Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, October 24, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 24, 2017

11:00 am in 345 Altgeld Hall,Tuesday, October 24, 2017

Categories for $K$-theory and Devissage

Jonathan Campbell (Vanderbilt)

Abstract: What sorts of categories can K-theory be defined for? We know that exact categories and Waldhausen categories can be used as appropriate input. However, there are geometric categories where we would like to define K-theory where we are only allowed to ``cut and paste" rather than quotient --- examples of these include the category of varieties, and the category of polytopes. I'll define a more general context where one may talk about the algebraic K-theory of these categories, and outline a proof of a general version of Quillen's devissage. I'll outline applications to studying "derived motivic measures" and the scissors congruence problem. This is joint work with Inna Zakharevich.

12:30 pm in 222 Loomis,Tuesday, October 24, 2017

Entropic A theorem and the Markov property of the vacuum

Eduardo Teste (Centro Atomico de Bariloche, Argentina)

Abstract: A state is said to be Markovian if it fulfil the important condition of saturating the Strong Subadditivity inequality. I will show how the vacuum state of any relativistic QFT is a Markov state when reduced to certain geometric regions of spacetime. For the CFT vacuum, the Markov property is the key ingredient to prove the A theorem (irreversibility of the RG flow in relativistic QFT in d=4 spacetime dimensions) using vacuum entanglement entropy. This extends the entropic proofs of the c and F theorems in dimensions d=2 and d=3, giving a unified picture. I will also comment on the relation of this Markov property with the unitarity bound and holography.

1:00 pm in 345 Altgeld Hall,Tuesday, October 24, 2017

Cancelled

2:00 pm in 243 Altgeld Hall,Tuesday, October 24, 2017

Greedy bases and Lebesgue-type constants

Pablo Berna (University of Murcia)

Abstract: Given a basis $\mathcal{B}:=(e_j)_{j=1}^\infty$ of a Banach space $\mathbb{X}$, the $Greedy$ $Algorithm$ $\lbrace \mathcal{G}_m\rbrace_{m=1}^\infty$ is an algorithm of approximation where the objective is to approximate each element $x\in\mathbb{X}$ by $\mathcal{G}_m(x) = \sum_{j\in A(x)}e_j^*(x)e_j$, where $A(x)$ is a set of $m$ indices associated with the largest coefficients of $x$ in absolute value. In this talk we will study the $Greedy$ $Bases$ and we will obtain new bounds of the called $Lebesgue-type$ $constants$ for the greedy approximation. These new bounds are given only in terms of the upper democracy functions of the basis and its dual and we will also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces.

2:00 pm in 347 Altgeld Hall,Tuesday, October 24, 2017

Multifractal analsyis of jump diffusion processes

Xiaochuan Yang (Michigan State)

Abstract: In this talk, I explain how to use tools from analysis and fractal geometry to describe the sample paths regularity of the solution of SDE with jumps. The key probabilistic argument is a new upper tail estimate for the increments of the solution. Examples include non degenerate stable driven SDEs and variable order stable-like processes in the sense of R. Bass.

3:00 pm in 241 Altgeld Hall,Tuesday, October 24, 2017

How to construct good voting trees

Adam Wagner (Illinois Math)

Abstract: A well-known question in social choice theory is the following: given a collection of n candidates which are each pairwise comparable, how should we select a winner? The pairwise comparisons between candidates may be encoded as a tournament of n vertices, with edges directed from the winner to the loser in each comparison. The subtlety of the problem lies in the fact that the tournament need not be transitive, and therefore there is no indisputable way to select a global winner. One natural way of selecting a winning candidate from a tournament is to use a voting tree, a complete binary tree with leaves labelled from [n] (with possible repeats). Given any tournament as input, a voting tree deterministically selects a winner from the tournament by running pairwise elections between the leaves until we arrive at the root node. We want the winning candidate to beat many other candidates, so we define the performance guarantee of a voting tree as the minimum out-degree of any winner that it produces. We survey the question of how to construct good voting trees using deterministic and random methods and what the limitations of our techniques are.

3:00 pm in 243 Altgeld Hall,Tuesday, October 24, 2017

Degree bounds for invariant rings of quivers

Visu Makam (University of Michigan)

Abstract: The ring of polynomial invariants for a rational representation of a reductive group is finitely generated. Nevertheless, it remains a difficult task to find a minimal set of generators, or even a bound on their degrees. Combining ideas originating from Hochster, Roberts and Kempf with the study of various ranks associated to linear matrices, we prove "polynomial" bounds for various invariant rings associated to quivers. The polynomiality of our bounds have strong consequences in algebraic complexity, notably a polynomial time algorithm for non-commutative rational identity testing. This is joint work with Derksen.

4:00 pm in 245 Altgeld Hall,Tuesday, October 24, 2017

Town Hall Meeting

Abstract: All members of the department are invited to this town hall meeting organized by the Department Chair Search Committee.