Speaker: Yiannis Sakellaridis (Rutgers Newark and IAS) -

Abstract: I will introduce a new paradigm for comparing relative trace formulas,

in order to prove instances of (relative) functoriality and relations

between periods of automorphic forms.

More precisely, for a spherical variety X=H\G of rank one, I will prove

that there is an explicit "transfer operator" which transforms the

orbital integrals of the relative trace formula for X x X/G to the

orbital integrals of the Kuznetsov formula for GL(2) or SL(2), equipped

with suitable non-standard test functions. The operator is determined by

the L-value associated to the square of the H-period integral, and the

proof uses a deep theory of Friedrich Knop on the cotangent bundles of

spherical varieties. This is part of an ongoing joint project with

Daniel Johnstone and Rahul Krishna, who are proving instances of the

fundamental lemma. Globally, this transfer will induce an identity of

relative trace formulas and global relative characters, translating to

an Ichino–Ikeda type formula that relates the square of the H-period to

the said L-value.

This can be viewed as part of the program of relative functoriality, a

generalization of the Langlands functoriality conjecture, predicting

relations between the automorphic spectra of two spherical varieties

when there is a map between their dual groups. The case under

consideration here is the simplest non-abelian case of this, when the

dual groups are equal and of rank one. If time permits, I will discuss

how the transfer operator here and in a few examples of higher rank

where it is known is a "deformation" of an abelian transfer operator

obtained by replacing the spherical variety by its asymptotic cone (or

boundary degeneration).

]]>

Speaker: Siming He (University of Maryland) -

Abstract: http://www.terpconnect.umd.edu/~lvrmr/2017-2018-F/Classes/RIT.shtml

]]>

Speaker: Jesse Han (McMaster University) -

Abstract: Suppose we have some process to attach to every model of a first-order theory some (permutation) representation of its automorphism group, compatible with elementary embeddings. How can we tell if this is "definable", i.e. really just the points in all models of some imaginary sort of our theory?

In the '80s, Michael Makkai provided the following answer to this question: a functor Mod(T) → Set is definable if and only if it preserves all ultraproducts and all "formal comparison maps" between them (generalizing e.g. the diagonal embedding into an ultrapower). This is known as strong conceptual completeness; formally, the statement is that the category Def(T) of definable sets can be reconstructed up to bi-interpretability as the category of "ultrafunctors" Mod(T) → Set.

Now, any general framework which reconstructs theories from their categories of models should be considerably simplified for ω-categorical theories. Indeed, we show:

If T is ω-categorical, then X : Mod(T) → Set is definable, i.e. isomorphic to (M \mapsto ψ(M)) for some formula ψ ∈ T, if and only if X preserves ultraproducts and diagonal embeddings into ultrapowers. This means that all the preservation requirements for ultramorphisms, which a priori get unboundedly complicated, collapse to just diagonal embeddings when T is ω-categorical.

This definability criterion fails if we remove the ω-categoricity assumption. We construct examples of theories and non-definable functors Mod(T) → Set which exhibit this.

]]>

Speaker: Sebastien Picard (Columbia) -

Abstract: The Anomaly flow is a geometric flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. Its stationary points satisfy the Hull-Strominger system of partial differential equations. The Anomaly flow allows metrics with torsion, and we hope to use it to study non-Kahler complex geometry. I will discuss general features of this flow, and describe its behavior on certain examples. This is joint work with D.H. Phong and X.-W. Zhang.

]]>

Speaker: Kasso Okoudjou (UMD) -

Abstract: I will describe a recent construction of multivariate Wilson orthonormal basis, that generalizes many of the existing ones.

]]>

Speaker: Dipendra Prasad (UMD and TIFR (India)) - http://www.math.tifr.res.in/~dprasad/

Abstract: Following the natural instinct that when a group operates on a number

field then every term in the class number formula should factorize ‘compatibly’ ac-

cording to the representation theory (both complex and modular) of the group, we are

led — in the spirit of Herbrand-Ribet’s theorem on the p-component of the class num-

ber of Q(\zeta_p) — to some natural questions about the p-part of the classgroup of any CM

Galois extension of Q as a module for Gal(K/Q), and about integrality of L-values.

This talk will attempt doing this in terms of precise conjectures. Talk is based on a recent paper

with the same title available in the arXiv.

]]>

Speaker: Prof. Jeff Calder (School of Mathematics, University of Minnesota) - http://www-users.math.umn.edu/~jwcalder/

Abstract: Semi-supervised learning refers to machine learning algorithms that make use of both labeled data and unlabeled data for learning tasks. Examples include problems such as speech recognition, website classification, and discovering folding structure of proteins. In many problems there is an abundance of unlabeled data, while labeled data often requires expert labeling and is expensive to obtain. This has led to a resurgence of semi-supervised learning techniques, which use the topological or geometric properties of large amounts of unlabeled data to aid the learning task. In this talk, I will discuss some new rigorous PDE scaling limits for existing semisupervised learning algorithms and their practical implications. I will also discuss how these scaling limits suggest new ideas for fast algorithms for semi-supervised learning.

]]>

Speaker: Adam Kanigowski

http://www.adkanigowski.cba.pl/en.php

Abstract: Parabolic dynamical systems are systems of intermediate (polynomial) orbit growth. Most important classes of parabolic systems are: unipotent flows on homogeneous spaces and their smooth time changes, smooth flows on compact surfaces, translation flows and IET's (interval exchange transformations). Since the entropy of parabolic systems is zero, other properties describing chaoticity are crucial: mixing, higher order mixing, decay of correlations.

One of the most important tools in parabolic dynamics is the Ratner property (on parabolic divergence), introduced by M. Ratner in the class of horocycle flows. This property was crucial in proving famous Ratner's rigidity theorems in the above class.

We will introduce generalisations of Ratner's property for other parabolic systems and discuss it's consequences for chaotic properties. In particular this allows to approach the Rokhlin problem in the class of smooth flows on surfaces and in the class of smooth time changes of Heisenberg nilflows.

]]>

Speaker: Yannis Kevrekidis (Chem and Biomolec Eng. and Applied Math and Stat, Johns Hopkins)

Abstract: Obtaining predictive dynamical equations from data lies at the heart of science and engineering modeling, and is the linchpin of our technology. In mathematical modeling one typically progresses from observations of the world (and some serious thinking!) first to equations for a model, and then to the analysis of the model to make predictions.

Good mathematical models give good predictions (and inaccurate ones do not) - but the computational tools for analyzing them are the same: algorithms that are typically based on closed form equations. While the skeleton of the process remains the same, today we witness the development of mathematical techniques that operate directly on observations -data-, and appear to circumvent the serious thinking that goes into selecting variables and parameters and deriving accurate equations. The process then may appear to the user a little like making predictions by "looking in a crystal ball". Yet the "serious thinking" is still there and uses the same -and some new- mathematics: it goes into building algorithms that "jump directly" from data to the analysis of the model (which is now not available in closed form) so as to make predictions. Our work here presents a couple of efforts that illustrate this ``new” path from data to predictions. It really is the same old path, but it is travelled by new means.

]]>

Speaker: Behrang Forghani (University of Connecticut) - https://sites.google.com/site/behrangforghani/

Abstract: TBA

]]>

Speaker: Fei Lu (John Hopkins University) - http://www.math.jhu.edu/~feilu/

Abstract: The need to develop reduced nonlinear statistical-dynamical models from time series of partial observations of complex systems arises in many applications such as geophysics, biology and engineering. The challenges come mainly from memory effects due to the nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete data.

We address these challenges by introducing a discrete-time stochastic parametrization framework, in which we infer nonlinear autoregression moving average (NARMA) type models to take the memory effects into account. We show by examples that the NARMA type stochastic reduced models that can capture the key statistical and dynamical properties, and therefore can improve the performance of ensemble prediction in data assimilation. The examples include the Lorenz 96 system (which is a simplified model of the atmosphere) and the Kuramoto-Sivashinsky equation of spatiotemporally chaotic dynamics.

]]>

Speaker: Matthew Yu (UMD) -

Abstract: We will introduce the concept of a topological field theory, namely Chern-Simons theory, and describe show how some results of the quantum Hall effect can be derived from the Chern-Simons action. I will be following Tong with some additional remarks on topological field theories.

]]>

Speaker: Abram Kagan (UMCP) -

]]>