Mathematics Dept.
  Events
USC
Wed 23 Jan 2019, 2:45pm
Topology and related seminars
ESB 4133
Fibrant resoultions of motivic Thom spectra
ESB 4133
Wed 23 Jan 2019, 2:45pm-3:45pm

Abstract

This is a joint work with G.Garkusha. In the talk I will discuss the construction of fibrant replacements for spectra consisting of Thom spaces (suspension spectra of varieties and algebraic cobordism MGL being the motivating examples) that uses the theory of framed correspondences. As a consequence we get a description of the infinite loop space of MGL in terms of Hilbert schemes.
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Frederic Paquin-Lefebvre
Mathematics, UBC
Wed 23 Jan 2019, 2:45pm
Mathematical Biology Seminar
ESB 4127
The dynamics of diffusively coupled oscillators
ESB 4127
Wed 23 Jan 2019, 2:45pm-3:45pm

Abstract


When two identical nonlinear oscillators are coupled through a 1-D bulk diffusion field, new patterns of synchronization occur that would be absent in the uncoupled system. Furthermore, if the two oscillators are quiescent, the effect of the coupling can be to turn the oscillations on. Mathematically, the models consist of systems of nonlinear ODEs coupled with linear diffusive PDEs. Through a detailed bifurcation analysis of three different examples, we reveal some of the underlying mechanisms behind phenomena as diverse as the diffusion sensing of reacting agents, the synchronization of chaotic oscillations and the formation of membrane-bound patterns at the cell-scale level.



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Dr Christian Hilbe
IST Austria
Wed 23 Jan 2019, 4:00pm
Department Colloquium
MATH 100
Modeling the dynamics of extortion and cooperation in iterated games
MATH 100
Wed 23 Jan 2019, 4:00pm-5:00pm

Abstract

Iterated games are the baseline model to explain how cooperation can evolve in repeated interactions. The basic idea is that individuals are more likely to cooperate if they can expect their beneficiaries to remember and to return their cooperative acts in future. However, six years ago, William Press and Freeman Dyson have shown that certain repeated games also allow individuals to employ extortionate strategies. By using an extortionate strategy, players can guarantee that they systematically outperform their opponent, irrespective of the opponent’s reaction. In this talk, I will first present a simple proof for the existence of these extortionate strategies. This proof applies to both, finitely and infinitely repeated games with arbitrarily many players. I will then discuss under which circumstances such strategies can emerge in an evolutionary process, and how successful they are against actual human players. In the end, I will discuss a few interesting generalizations and open problems.

Note for Attendees

Pre-colloquium refreshments will be served in MATH 125 at 3:45 p.m.
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Dr Heyrim Cho
University of Maryland, College Park
Thu 24 Jan 2019, 2:15pm
MATH 126
Applied Stochastics Seminar: Numerical methods for uncertainty quantification - from noise parameterization to efficient simulation of parameterized stochastic systems
MATH 126
Thu 24 Jan 2019, 2:15pm-3:15pm

Details

Abstract:
For a reliable simulation of systems subject to noise, it is necessary to characterize the noise properly and develop efficient algorithms. In the first part of this talk, I will present an extension of Karhunen-Loeve (K-L) expansion to model and simulate multiple correlated random processes. The method finds the appropriate expansion for each correlated random process by releasing the bi-orthogonal condition of the K-L expansion. I will address the convergence and computational efficiency, in addition to some explicit formulae and analytical results. In the remaining talk, I will discuss numerical methods to effectively compute the propagation of uncertainty in parameterized stochastic differential equations. Joint response-excitation PDF equation generalizes the existing PDF equations and enables us to compute the PDF of the solution to system subject to non-Gaussian colored noise. An adaptive discontinuous Galerkin method combined with probabilistic collocation method is developed to resolve both local and discontinuous dynamics, while low-rank tensor method is employed in case of high-dimensionality. For anisotropic parameterized stochastic PDEs, we develop a reduced basis method using ANOVA decomposition to automatically identify the important dimensions and appropriate resolution in each dimension. The effectiveness of the methods is demonstrated in high-dimensional stochastic PDEs.
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Dr Heyrim Cho
University of Maryland, College Park
Fri 25 Jan 2019, 3:00pm
Department Colloquium
ESB 2012
Mathematical modeling from single-cell data and its implications in cancer development and drug resistance
ESB 2012
Fri 25 Jan 2019, 3:00pm-4:00pm

Abstract

Recent advances in single-cell gene sequencing data and high-dimensional data analysis techniques are bringing in new opportunities in modeling biological systems. In this talk, I discuss different approaches to develop mathematical models from single-cell data. For high-dimensional single-cell gene sequencing data, dimension reduction techniques are applied to find the trajectories of cell states in the reduced differentiation space, then modeled as directed and random movement on the abstracted graph with PDEs. Normal hematopoiesis differentiation and abnormal processes of acute myeloid leukemia (AML) progression are simulated, and the model can predict the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of AML. In addition, we develop representations of multi-correlated stochastic processes for correlated time series cell data, by releasing the bi-orthogonal condition of Karhunen-Loeve expansion. Convergence and computational efficiency of the methods are addressed. Finally, for fluorescence in situ hybridization data that provides spatial-temporal patterns of cells, we develop tumor growth model incorporating dynamics of drug resistance. It is demonstrated that assuming continuous cell state may result in different dynamics of anti-cancer drug resistance when compared with the predictions of classical discrete models, and its implications in designing therapies are studied.

Note for Attendees

Pre-colloquium refreshments will be served in ESB 4133 at 2:30 p.m.
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University of Alberta
Mon 28 Jan 2019, 4:00pm
Algebraic Geometry Seminar
MATH 126
Multiplicity-free products of Schubert divisors
MATH 126
Mon 28 Jan 2019, 4:00pm-5:00pm

Abstract

Let G/B be a flag variety over C, where G is a simple algebraic group
with a simply laced Dynkin diagram, and B is a Borel subgroup. The
Bruhat decomposition of G defines subvarieties of G/B called Schubert
subvarieties. The codimension 1 Schubert subvarieties are called
Schubert divisors. The Chow ring of G/B is generated as an abelian
group by the classes of all Schubert varieties, and is "almost"
generated as a ring by the classes of Schubert divisors. More
precisely, an integer multiple of each element of G/B can be written
as a polynomial in Schubert divisors with integer coefficients. In
particular, each product of Schubert divisors is a linear combination
of Schubert varieties with integer coefficients.

In the first part of my talk I am going to speak about the
coefficients of these linear combinations. In particular, I am going
to explain how to check if a coefficient of such a linear combination
is nonzero and if such a coefficient equals 1. In the second part
of my talk, I will say something about an application of my result,
namely, how it makes it possible estimate so-called canonical
dimension of flag varieties and groups over non-algebraically-closed
fields.
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Dr Nicholas Cook
Stanford University
Mon 28 Jan 2019, 4:00pm
Department Colloquium
MATH 100
Large deviations for sparse random graphs
MATH 100
Mon 28 Jan 2019, 4:00pm-5:00pm

Abstract

Let G=G(N,p) be an Erd\H{o}s--R\'enyi graph on N vertices (where each pair is connected by an edge independently with probability p). We view N as going to infinity, with p possibly going to zero with N. What is the probability that G contains twice as many triangles as we would expect? I will discuss recent progress on this ``infamous upper tail" problem, and more generally on tail estimates for counts of any fixed subgraph. These problems serve as a test bed for the emerging theory of \emph{nonlinear large deviations}, and also connect with issues in extending the theory of \emph{graph limits} to handle sparse graphs. In particular, I will discuss our approach to the upper tail problems via new versions of the classic regularity and counting lemmas from extremal combinatorics, specially tailored to the study of random graphs in the large deviations regime. This talk is based on joint work with Amir Dembo.

Note for Attendees

Pre-colloquium refreshments will be served in MATH 125 at 3:45 p.m.
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Aaron Berk
Mathematics, UBC
Tue 29 Jan 2019, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS lounge)
Parameter Instability Regimes in Sparse Proximal Denoising Programs
ESB 4133 (PIMS lounge)
Tue 29 Jan 2019, 12:30pm-1:30pm

Abstract

Compressed sensing theory explains why Lasso programs recover structured high-dimensional signals with minimax order-optimal error. Yet, the optimal choice of the program’s governing parameter is often unknown in practice. It is still unclear how variation of the governing parameter impacts recovery error in compressed sensing, which is otherwise provably stable and robust. We establish a novel notion of instability in Lasso programs when the measurement matrix is identity. This is the proximal denoising setup. We prove asymptotic cusp-like behaviour of the risk as a function of the parameter choice, and illustrate the theory with numerical simulations. For example, a 0.1% underestimate of a Lasso parameter can increase the error significantly; and a 50% underestimate can cause the error to increase by a factor of 109. We hope that revealing parameter instability regimes of Lasso programs helps to inform a practitioner’s choice. Finally, we discuss how these results extend to their more general Lasso counterparts.

Note for Attendees

A light lunch (sushi) will be served.
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Dr Nicholas Cook
Stanford University
Tue 29 Jan 2019, 3:30pm
Probability Seminar
MATH 126
Maximum of the characteristic polynomial for a random permutation matrix
MATH 126
Tue 29 Jan 2019, 3:30pm-4:30pm

Abstract

Statistics of the characteristic polynomial for large Haar unitary matrices U at points on the unit circle have received considerable attention due to similarities with the statistics of the Riemann zeta function far up the critical axis. While the best-known instances of this universality phenomenon concern statistics of \emph{zeros} for these functions (eigenvalues of U), there is strong evidence that the analogy also applies to extreme values.

Towards the more modest goal of understanding this universality phenomenon within the class of distributions on the unitary group, in this talk we consider the characteristic polynomial \chi_N(z) for an N\times N Haar permutation matrix. Our main result is a law of large numbers for (the logarithm of) the maximum modulus of \chi_N(z) over the unit circle. The main idea is to uncover a multi-scale structure in the distribution of the random field \chi_N(z), and to adapt a well-known second moment argument for the maximum of a branching random walk. Unlike the analogous problem for the Haar unitary, the distribution of \chi_N(z) is sensitive to Diophantine properties of the argument of z. To deal with this we borrow tools from the Hardy--Littlewood circle method in analytic number theory. Based on joint work with Ofer Zeitouni.
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