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 Events
Johns Hopkins University
Thu 21 Mar 2019, 12:30pm
Institute of Applied Mathematics
ESB 4133
Minimal Sublinear Representations of Convex Sets
ESB 4133
Thu 21 Mar 2019, 12:30pm-1:30pm

Abstract

It is well-known that a closed convex set C containing the origin in its interior can be represented as the 1-sublevel set of its gauge function. If the set C is compact, then the gauge is the unique sublinear function whose 1-sublevel coincides with C. However, if C is not compact, there can be multiple different sublinear functions whose 1-sublevels coincide with C. We call any such function a sublinear representation of C. It is not hard to see that the gauge of C is the largest sublinear representation of C, with respect to pointwise dominance. We show that there is a unique smallest sublinear representation f^ of C, i.e., f <= f for any other sublinear representation f of C. The gauge, which is the largest sublinear representation of C, is well-known to be equal to the support function of the polar of C. We associate the notion of a “prepolar” with other sublinear representations and show that the geometric analog of the smallest sublinear representation is the concept of the smallest “prepolar”, with respect to set inclusion. This smallest “prepolar” has an explicit description, just like the classical polar.


Note for Attendees

Sushi served for lunch.
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UBC Mathematics
Fri 22 Mar 2019, 3:00pm
Department Colloquium
ESB 2012
Graduate Research Award: Essential dimension of representations of algebras
ESB 2012
Fri 22 Mar 2019, 3:00pm-4:00pm

Abstract

Let A be a finite-dimensional algebra. A fundamental theorem of Drozd shows that the complexity of the representation theory of A belongs to exactly one of three rather distinct classes, called finite, tame or wild representation type. I will explain how the notion of essential dimension determines the representation type of A. I will go further and define new numerical invariants of A that refine the representation type of A. I will then determine these invariants explicitly in the special case of quiver algebras.

Note for Attendees

Refreshments will be served at 2:30 p.m. in ESB 4133 (Lounge).
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UBC
Mon 25 Mar 2019, 4:00pm
Algebraic Geometry Seminar
MATH 126
Minimal number of generators of an étale algebra
MATH 126
Mon 25 Mar 2019, 4:00pm-5:00pm

Abstract

O. Forster proved that over a ring R of Krull dimension d a finite module M of rank at most n can be generated by n+d elements. Generalizing this in great measure U. First and Z. Reichstein showed that any finite R-algebra A can be generated by n+d elements if each A\otimes_R k(\mathfrak{p}), for \mathfrak{p}\in \mathrm{MaxSpec}(R), is generated by n elements. It is natural to ask if the upper bounds can be improved. For modules over rings R. Swan produced examples to match the upper bound. Recently B. Williams obtained weaker lower bounds in the context of Azumaya algebras. In this paper we investigate this question for étale algebras. We show that the upper bound is indeed sharp. Our main result is a construction of universal varieties for degree-2 étale algebras equipped with a set of r generators and explicit examples realizing the upper bound of First & Reichstein. This is joint work with Ben Williams.
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University of Durham
Wed 27 Mar 2019, 2:45pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
TBD
ESB 4133 (PIMS Lounge)
Wed 27 Mar 2019, 2:45pm-3:45pm

Abstract

 
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Catherine Byrne
Department of Microbiology and Immunology, UBC
Wed 27 Mar 2019, 2:45pm
Mathematical Biology Seminar
ESB 4127
Understanding the drivers of Epstein-Barr virus shedding with HIV-1 co-infection
ESB 4127
Wed 27 Mar 2019, 2:45pm-3:45pm

Abstract

Epstein-Barr virus (EBV) is a ubiquitous infection worldwide and is associated with the development of several kinds of cancers. Rates of EBV replication and disease are higher in individuals who are co-infected with HIV-1; however, the causes of this remain unknown. Here, we developed a mathematical model to describe the dynamics of EBV infection within the tonsils and analyzed oral EBV shedding data in a cohort of adults from Uganda to predict the role of HIV-1 in determining infection severity. 
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