the core of modern data analysis. One particular interest in
medicine is the extraction of hidden dynamics from an observed time series
composed of multiple oscillatory signals. The mathematical and statistical
problems are made challenging by the structure of the signal which
consists of non-sinusoidal oscillations with time varying amplitude and
time varying frequency, and by the heteroscedastic nature of the noise. In
this talk, I will discuss recent progress in solving this kind of problem.
Based on the cepstrum-based nonlinear time-frequency analysis and manifold
learning technique, a particular solution will be given along with its
theoretical properties. I will also discuss the application of this method
to two medical problems – (1) the extraction of a fetal ECG signal from a
single lead maternal abdominal ECG signal; (2) the simultaneous extraction
of the instantaneous heart rate and instantaneous respiratory rate from a
PPG signal during exercise. If time permits, an extension to multiple-time
series will be discussed.

'Knot theory' is the study of closed, embedded curves in
three-dimensional space. Classically, knots can be studied via a
various computable polynomial invariants, such as the Alexander
polynomial. In this first talk, I will recall the basics of knot
theory and the Alexander polynomial, and then move on to a more modern
knot invariant, 'knot Floer homology', a knot invariant with more
algebraic structure associated to a knot. I will describe applications
of knot Floer homology to traditional questions in knot theory, and
sketch its definition. This knot invariant was originally defined in
2003 in joint work with Zoltan Szabo, and independently by Jake
Rasmussen. A combinatorial formulation was given in joint work with
Ciprian Manolescu and Sucharit Sarkar in 2006.

This talk is the third and final part of our working through the paper by Karp and Williams. We will discuss (finally) the poset structure that you can put on the elements of the m=1 amplituhedron and how they can be made into a hyperplane arrangement.

Title: Log-Canonical Coordinates for Poisson Brackets

Date: 03/21/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In symplectic geometry, Darboux's theorem gives so-called 'canonical' local coordinates, in which the Poisson bracket (obtained from the symplectic structure) takes a particularly simple form: all brackets of coordinate functions are constants. We investigate whether something analogous is true for Poisson varieties (when the coordinate change is only allowed to be rational functions), and give a negative answer in the case of log-canonical coordinates.

Title: Bordered techniques in Heegaard Floer homology.

Date: 03/21/2017

Time: 5:20 PM - 6:20 PM

Place: 115 International Center

Heegaard Floer homology is a closed three-manifold invariant, defined
in joint work with Zoltan Szabo, using methods from symplectic
geometry (specifically, the theory of pseudo-holomorphic disks). The
inspiration for this invariant comes from gauge theory. In joint work
with Robert Lipshitz and Dylan Thurston from 2008, the theory was
extended to an invariant for three-manifolds with boundary,
'bordered
Floer homology'. I will describe Heegaard Floer homology, motivate
its construction, list some of its key properties and applicat

I begin with solving the loop equations in matrix models. The TR allows constructing in a very algorithmic way all correlation functions $W_s^{(g)}(x_1,\dots,x_s)$ of a model on the base of the spectral curve $\Sigma(x,y)=0$ obtained as a solution of a master loop equation in the planar approximation; the variable $y$ is identified with $W_1^{(0)}$. We are then able to construct all $W_s^{(g)}$ out of this, not very abundant, set of data supplied with the two-point correlation function $W_2^{(0)}(x_1,x_2)dx_1dx_2$, which is a universal Bergmann 2-differential on the spectral curve. We are also able to construct terms of the free energy $F^g$ using the Ch.-Eynard integration formula applied to $W_1^{(g)}$. Finally, I will describe a Feynman-like diagrammatic technique for evaluation all $W_s^{(g)}$.

I will describe a bordered construction of knot Floer homology,
defined as a computable, combinatorial knot invariant. Generators
correspond to Kauffman states, and the differentials have an algebraic
interpretation in terms of a certain derived tensor product. I will
also explain how methods from bordered Floer homology prove that this
invariant indeed computes the holomorphically defined knot Floer
homology. This is joint work with Zoltan Szabo.

In this talk, we introduce Cartan geometries which are spaces
that are locally modeled on homogeneous spaces. We express what
conformal geometries are in this language and then move on to study
signature (2,3) conformal geometry with holonomy in the split form of
G_2 . The exposition will mainly follow https://arxiv.org/abs/1002.1767
chapters 4 and 5.

Title: Math Learning Center and Supplemental Instruction in 2017-18

Date: 03/27/2017

Time: 4:10 PM - 5:00 PM

Place: C109 Wells Hall

We are going to discuss the expected changes to the MLC to be implemented this coming fall. We would also welcome input from faculty and GTAs regarding alternative formats to assist students, including problem/review sessions, etc.
The team on Supplemental Instruction will also be join us to acquaint us with plans for SI in Fall '17 and Spring '18. We will discuss possible ways to coordinate these efforts with instructors of large lectures and course coordinators.

Title: A new physical space approach to improved decay for linear waves with applications

Date: 03/27/2017

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

I will present a vector field method for the linear wave equation on a wide class of spacetimes through which one can obtain almost optimal energy and pointwise decay results, as well as precise asymptotics. I will also talk about applications of the above method to problems in general relativity. This is joint work with S. Aretakis and D. Gajic.

Title: What polytopes tell us about toric varieties

Date: 03/28/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Polytopes are among the oldest mathematical objects that have been studied. Often, people want to find their volumes, identify triangulations, and describe their lattice points, and more. But why bother doing this? From a combinatorial perspective, the data often answer counting questions that one might have. However, there is much more depth from an algebro-geometric standpoint: this information is often useful for learning about certain toric varieties.
In the first of these talks, I will give the background needed to understand what a normal projective toric variety is and how to model them using polytopes. In the second talk, I will define several properties that an algebraic geometer may want to know about a toric variety, and explain how to detect these properties from a purely polytopal perspective.

I will describe the relation between global and local models. Local models are those based on a signle branchpoint. Although a zoo of global models is huge, we have just two types of local models: the Kontsevich model (the Airy kernel) associated with ``dynamic' branchpoints (zeros $\mu_\alpha$ of $dx$ such that $y(\mu_\alpha)\ne \{0,infty\}$) and the Brezin-Gross-Witten model (the Bessel kernel) associated with a 'hard' edge (a zero $a_\alpha$ of $dx$ such that $ydx$ is regular at $a_\alpha$). I will describe the canonical action of quadratic operators establishing exact relations of Ch.-Givental form between a global model and the direct product of local models.

Title: Hedgehog Bases: A taste of the mathematics and physics of scattering amplitudes

Date: 03/30/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

Scattering amplitudes are the heart of particle physics, forming a bridge between theory and experiment. The last decade has seen the rise of 'amplitudeology', a program for uncovering the hidden mathematical structures of scattering amplitudes. By importing recent results in mathematics, amplitudeology has produced new insights into scattering amplitudes - such as the 'amplitudehedron' - and translated them into practical computational techniques. In this talk, I will discuss a cluster algebra structure of scattering amplitudes in N=4 Super Yang-Mills theory and deep connections with Goncharov polylogarithms. In order to develop a computational framework which exploits this connection, I show how to construct bases of Goncharov polylogarithms that can be used to describe 6-particle scattering at any order in perturbation theory.

Speaker: Jianfeng Zhang, University of Southern California

Title: A Martingale Approach for Fractional Brownian Motions

Date: 03/30/2017

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

Empirical studies show that the volatilities could be rough, which typically go beyond the semimartinagle framework and the fractional Brownian Motion (fBM) becomes a natural tool. Compared with BM, fBM has two features: (i) non-Markoivan; (ii) non-semimartingale (when the Hurst parameter $H< {1\over 2}$). We shall show that the recent development of path dependent PDEs provides a convenient tool to extend the standard literature of pricing/hedging derivatives to an fBM framework. This is a joint work with Frederi Viens.

In this talk we briefly go through harmonic map and Dirac-harmonic map.
Especially we see the bubble behavior of harmonic and Dirac-harmonic map and estimates needed.