Department of Mathematics and Statistics,

University of Ottawa

**Coffee:** Common Room, before the colloquium at 3:30pm.

(Carleton-uOttawa transportation information)

- Speaker: David McDonald (UOttawa)
- Title: Yaglom limits can depend on the initial state
- Abstract: To quote the economist John Maynard Keynes: "The long run is a misleading guide to current affairs. In the long run we are all dead." It makes more sense to study the state of an evanescent system given it has not yet expired. For a substochastic Markov chain with kernel $K$ on a state space $S$ with killing this amounts to the study of the Yaglom limit; that is the limiting probability the state at time $n$ is $y$ given the chain has not been absorbed; i.e. $\lim_{n\to\infty}K^n(x,y)/K^n(x,S)$. We give an example where the Yaglom limit depends on the starting state $x$. We explain this phenomenon by showing that when the $\rho$-Martin entrance boundary is non-trivial the Yaglom limit may depend on the starting state of the Markov chain. The proof involves an analysis of the space-time $\rho$-Martin entrance boundary.

- Speaker: Elena Fimmel (Mannheim University of Applied Sciences, Germany)
- Title: Error-Detecting Genetic Codes
- Abstract: The genetic code is the major tool that nature uses for the transmission
of information. Several approaches from communication theory, mathematics
and physics have been proposed to explain the structure and the
functionality of the genetic code. One of the approaches is based on the
finding in 1996 of circular codes in large genes of prokaryotes and
eukaryotes by Arqus and Michel. Circular codes are subsets of the set of
codons that seem to be used by nature in order to eventually detect
frame-shift errors in the translation process. They are weaker versions of
the comma-free codes, proposed by Crick (co-discoverer of the helical
structure of the DNA) in 1957, that can detect frame-shift errors
immediately.

In the present talk, after a necessary biological preliminary, five hierarchically ordered classes of trinucleotide codes, including comma-free and circular codes, will be introduced. This hierarchical representation is based on a useful criterion for circularity. As a further application of this criterion, it will be shown that the so-called RNY-primeval code, from which the modern genetic code is believed to have originated, is circular (and even comma-free). Besides, it will be shown that it is impossible to encode all twenty amino acids with codes from four out of five hierarchical classes that have the strongest error-correcting properties, a fact previously known only empirically for comma-free and circular codes.

- Speaker: Alexander Premet (University of Manchester, UK)
- Title: Classification of maximal subalgebras of exceptional simple Lie algebras
- Abstract: I will speak on the classification of maximal subalgebras of exceptional simple Lie algebras (such as E8) over fields of good characteristic. I will discuss the rich history of the problem of classifying maximal subalgebras and subgroups (Sophus Lie, Dynkin, Seitz, etc). The subject is suitable for a general mathematical audience.
- Short bio: Alexander Premet is a full professor at the University of Manchester, UK.
He received his PhD in 1984 from one of the top Mathematical Institutes of the former Soviet Union - the Institute of Mathematics of BSSR, Minsk. His thesis 'Quadratic elements of algebraic groups and Lie algebras of Cartan type' was written under the supervision of A.E. Zalesskii. From 1984 till 1992 he was a permanently employed at the
Institute of Mathematics of BSSR as a research fellow. From 1992-1994 he joined the faculty of the University of California, Riverside, as an assistant professor.
In 1995 he moved to the University of Manchester, UK.

Professor Premet have held a number of visiting positions at leading Universities and Institutes, e.g., MSRI-Berkeley, Cambridge, MPIM-Bonn, Hamburg, University of Wisconsin Madison, etc. He published 52 papers in peer-reviewed journals; 4 of them in Inventiones Math., several in other top journals, like USSR Izvestiya, Crelles, Compositio, Advances, etc. He served on the editorial boards of Journal of Algebra, London Math. Society Journals, Transformation Groups. He refereed numerous grant proposals for American, British, Canadian, Israeli and Swiss Research Councils. He is currently a member of the EPSRC Peer Review College.

- Speaker: Dr. Hsien-Kuei Hwang, Institute of Statistical Science, Academia Sinica, Taiwan.
- Title: Maxima in multivariate samples and related structures
- Abstract: Dominance-based maxima of multidimensional samples have been introduced in many scientific and engineering disciplines with different names and under diverse guises: they are sometimes referred to as nondominance, records, outer layers, efficiency, or noninferiority but are more frequently known as Pareto optimality or Pareto efficiency in econometrics, engineering, multi-objective optimization, decision making, etc. Other terms used with essentially the same denotation include admissibility in statistics, Pareto front (and the corresponding notion of elitism) in evolutionary algorithms, and skyline in database language. In this talk, I will give a brief survey on the diverse aspects (applications, algorithms and theory) of maxima in multivariate samples with a special highlight on probabilistic properties. Graduate students are encouraged to attend.

- Speaker: Konstantina Trivisa (University of Maryland)
- Title: On mechanical models for tumor growth: modeling, analysis and simulations
- Abstract: Tumor growth modeling is the investigation of the complex dynamics of cancer progression using a mathematical formulation. Internal dynamics of tumor cells, their interactions with each other and with their surrounding tissue, transfer of chemical substances and many phenomena are typically encoded in mathematical models. This mathematical formulation relies on biological and clinical observations. At the same time, it is currently clear that mathematics could make a huge contribution to many areas of experimental cancer investigation since there is now a wealth of experimental data which requires systematic analysis.

In this research project we investigate the evolution of tumor growth relying on a non-linear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the role of nutrient and drug application in the evolution of cancerous cells. We construct an explicit convergent numerical scheme to approximate solutions of the nonlinear system by employing compactness methods in the spirit of P.L. Lions. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The talk will present results obtained in collaboration with D. Donatelli and F. Weber. - Short bio: Prof. Trivisa is an applied mathematician holding a joint position at the Department of Mathematics and the Institute for Physical Science and Technology at the University of Maryland. Her research lies on the interface between nonlinear partial differential equations and continuum physics and focuses on applications in fluid dynamics, multiphase flows, continuum mechanics, materials science and math biology. She has been recognized by a series of awards including an Alfred P. Sloan Research Fellowship, The Faculty Early Career Award, The Presidential Early Career Award for Scientists and Engineers (PECASE) as well as a Simons Foundation Fellowship. She was also selected as ADVANCE Professor and Leadership Fellow at the University of Maryland for her work on issues of diversity and inclusion. Prof. Trivisa is a director of the Applied Mathematics & Statistics, and Scientific Computation Program (AMSC), which is the largest interdisciplinary program in the country and is ranked among the top 10 applied math programs in the US. She is currently serving as Associate Director of the Institute for Physical Science and Technology (IPST), an internationally recognized center for interdisciplinary research in emerging areas at the boundaries between physical, mathematical and life sciences, and engineering.

- Speaker: Rita Gitik (University of Michigan)
- Title: On Intersection of Conjugate Subgroups
- Abstract: We present an algorithm deciding if the intersection of a quasiconvex subgroup of a negatively curved group with a conjugate is infinite. We will give the necessary background and discuss some relevant mathematical history. The talk will be accessible for graduate students.

- Speaker: Mihai Halic (CRM)
- Title: Relative semi-stability for vector bundles
- Abstract: We investigate the relationship between absolute and relative semi-stability for vector bundles on projective varieties. The results are used to explicitly describe the moduli spaces of semi-stable vector bundles on Hirzebruch surfaces. The talk is aimed at a general audience.

- Speaker: Zenghu Li (Beijing Normal University)
- Title: Stochastic equations for branching processes
- Abstract: A continuous-state branching process is the mathematical model for the evolution of a large population of small individuals. The process can be constructed as the strong solution to a stochastic integral equation driven by Gaussian and Poisson time-space noises. The genealogical structures of the population are represented by continuum random trees. More general population models take into consideration the influence of immigration, competition, environments and so on. The research in the subject has been undergoing rapid development and has led to better understanding of deep structures including Brownian excursions, stochastic flows, Levy trees and planar maps. In this talk, we present a number of stochastic integral equations in the theory of continuous-state branching processes. We explain how the equations can be used in the study the structural properties of the model.
- Short bio: Professor Li is currently Head of the School of Mathematical Sciences, Beijing Normal University. He has been a frequent visitor to Carleton University and has published extensively in probability including his book Measure-valued Branching Markov Processes, Springer, 2011.

- Speaker: Leah Edelstein-Keshet (UBC)
- Title: Models for cell signaling and regulation of cell shape
- Abstract: The shape of eukaryotic cells is determined by structural proteins (such as actin) collectively known as the cytoskeleton. For example, localized assembly of filamentous actin (F-actin) leads to protrusion of parts of the cell outwards, whereas activation of myosin powers contraction. Here I will talk about our recent mathematical modeling of cellular signaling. We study the signaling that regulates the assembly of F-actin, and the resultant effects on dynamics of cell shape, cell motility, and single-cell wound healing. I will describe how we used ordinary and partial differential equations to investigate the activities of regulatory proteins (small GTPases) that regulate the cytoskeleton, and how our methods helped to gain insight into mechanisms underlying experimentally observed cellular behaviour.
- Short bio: Leah Edelstein-Keshet is a full-time professor at the University of British Columbia. She wrote the SIAM book Mathematical Models in Biology. In 1995 she became the president of the Society for Mathematical Biology. In 2003 she was awarded the Krieger-Nelson Prize of the Canadian Mathematical Society.

- Speaker: Henri Darmon (McGill University)
- Title: Modular forms and their special values
- Abstract: I will discuss some basic ideas in the theory of modular forms and of complex multiplication, leading to the explicit construction of class fields (i.e., abelian extensions) of imaginary quadratic fields from special values of modular functions at quadratic imaginary irrationalities. I will then describe a conjectural approach for extending this theory to the setting of real quadratic fields, which grew out of joint work with Dasgupta, Charollois, Logan, Trifkovic some 15 years ago.
- Short bio:
Professor Henri Darmon is specializing in number theory. He works on Hilbert's 12th problem and its relation with the Birch-Swinnerton-Dyer conjecture. He is currently a James McGill Professor of Mathematics at McGill University.
He received his B.Sc from McGill University in 1987 and his Ph.D from Harvard University in 1991 under supervision of Benedict Gross. From 1991 to 1996, he held positions in Princeton University. Since 1994, he has been a professor at McGill University.
Research-based distinctions:

1990-91. Sloan Doctoral Dissertation Fellowship.

1996-98. Alfred P. Sloan Research Award.

1996. G. De B. Robinson Award.

1997. Prix Andre Aisenstadt.

1998. Coxeter-James Prize of the Canadian Mathematical Society.

2002. E.W.R. Steacie Memorial Fellowship

2002. Ribenboim Prize, Canadian Number Theory Association.

2003. Earle Raymond Hedrick Lecturer of the MAA

2003. Elected fellow of the Royal Society of Canada

2008. Killam Fellowship of the Canada Council of the Arts.

2008. John L. Synge award of the Royal Society of Canada.Member of Editorial Boards:

1999-2005: Journal of Number Theory.

2001-2005: Editor-in-chief, Canadian Journal of Mathematics

2003-present. Commentari Mathematici Helvetici

2005-2010. International Journal of Number Theory.

2015-present. The Ramanujan Journal.

2016-present. Transactions and Memoirs of the AMS.

- Speaker: Gregory Smith (Queens University)
- Title: Sum-of-Squares Certificates on Real Curves
- Abstract: How can one use sums of squares to characterize nonnegative polynomials? In this talk, we will review some general methods for certifying that a polynomial is nonnegative on a real projective subvariety. We will then present new optimal degree bounds for certificates on real projective curves. This talk is based on joint work with Grigoriy Belkherman and Mauricio Velasco.
- Short bio:
Prof. G. Smith's research focuses on combinatorial varieties, the fundamental objects at the interface between algebra, combinatorics and geometry. Combinatorial varieties account for a large number of the important geometric objects that arise in commutative algebra, representation theory, and mathematical physics, and their explicit nature makes them a good testing ground for general theories and conjectures, as well as computational experimentation. Prof. Smith is also noted for his many contributions to Macaulay2, a software system that supports research in algebraic geometry and commutative algebra. The research tools he has developed for the system are particularly valuable for collecting heuristic evidence, establishing patterns, and exploring pathologies, and they have found a broad range of users including physicists, algebraists and geometers.

Prof. Gregory Smith received his BSc from Queen's University in 1995, his MA from Brandeis University in 1997, and his PhD from the University of California at Berkeley in 2001. He is presently a professor in the Department of Mathematics and Statistics at Queen's University. He has held postdoctoral and visiting positions at Columbia University in New York, the Mathematical Sciences Research Institute (MSRI) in Berkeley, the Royal Institute of Technology (KTH) in Stockholm, and the Mittag-Leffler Institute in Sweden. In 2007, he was the recipient of the Andre-Aisenstadt Prize from the Centre de Recherches Mathematiques (CRM) and the 2012 Coxeter-James Prize of the Canadian Mathematical Society.

- Speaker: Nassif Ghoussoub (UBC)
- Title: Optimal Mass transport as a natural extension of classical mechanics to the manifold of probability measures
- Abstract: I will describe how deterministic and stochastic dynamic optimal mass transports are to Mean Field Games what the classical calculus of variations offers to classical mechanics.
- Short bio: Nassif Ghoussoub obtained his Doctorat d'Žtat in 1979 from the UniversitŽ Pierre et Marie Curie in Paris, France. His present research interests are in non-linear analysis and partial differential equations. He is currently a Professor of Mathematics, a "Distinguished University Scholar", and an elected member of the Board of Governors of the University of British Columbia for the period 2008-2013. He was the founding Director of PIMS (Pacific Institute for the Mathematical Sciences) for the period 1996-2003, a co-founder of the MITACS Network of Centres of Excellence (Mathematics of Information Technology and Complex Systems) and a member of its Board of Directors for the periods 1998-2003 and 2008-16. He is also the founder of BIRS (Banff International Research Station) and has been its Scientific Director since 2004. In June 2011, he became the Scientific Director of the MPrime network of Centres of Excellence. He was elected Fellow of the Royal Society of Canada in 1993, and was appointed Officer of the Order of Canada in December 2015.

- Speaker: Vladimir Chernousov (University of Alberta)
- Title: Can one hear the shape of a drum?
- Abstract: The title of my talk is due to Mark Kac who published the famous paper with the same title in the American Mathematical Monthly in 1966. In differential geometry, given a Riemannian manifold M one considers the following sets of data: spectrum of the Laplace operator (eigenvalues with multiplicities) and length spectrum (lengths of closed geodesics). Then it is naturally to ask if these data determines our manifold M uniquely up to isometry or at least up to a finite-sheeted cover. ``Identifying a Riemannian manifold with a drum'' one can think of spectrum of its Laplace operator and length spectrum as discrete characteristics (frequency, amplitude and etc.) of the sound wave produced by the drum when you hit it with a hammer. Hence we come in a naural way to the question stated in the title of the talk: can one hear the shape of a drum? This question received considerable attention in geometry. In the talk we first explain how it leads us to some natural problems and conjectures in algebra related to a question whether a noncommutative object (like a quaternion algebra) can be determined by its abelian subobjects. Then we will survey recent developments.
- Short bio: Prof. Chernousov has received his PhD from the University of Minsk in 1983. During 1990's he held numerous visiting position at EPFL, Lausanne, ETH Zurich, Alexander von Humboldt Fellow at the University of Bielefeld, Germany; University of Munster. Since 2003 he is a professor at the University of Alberta and a Canada Research Chair Tier I in Algebra.

- Speaker: Peter Latham (King's College London)
- Title: Recent progress in the local Langlands programme
- Abstract: The main objective of modern algebraic number theory is to understand the absolute Galois group of the rational numbers, which has an incredibly rich structure. As with many problems in number theory, this breaks into a family of local problems, namely the study of a certain decomposition subgroup for each prime number p. The local Langlands programme is an attempt to understand each of these local problems, by comparing the representation theory of these decomposition groups to that of p-adic Lie groups. I?ll give an introduction to the motivation behind and history of this programme, and explain some spectacular recent progress in using it to obtain explicit new results on the structure of Galois groups.

- Speaker: Martina Lanini (Università di Roma Tor Vergata)
- Title: Combinatorics of Schubert varieties in representation theory
- Abstract: Born in the 19th century to answer questions from enumerative geometry, Schubert varieties have been extensively studied since then. Lying in the intersection of geometry, combinatorics and algebra, these varieties naturally bridge problems in these different areas. I will discuss some instances in which geometric and topological properties of Schubert varieties encode important representation theoretical information. In particular, I will explain how "moment graphs", certain combinatorial gadgets attached to any Schubert variety, help in the study of these relevant properties and, hence, provide new tools to attack representation theoretical questions.