Department of

# Mathematics

Seminar Calendar
for events the day of Monday, September 21, 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, September 21, 2015

11:00 am in 443 Altgeld Hall,Monday, September 21, 2015

#### Cohomology theories associated to infinity-topoi, globally equivariant spectra, and elliptic cohomology

###### David Gepner (Purdue)

Abstract: In this talk I will explain how an infinity-topos equipped with a suitable ring or module object gives rise to a (co)homology theory, locally defined on the infinity-topos, and show how this recovers various versions of (co)homology. As an application, we will recover Schwede's category of global spectra as well as Lurie's construction of equivariant elliptic cohomology. Finally, in the presence of a ring structure, we will see how to find invertible and dualizable objects as well as maps which admit transfers with respect to these theories. This is joint work in progress with Thomas Nikolaus.

4:00 pm in 141 Altgeld Hall,Monday, September 21, 2015

#### Compact spaces as quotients of projective Fraïssé limits

###### Aristotelis Panagiotopoulos (UIUC Math)

Abstract: Projective Fraïssé structures were introduced by T. Irwin and S. Solecki and they were used to provide a very useful construction of a certain compact space known as the pseudo-arc. We develop a theory of projective Fraïssé limits in the Irwin-Solecki spirit which moreover support a dual structure. Let $\mathbf{K}$ be a totally disconnected, second countable, compact space. We prove that a subgroup G of $\mathrm{Homeo}(\mathbf{K})$ is closed in the compact-open topology if and only if it is the automorphism group of some dual topological Fraïssé limit $\mathbf{K}$ on domain $\mathbf{K}$. As an application we prove that every second countable, compact space is the the quotient of topological Fraïssé limit $\mathbf{K}$ with a closed equivalence relation on $\mathbf{K}$ that is definable in $\mathbf{K}$.

4:00 pm in 145 Altgeld Hall,Monday, September 21, 2015

#### What is...the Springer resolution?

###### Josh Wen (UIUC Math)

Abstract: The Springer resolution is a classical object in geometric representation theory. I'll first give a 'Morse-theoretic' primer on the flag variety and then delve into the basic constructions. Along the way, I hope to keep things somewhat concrete by illustrating what these constructions look like for the special linear group.

4:00 pm in 245 Altgeld Hall,Monday, September 21, 2015

#### Multiplying elements in group rings: algebra, analysis, and group theory

###### Igor Mineyev (Department of Mathematics, University of Illinois)

Abstract: Multiply two nonzero numbers. The result is always nonzero. What if we multiply two nonzero elements of a field? (An exercise to do before the talk.) What about the product of two nonzero elements of a ring? (Another exercise.) The humble "borderline question" is: what happens in the group ring RG, where G is a group and R is the field of real numbers? When G is torsion-free, this "basic" question is known as the Kaplansky zero-divisor conjecture. It is a major unsolved problem in algebra, widely open since 1950's. I will describe what little is known about the problem and discuss some related conjectures on the algebraic side. There is also a strong analytic version of this question, known as the Atiyah problem, stated in terms of the Murray-von Neumann dimension of certain Hilbert modules.

5:00 pm in 241 Altgeld Hall,Monday, September 21, 2015