Department of

Mathematics


Seminar Calendar
for Graduate Student Homotopy Seminar events the year of Thursday, December 7, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    November 2017          December 2017           January 2018    
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Monday, September 11, 2017

3:00 pm in 141 Altgeld Hall,Monday, September 11, 2017

What is tmf

Ningchuan Zhang   [email] (UIUC)

Abstract: In short, Topological Modular Forms (tmf) is a “universal elliptic cohomology theory”. More precisely, it is the global section of a sheaf of $E_\infty$-ring spectra over the moduli stack of (generalized) elliptic curves. In this talk, I’ll introduce tmf and sketch the construction of it (or really this sheaf of $E_\infty$ ring spectra). If time allows, I’ll also explain its relationship to classical modular forms.

Monday, September 18, 2017

3:00 pm in 141 Altgeld Hall,Monday, September 18, 2017

Framed Cobordism and Stable Homotopy Groups

Brian Shin   [email] (UIUC)

Abstract: I will introduce the notion of framed cobordism describe its connection to the homotopy groups of spheres. Using this connection, I will calculate several of these groups using geometry.

Monday, September 25, 2017

3:00 pm in 141 Altgeld Hall,Monday, September 25, 2017

An Invitation to Motivic Homotopy Theory

Daniel Carmody   [email] (UIUC)

Abstract: In this talk I’ll introduce some of the basic constructions in motivic homotopy theory while trying to give motivations for some of the more complex definitions. This will be largely based on Dan Dugger’s $Universal$ $Homotopy$ $Theories$ paper.

Monday, October 2, 2017

3:00 pm in 141 Altgeld Hall,Monday, October 2, 2017

Why Higher Categories come into Topological Field Theories

Nima Rasekh   [email] (UIUC)

Abstract: In this talk we introduce topological field theories and give various examples. The eventual goal of the talk is to see why it makes sense to use higher categories when studying topological field theories. No knowledge of any of these subjects is assumed.

Monday, October 9, 2017

3:00 pm in 141 Altgeld Hall,Monday, October 9, 2017

Cohomology operations for Landweber exact spectra

William Balderrama   [email] (UIUC)

Abstract: If $E$ is a complex-oriented spectrum, then we can obtain a formal group law on $\pi_\ast E$ from the map $\mathbb{CP}^\infty\times\mathbb{CP}^\infty\rightarrow\mathbb{CP}^\infty$ classifying the tensor product of line bundles. If $E$ is Landweber exact, this formal group law completely controls the structure of stable operations in $E$-cohomology. There are, however, many useful unstable cohomology operations. In this talk, I will try to say something about structures related to these.

Monday, October 16, 2017

3:00 pm in 141 Altgeld Hall,Monday, October 16, 2017

Motivating Higher Topos Theory

Joseph Rennie   [email] (UIUC)

Abstract: In this talk I will try to argue for the necessity of higher topos theory without using the word “derived” once. I will very briefly discuss the various demonstrations of possibility, but will mostly focus on two aspects which make them interesting: object classifiers, and descent. If time permits, I will give a Weil’ld outline of a connection to physics.

Monday, October 23, 2017

3:00 pm in 141 Altgeld Hall,Monday, October 23, 2017

Loop Spaces and Chiral Differential Operators

Josh Wen   [email] (UIUC)

Abstract: The Witten genus is an invariant of a manifold $M$ that is morally gotten by doing index theory on $LM$. It would be lovely to turn ‘morally’ into ‘actually’, and hence the interest in geometric constructions of the Witten genus. One possible approach is via the ring of chiral differential operators (CDOs) on $M$, introduced for complex $M$ by Malikov, Schechtman, and Vaintrob. This is a sheaf of vertex algebras on $M$ whose character yields the Witten genus. Of course, any old bundle with the right numerical data can do this—the interesting part is to relate the construction to loop space. I’ll introduce the bare minimum about vertex algebras to discuss the obstruction theory to gluing together a global sheaf of CDOs, wherein loopy things like local lifts of Wess-Zumino forms and ‘curvature' 3-forms appear.

Monday, November 6, 2017

3:00 pm in 141 Altgeld Hall,Monday, November 6, 2017

A Vietoris-Smale Mapping Theorem for the Homotopy of Hyperdefinable Sets

Elliot Kaplan   [email] (UIUC)

Abstract: I will discuss the paper $``$A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets$''$ by Alessandro Achille and Alessandro Berarducci $[\href{https://arxiv.org/abs/1706.02094}{arXiv}]$. In this paper, they compare the definable homotopy groups of a definable space $X$ (denoted $\pi_n(X)^\mathrm{def}$) with the (usual) homotopy groups of the quotient $X/E$, where $E$ is a certain kind of type-definable equivalence relation on $X$. Under certain assumptions, these groups are actually isomorphic. These types of quotients were first studied by Pillay in the case that $X$ is actually a definable group. No knowledge of model theory will be assumed. In particular ``definable,'' ``type-definable'' and ``o-minimal'' will all be defined.

Monday, November 13, 2017

3:00 pm in 141 Altgeld Hall,Monday, November 13, 2017

Miller’s Proof of the Height 1 Telescope Conjecture

Dominic Culver   [email] (UIUC)

Abstract: In their attempt to find a better qualitative understanding of the stable homotopy groups of spheres, Morava and Miller-Ravenel-Wilson found that there are pervasive periodic phenomena in the Adams-Novikov E_2-term. These phenomena were originally understood only algebraically, but in 1984, Doug Ravenel attempted to provide topological origins for these algebraic periodicities. He asked, the now famous, telescope conjecture, which in essence, asks to what degree the topological periodic phenomena is controlled by the geometry of formal groups. In this talk, I will give a precise formulation of the telescope conjecture and sketch Miller’s proof of the height 1 telescope conjecture at odd primes. I will also sketch the basic ideas of chromatic homotopy theory along the way.

Monday, November 27, 2017

3:00 pm in 141 Altgeld Hall,Monday, November 27, 2017

Nilpotence in Motivic Homotopy Theory

Tsutomu Okano   [email] (UIUC)

Abstract: This talk will be in two (currently unrelated) parts. First I will construct the stable motivic homotopy category by defining the category of GL spectra which is an analogue of the category of orthogonal spectra from topology. I will mention some nice properties of this category and give examples of motivic GL spectra. In the second part of the talk I will introduce the question of nilpotence in motivic homotopy theory. This seems to be a difficult problem since Nishida nilpotence and the Devinatz-Hopkins-Smith theorems do not carry over immediately. I will talk about some approaches that are being used to shed light on these questions.

Monday, December 4, 2017

3:00 pm in 141 Altgeld Hall,Monday, December 4, 2017

Oriented Ring Spectra and Formal Group Laws

Liz Tatum   [email] (UIUC)

Abstract: Each complex-oriented ring spectrum has a corresponding formal group law. I will discuss some properties of these formal group laws, and give some examples of how they are used for computations involving complex-oriented cohomology theories.