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Series: School of Mathematics Colloquium

Convex algebraic geometry is an emerging ﬁeld at the interface of convex optimizationand algebraic geometry. A primary focus lies on the mathematical underpinnings ofsemidefinite programming. This lecture oﬀers a self-contained introduction. Startingwith elementary questions concerning multifocal ellipses in the plane, we move on todiscuss the geometry of spectrahedra and orbitopes, and we end with recent resultson the convex hull of a real algebraic variety.

Series: School of Mathematics Colloquium

In the 1930's, Tarski introduced his plank problem at a time when the field
Discrete Geometry was about to born.
It is quite remarkable that Tarski's question and its variants continue to generate
interest in the geometric and analytic
aspects of coverings by planks in the present time as well. The talk is of a survey
type with some new results and with
a list of open problems on the discrete geometric side of the plank problem.

Series: School of Mathematics Colloquium

Euler's celebrated pentagonal numbers theorem is one themost fundamental in the theory of partitions and q-hypergeometric series.The recurrence formula that it yields is what MacMahon used to compute atable of values of the partition function to verify the deep Hardy-Ramanujanformula. On seeing this table, Ramanujan wrote down his spectacular partition congruences. The author recently proved two new companions to Euler'stheorem in which the role of the pentagonal numbers is replaced by the squares.These companions are deeper in the sense that lacunarity can be achievedeven with the introduction of a parameter. One of these companions isdeduced from a partial theta identity in Ramanujan's Lost Notebook and theother from a q-hypergeometric identity of George Andrews. We will explainconnections between our companions and various classical results such asthe Jacobi triple product identity for theta functions and the partitiontheorems of Sylvester and Fine. The talk will be accessible to non-experts.

Series: School of Mathematics Colloquium

Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as "chemotaxis" or "phototaxis." Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. Following these observations, in this talk we will present a hierarchy of mathematical models for phototaxis: a stochastic model, an interacting particle system, and a system of PDEs. We will discuss the models, their simulations, and our theorems that show how the system of PDEs can be considered as the limit dynamics of the particle system. Time-permitting, we will overview our recent results on particle, kinetic, and fluid models for phototaxis. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute), Tiago Requeijo (Math, Stanford), and Seung-Yeal Ha (Seoul, Korea).

Series: School of Mathematics Colloquium

n his 1994 ICM lecture, Borcherds famously introduced an entirely new conceptin the theory of modular forms. He established that modular forms with very specialdivisors can be explicitly constructed as infinite products. Motivated by problemsin geometry, number theorists recognized a need for an extension of this theory toinclude a richer class of automorphic form. In joint work with Bruinier, the speakerhas generalized Borcherds's construction to include modular forms whose divisors arethe twisted Heegner divisors introduced in the 1980s by Gross and Zagier in theircelebrated work on the Birch and Swinnerton-Dyer Conjecture. This generalization,which depends on the new theory of harmonic Maass forms, has many applications.The speaker will illustrate the utility of these products by resolving open problemson the following topics: 1) Parity of the partition function 2) Birch and Swinnerton-Dyer Conjecture and ranks of elliptic curves.

Series: School of Mathematics Colloquium

The purpose of this talk is to describe a variational approach to the problemof A.D. Aleksandrov concerning existence and uniqueness of a closed convexhypersurface in Euclidean space $R^{n+1}, ~n \geq 2$ with prescribed integral Gauss curvature. It is shown that this problem in variational formulation is closely connected with theproblem of optimal transport on $S^n$ with a geometrically motivated cost function.

Series: School of Mathematics Colloquium

There are presently different approaches to definealgebraic geometry over the mysterious "field with one element".I will focus on two versions, one by Soule' and one by Borger,that appear to have a direct connection to NoncommutativeGeometry via the quantum statistical mechanics of Q-latticesand the theory of endomotives. I will also relate to endomotivesand Noncommutative Geometry the analytic geometry over F1,as defined by Manin in terms of the Habiro ring.

Series: School of Mathematics Colloquium

The lecture will outline how the method of characteristics can
be used in the context of solutions to hyperbolic conservation laws that
are merely continuous functions. The Hunter-Saxton equation will be used
as a vehicle for explaining the approach.

Series: School of Mathematics Colloquium

Light refreshments will be available in Room 236 at 10:30 am.

A single round soap bubble provides the least-area way to enclose a given volume. How does the solution change if space is given some density like r^2 or e^{-r^2} that weights both area and volume? There has been much recent progress by undergraduates. Such densities appear prominently in Perelman's paper proving the Poincare Conjecture. No prerequisites, undergraduates welcome.

Series: School of Mathematics Colloquium

Much research in modern, quantitative seismology is motivated -- on
the one hand -- by the need to understand subsurface structures and
processes on a wide range of length scales, and -- on the other hand
-- by the availability of ever growing volumes of high fidelity
digital data from modern seismograph networks or multicomponent
acquisition systems developed for hydro-carbon exploration, and access
to increasingly powerful computational facilities. We discuss
(elastic-wave) inverse scattering of reflection seismic data,
wave-equation tomography, and their interconnection using techniques
from microlocal analysis and applied harmonic analysis. We introduce a
multi-scale approach and present a framework of partial reconstruction
in connection with limited boundary acquisition geometry. The formation of caustics
leads to one of the complications which will be discussed. We illustrate various
aspects of this research program with examples from global seismology and mineral
physics coupled to thermo-chemical convection.