Department of

# Mathematics

Seminar Calendar
for events the day of Monday, October 19, 2015.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, October 19, 2015

4:00 pm in 245 Altgeld Hall,Monday, October 19, 2015

#### Descriptive set theory and classification problems in mathematics

###### Anush Tserunyan (Department of Mathematics, University of Illinois)

Abstract: Descriptive set theory combines techniques from set theory, topology, analysis, recursion theory, and combinatorics, to study definable subsets of Polish spaces. Examples of such sets include Borel, analytic (projection of Borel), co-analytic (complement of analytic), etc. For the past 25 years, a major focus of descriptive set theory has been the study of equivalence relations on Polish spaces that are definable when viewed as sets of pairs; e.g. orbit equivalence relations induced by continuous actions of Polish groups are analytic. This study provides appropriate framework and tools for understanding the nature of classification of mathematical objects (measure-preserving transformations, unitary operators, Riemann surfaces, etc.) up to some notion of equivalence (isomorphism, conjugacy, conformal equivalence, etc.), and measuring the complexity of such classification problems. Due to its broad scope, it has natural interactions with other areas of mathematics, such as ergodic theory and topological dynamics, functional analysis and operator algebras, representation theory, topology, etc. In this talk, I will give an introduction to this fascinating subject.

4:00 pm in 145 Altgeld Hall,Monday, October 19, 2015

#### What is ... tight closure?

###### Matthew Mastroeni (UIUC Math)

Abstract: Tight closure is a closure operation on the ideals of a ring of prime characteristic introduced by Hochster and Huneke in the late 1980's. Since its introduction, tight closure has been extremely successful in providing a unifying framework for results that had previously seemed unrelated while simultaneously greatly simplifying their proofs. The aim of this talk is to discuss some of the basic applications of tight closure theory, which are significant even if one is only interested in rings containing a field of characteristic zero.

5:00 pm in 245 Altgeld Hall,Monday, October 19, 2015

#### Some Recent Developments in Boundary Representations of Operator Systems

###### Stephen Longfield (UIUC Math)

5:00 pm in 141 Altgeld Hall (note the temporary time change!),Monday, October 19, 2015

#### Forcing and Baire category: an introduction (Part IV)

###### Anton Bernshteyn (UIUC Math)

Abstract: We continue the series of talks introducing the method of forcing and its interpretation through the lens of Baire category. Previously we discussed how, using a countable model $M$ of ZFC, one can define objects that are "generic" relative to $M$. Now we are going to learn how to adjoin such objects to $M$ in order to construct models of ZFC with certain desirable properties (for instance, models satisfying the Continuum Hypothesis or its negation).