Department of

Mathematics


Seminar Calendar
for Topology events the year of Monday, April 16, 2018.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, January 19, 2018

4:00 pm in 241 Altgeld Hall,Friday, January 19, 2018

Organizational Meeting

Abstract: We'll get the schedule of talks for the semester worked out. There will be cookies.

Friday, January 26, 2018

4:00 pm in 241 Altgeld Hall,Friday, January 26, 2018

Ice cream geometry: a mathematical activity and coloring book

Melinda Lanius (UIUC)

Abstract: Come explore metrics I use in my dissertation research. In a souped-up color-by-numbers, we'll develop a general notion of circle and ball. In geodesic connect-the-dots, we'll see what happens when straight lines curve. In metric mazes, we'll come to appreciate the wonky ways of nonhomogeneous spaces. After exploring the geometries of the real and hyperbolic plane, sphere, cone, and cylinder, we'll conclude by building less familiar objects: surfaces with a Euclidean, cylindrical, conical, or hyperbolic-funnel end. Please bring your fun office supplies!

Tuesday, January 30, 2018

11:00 am in 345 Altgeld Hall,Tuesday, January 30, 2018

Localizing the E_2 page of the Adams spectral sequence

Eva Belmont (MIT)

Abstract: The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. In this talk, I will discuss an approach for computing the Adams E_2 page at p = 3 in an infinite region, by computing its localization by the non-nilpotent element b_{10}. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory. This computation fits in the framework of chromatic homotopy theory in the stable category of comodules.

Friday, February 2, 2018

4:00 pm in 241 Altgeld Hall,Friday, February 2, 2018

Integrable systems in algebraic geometry

Matej Penciak (UIUC)

Abstract: In this talk I'll introduce the definition of an algebraic integrable system. The definition axiomatizes what it means to have an integrable system in algebraic geometry. After connecting the definition to the more common notion in differential geometry, I'll give a few examples of my favorite integrable systems. Possible examples are the Hitchin system on the space of Higgs bundles, the Calogero-Moser system, the Toda lattice hierarchy, and, if time permits, I'll try to give a hint of how all these systems are related via the gauge theory in physics.

Friday, February 9, 2018

4:00 pm in 241 Altgeld Hall,Friday, February 9, 2018

What are spectra?

Tsutomu Okano (UIUC)

Abstract: This is an introductory talk on stable homotopy theory. I will begin by discussing stable phenomena in homotopy theory that led to the definition of spectra. Spectra represent generalized cohomology theories, such as various kinds of K-theories and cobordism theories. This implies that results about cohomology theories can be proven in the category of spectra, using analogies from topological spaces. Unfortunately, the earlier category of spectra was defective in formal properties and it was not until the 1990's that suitable categories of spectra were defined. This formality leads to the study of other sorts of homotopy theories, such equivariant and motivic homotopy theories.

Tuesday, February 13, 2018

11:00 am in 345 Altgeld Hall,Tuesday, February 13, 2018

Factorization homology and topological Hochschild cohomology of Thom spectra

Inbar Klang (Stanford)

Abstract: By a theorem of Lewis, the Thom spectrum of an n-fold loop map to BO is an E_n-ring spectrum. I will discuss a project studying the factorization homology and the E_n topological Hochschild cohomology of such Thom spectra, and talk about some applications, such as computations, and a duality between topological Hochschild homology and cohomology of certain Thom spectra. Time permitting, I will discuss connections to topological field theories. This talk will include an introduction to factorization homology via labeled configuration spaces.

Friday, February 16, 2018

4:00 pm in 241 Altgeld Hall,Friday, February 16, 2018

Canceled

(UIUC)

Wednesday, February 21, 2018

12:00 pm in 443 Altgeld Hall,Wednesday, February 21, 2018

Bounding Betti numbers of patchworked real hypersurfaces by Hodge numbers

Kristin Shaw (MPI Leipzig)

Abstract: The Smith-Thom inequality bounds the sum of the Betti numbers of a real algebraic variety by the sum of the Betti numbers of its complexification. In this talk I will explain our proof of a conjecture of Itenberg which refines this bound for a particular class of real algebraic projective hypersurfaces in terms of the Hodge numbers of its complexification. The real hypersurfaces we consider arise from Viro’s patchworking construction, which is a powerful combinatorial method for constructing topological types of real algebraic varieties. To prove the bounds conjectured by Itenberg, we develop a real analogue of tropical homology and use spectral sequences to compare it to the usual tropical homology of Itenberg, Katzarkov, Mikhalkin, Zharkov. Their homology theory gives the Hodge numbers of a complex projective variety from its tropicalisation. Lurking in the spectral sequences of the proof are the keys to controlling the topology of the real hypersurface produced from a patchwork. This is joint work in preparation with Arthur Renaudineau.

Friday, February 23, 2018

4:00 pm in 241 Altgeld Hall,Friday, February 23, 2018

Unoriented Bordism and the Algebraic Geometry of Homotopy Theory

Brian Shin (UIUC)

Abstract: The classification of manifolds up to bordism is an interesting geometric problem. In this talk, I will discuss how this problem connects to homotopy theory. In particular, I will demonstrate that viewing the homotopy theorist's toolbox through the lens of algebraic geometry leads naturally to the solution for the classification of manifolds up to bordism.

Tuesday, March 6, 2018

11:00 am in Psychology Building 21,Tuesday, March 6, 2018

The generalized homology of $BU\langle 2k\rangle$

Phillip Jedlovec

Abstract: In their 2001 paper, ``Elliptic spectra, the Witten genus and the theorem of the cube,'' Ando, Hopkins, and Strickland use an algebro-geometric perspective to give a partial description of the generalized homology of the connective covers of BU. For any complex-orientable cohomology theory, E, they define homology elements $b_{i_1, ..., i_k}$ in $E_*BU\langle 2k\rangle$, prove the so called ``cocycle relations'' and ``symmetry relations'' on these elements, and show that when $E=H\mathbb{Q}$ or $k=1, 2,$ or $3$, these are in fact the defining relations for $E_*BU\langle 2k\rangle$. In this talk, I will sketch a new proof of these results which uses very little algebraic geometry, but instead uses facts about Hopf rings and the work of Ravenel and Wilson on the homology of the spaces in the $\Omega$-spectrum for Brown-Peterson cohomology. Time permitting, I will also discuss how this approach might be used to prove the Ando-Hopkins-Strickland theorem for $k>3$ and $E=H\mathbb{Z}_{(2)}$.

Friday, March 9, 2018

4:00 pm in 241 Altgeld Hall,Friday, March 9, 2018

Cancelled

Friday, March 16, 2018

4:00 pm in 241 Altgeld Hall,Friday, March 16, 2018

Operations in complex K theory

William Balderrama (UIUC)

Abstract: In nice cases, the correct algebraic setup can exert a large amount of control over a geometric situation. I will illustrate this with the example of complex K theory.

Friday, March 30, 2018

4:00 pm in 241 Altgeld Hall,Friday, March 30, 2018

Optimizing mesh quality

Sarah Mousley (UIUC)

Abstract: I will talk about a project in computational geometry I worked on during a summer internship at Sandia National Lab. Our work builds on that of Mullen et al., who introduced a new energy function for meshes (triangulations) and an algorithm for finding low energy meshes. The energy is a measure of the mesh’s quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. While the motivation for this work is to obtain better solutions to PDEs, do not be turned off. I didn't solve a single PDE all summer. This is a geometry talk.

Monday, April 2, 2018

4:00 pm in 245 Altgeld Hall,Monday, April 2, 2018

Why topology is geometry in dimension 3

Nathan Dunfield   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: After setting the stage by sketching a few facts about the topology and geometry of surfaces, I will explain why the study of the topology of 3-dimensional manifolds is inextricably linked to the study of homogenous geometries such as Euclidean, spherical, and (especially) hyperbolic geometry. This perspective, introduced by Thurston in the 1980s, was stunningly confirmed in the early 2000s by Perelman's deep work using geometric PDEs, and lead to the solution of the 100 year-old Poincaré conjecture. I will hint at how this perspective brings other areas of mathematics, specifically algebraic geometry and number theory, to bear on problems that initially appear purely topological in nature, and conclude with a live computer demonstration of how geometry can be used to tell different 3-manifolds apart in practice.

Friday, April 13, 2018

4:00 pm in 241 Altgeld Hall,Friday, April 13, 2018

What is group cohomology?

Elizabeth Tatum (UIUC)

Abstract: Group cohomology has many uses in algebra, topology, and number theory. In this talk, I will introduce group cohomology, give some basic examples, and discuss some applications to topology.

Tuesday, April 17, 2018

11:00 am in 345 Altgeld Hall,Tuesday, April 17, 2018

About Bredon motivic cohomology of a field

Mircea Voineagu

Abstract: We introduce Bredon motivic cohomology and show that complexes of equivariant equidimensional cycles compute this cohomology. We use this and other methods to identify the Bredon motivic cohomology of a field in weight 0 and 1 as well as the Bredon motivic cohomology of the field of complex numbers. This is a joint work with J. Heller and P.A. Ostvaer.

Friday, April 20, 2018

4:00 pm in 241 Altgeld Hall,Friday, April 20, 2018

The Volume Conjecture

Xinghua Gao (UIUC)

Abstract: It is a fundamental goal of modern knot theory to “understand” the Jones polynomial. The volume conjecture, which was initially formulated by Kashaev and later generalized by Murakami^2, relates quantum invariants of knots to the hyperbolic geometry of knot complements. In this talk, I will briefly explain the volume conjecture. No background in knot theory/hyperbolic geometry/physics required.

Wednesday, April 25, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, April 25, 2018

Randomness in 3-Dimensional Geometry and Topology

Malik Obeidin (Illinois Math)

Abstract: The probabilistic method, pioneered by Paul Erdos, has proven to be one of the most powerful and versatile tools in the field of combinatorics. Mathematicians working in diverse fields, from graph theory to number theory to linear algebra, have found the probabilistic toolset valuable. However, 3-manifold topology has only been recently approached from this angle, though the field itself is full of intricate combinatorics. In this talk, I'll describe some of the ways one might define a "random 3-manifold" and the subtleties that arise in the definition. I'll also talk about how one can use these ideas to experiment computationally with 3-manifolds, to help us get a handle on what is "common", and what is "rare".

Friday, April 27, 2018

4:00 pm in 241 Altgeld Hall,Friday, April 27, 2018

Cannon-Thurston Maps

Elizabeth Field (UIUC)

Abstract: When a map between two objects extends to a continuous map between their boundaries, this boundary map is called the Cannon-Thurston map. When the objects at play are a hyperbolic group and a hyperbolic subgroup, there are two main questions which arise from this definition. First, when does the Cannon-Thurston map exist? Further, if such a map exists, can the fibers of this map be described in some nice algebraic or geometric way? In this talk, we will introduce the background needed to understand these questions. We will then briefly explore some of what is known about their answers, as well as some open questions which remain.

Friday, August 10, 2018

4:00 pm in Altgeld Hall,Friday, August 10, 2018

Definable topological spaces in o-minimal structures

Pablo Andujar Guerrero (Konstanz)

Abstract: In model theory a topological space can be interpreted naturally as a second order structure, namely a set and the unary relation of subsets that corresponds to the topology. A different approach is that of a first order structure in which there exists a topological space with a basis that is (uniformly) definable. We call this a definable topological space. A natural example corresponds to the order topology on any linearly ordered structure. During this talk I’ll present results on definable topological spaces in an o-minimal structure $\mathcal{R} = (R, ...)$. In particular we will classify Hausdorff definable topological spaces $(X, \tau )$ where $X \subseteq R$. We’ll consider a number of first order properties of these spaces that resemble topological properties and note how, in the o-minimal setting, the induced framework, which we might call “definable topology”, resembles general topology. This is joint work with Margaret Thomas and Erik Walsberg.

Friday, August 31, 2018

4:00 pm in Altgeld Hall 241,Friday, August 31, 2018

Organizational Meeting

Jesse Huang (UIUC)

Abstract: We will draft a schedule of the seminar talks this semester. Please join us this afternoon and sign up if you have a topic in mind. As usual, cookies will be provided. All are welcome!

Friday, September 7, 2018

4:00 pm in Altgeld Hall 241,Friday, September 7, 2018

A generalization of pair of pants decompositions

Jesse Huang (UIUC)

Abstract: We will talk about higher dimensional pair of pants decompositions for smooth projective hypersurfaces.

Tuesday, September 11, 2018

11:00 am in 345 Altgeld Hall,Tuesday, September 11, 2018

Some Plethyistic Algebra

Charles Rezk (UIUC)

Abstract: This is a talk about an algebraic notion of a plethory. A plethory P determines a category of "P-rings", objects of which are commutative rings R equipped with a collection of functions $f_i : R \to R$ satisfying a list of axioms. Many interesting cohomology theories take values in a category of P-rings for some plethory P. The motivating example is K-theory, which takes values in "Lambda-rings", which is precisely the category of rings for the Lambda plethory. This talk will be expository, concentrating first on interesting examples of P-rings, then working backward to the definition of plethory. Then I'll talk about the "Witt ring" construction associated to any plethory, which includes and generalizes the classical construction of "Witt vectors".

Friday, September 14, 2018

4:00 pm in 241 Altgeld Hall,Friday, September 14, 2018

3 invariants of manifolds you won’t believe are the same!

Hadrian Quan

Abstract: We’ll start by discussing everyone’s favorite invariant: the determinant of a linear map. After generalizing this to an invariant of a chain complex, we’ll talk about three different different ways to get a number from a representation of $\pi_1(M)$: topological, analytic, and dynamical. Number 3 might surprise you!

Monday, September 17, 2018

3:00 pm in 243 Altgeld Hall,Monday, September 17, 2018

Pre-Calabi-Yau structures and moduli of representations

Wai-kit Yeung (Indiana University)

Abstract: Pre-Calabi-Yau structures are certain structures on associative algebras introduced by Kontsevich and Vlassopoulos. This incorporates as special cases many other algebraic structures of diverse origins. Elementary examples include double Poisson algebras introduced by Van den Bergh, as well as infinitesimal bialgebras studied by Aguiar. Other examples also arise from symplectic topology as well as from string topology, whose relation with topological conformal field theory can be formulated in terms of pre-Calabi-Yau structures. In this talk, we will define pre-Calabi-Yau structures, and study it in the context of noncommutative algebraic geometry. In particular, we show that Calabi-Yau structures, introduced by Ginzburg and Kontsevich-Vlassopoulos, can be viewed as noncommutative analogue of symplectic structures. Pushing this analogy, one can show that pre-Calabi-Yau structures are noncommutative analogue of Poisson structures. As a result, we indicate how a pre-Calabi-Yau structure on an algebra induces a (shifted) Poisson structure on the moduli space of representations of that algebra.

4:00 pm in 239 Altgeld Hall,Monday, September 17, 2018

Building your own Möbius kaleidocycles

Eliot Fried (Mathematics, Mechanics, and Materials Unit, Okinawa Institute of Science and Technology Graduate University)

Abstract: Kaleidocycles are internally mobile ring linkages consisting of tetrahedra which are connected by revolute hinges. Instead of the classical, very popular, kaleidocycles with even numbers (usually six, sometimes eight) of elements, we will build kaleidocycles with seven or nine elements and several different surface designs. Moreover, we will learn that it is possible to construct kaleidocycles with any number of elements greater than or equal to six. These new kaleidocycles have two novel properties: they have only a single degree of freedom and they have the topology of a Möbius band.

Tuesday, September 18, 2018

11:00 am in 345 Altgeld Hall,Tuesday, September 18, 2018

Unstable $v_1$-periodic Homotopy Groups through Goodwillie Calculus

Jens Kjær (Notre Dame Math)

Abstract: It is a classical result that the rational homotopy groups, $\pi_*(X)\otimes \mathbb{Q}$, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on $X$. The closest we can get to a similar statement for general homotopy groups is the Goodwillie spectral sequence, which computes the homotopy group of a space from its "spectral Lie algebra". Unfortunately both input and differentials are hard to get at. We therefore simplify the homotopy groups by taking the unstable $\nu_h$-periodic homotopy groups, $\nu_h^{-1}\pi_*(\ )$ (note $h=0$ recovers rational homotopy groups). For $h=1$ we are able to compute the $K$-theory based $\nu_1$-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute $\nu_1^{-1}\pi_*SU(d)$ in a very different way from the original computation by Davis.

4:00 pm in 245 Altgeld Hall ,Tuesday, September 18, 2018

Möbius kaleidocycles: a new class of everting ring linkages

Eliot Fried (Mathematics, Mechanics, and Materials Unit, Okinawa Institute of Science and Technology Graduate University)

Abstract: Many of Escher’s works have become mainstays of popular culture. Famous examples include his Möbius bands and kaleidocycles. Made from six identical regular tetrahedra joined by revolute hinges, a kaleidocycle possesses a single internal degree-of-freedom that is manifested by a cyclic everting motion that brings different faces of the tetrahedra into view. We will describe "Möbius Kaleidocycles," a previously undiscovered class ring linkages made from seven or more identical links joined by revolute hinges. For each number of links, there exists a specific twist angle between neighboring hinges for which the associated Möbius Kaleidocycle possesses only a single internal degree-of-freedom, allowing for cyclic eversion, and the hinge orientations induce a nonorientable topology equivalent to that of a 3π twist Möbius band. Apart from technological applications, including perhaps the design of new organic ring molecules with peculiar electronic properties, Möbius kaleidocycles generate a myriad of intriguing questions in geometry and topology, some of which will be addressed in this talk. This is joint work with postdoctoral scholar Johannes Schönke.

Thursday, September 20, 2018

12:00 pm in 243 Altgeld Hall,Thursday, September 20, 2018

The generalization of the Goldman bracket to three manifold and its relation to Geometrization

Moira Chas (Stony Brook)

Abstract: In the eighties, Bill Goldman discovered a Lie algebra structure on the free abelian group with basis the free homotopy classes of closed oriented curves on an oriented surface S. In the nineties, jointly with Dennis Sullivan, we generalized this Lie algebra structure to families of loops (defining the equivariant homology of the free loop space of a manifold). This Lie algebra, together with other operations in spaces of loops is now known as String Topology. The talk will start with a discussion of the Goldman Lie bracket in surfaces, and how it "captures" the geometric intersection number between curves. It will continue with the description of the string bracket, which generalizes of the Goldman bracket to oriented manifolds of dimension larger than two, and the space of families of loops where the string bracket is defined. The second part of the lecture describes how this structure in degrees zero and one plus the power operations in degree zero recognizes key features of the Geometrization, the above mentioned joint work. The lions share of effort concerns the torus decomposition of three manifolds which carry mixed geometry. This is joint work with Siddhartha Gadgil and Dennis Sullivan.

4:00 pm in 245 Altgeld Hall,Thursday, September 20, 2018

Computer Driven Questions, Theorems and Pre-theorems in Low Dimensional Topology

Moira Chas (Stony Brook University)

Abstract: Consider an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S. Then each unbased homotopy class C of closed oriented curves on S determines three numbers: the word length (that is, the minimal number of letters needed to express C as a cyclic word in the generators and their inverses), the minimal geometric self-¬intersection number, and finally the geometric length. Also, the set of free homotopy classes of closed directed curves on S (as a set) is the vector space basis of a Lie algebra discovered by Goldman. This Lie algebra is closely related to the intersection structure of curves on S. These three numbers, as well as the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) using a computer. These computations led us to counterexamples to existing conjectures, to formulate new conjectures and (sometimes) to subsequent theorems.

Friday, September 21, 2018

4:00 pm in 241 Altgeld Hall,Friday, September 21, 2018

The geometry of some low dimensional Lie groups

Ningchuan Zhang (UIUC)

Abstract: In this talk, I'll give explicit geometric descriptions of some low dimensional matrix groups. The goal is to show $\mathrm{SU}(2)\simeq S^3$ is a double cover of $\mathrm{SO}(3)$ and $\mathrm{SL}_2(\mathbb{C})$ is a double cover of the Lorentz group $\mathrm{SO}^+(1,3)$. Only basic knowledge of linear algebra and topology is assumed.

Tuesday, September 25, 2018

11:00 am in 345 Altgeld Hall,Tuesday, September 25, 2018

$A_n$ spaces in homotopy type theory

Egbert Rijke (UIUC)

Abstract: We will propose a definition of $A_n$ spacesin homotopy type theory.

Friday, September 28, 2018

4:00 pm in 241 Altgeld Hall,Friday, September 28, 2018

Integrability of the Toda lattice

Matej Penciak (UIUC)

Abstract: I will introduce topics such as the Toda lattice, the Lax matrices, and integrability.

Friday, October 5, 2018

4:00 pm in 241 Altgeld Hall,Friday, October 5, 2018

An introduction to Ratner's theorem

Venkata Sai Narayana Bavisetty

Abstract: This talk will be an introduction to ergodic theory. I will start out by explaining what ergodicity means and state Ratner's theorem. I will conclude by sketching the proof of Oppenheim conjecture(now a theorem).

Tuesday, October 9, 2018

11:00 am in 345 Altgeld Hall,Tuesday, October 9, 2018

Chromatic homotopy is algebraic when $p > n^2+n+1$

Piotr Pstrągowski (Northwestern Math)

Abstract: It is generally accepted that the structure of the $E(n)$-local homotopy theory, where $E(n)$ is the p-local Johnson-Wilson spectrum at height $n$, becomes increasingly algebraic when $p$ is large with respect to $n$. To give two examples of this phenomena, when the prime is large, the $E(n)$-based Adams spectral sequence collapses on the second page for degree reasons, giving an algebraic description of the homotopy of the $E(n)$-local sphere. Moreover, it is a result of Hovey and Sadofsky that the only invertible spectra in this range are the spheres, showing that the $E(n)$-local Hopkins' Picard group is isomorphic to the integers at large primes, in stark contrast to what happens when $p$ is small. In this talk, we show that when $p > n^2+n+1$, the homotopy category of $E(n)$-local spectra is equivalent to the homotopy category of differential $E(n)_*E(n)$-comodules, giving a precise sense in which chromatic homotopy is "algebraic" at large primes. This extends the work of Bousfield at $n = 1$ to all heights, and affirms a conjecture of Franke.

Friday, October 12, 2018

4:00 pm in 241 Altgeld Hall,Friday, October 12, 2018

Supersymmetry and Morse theory

Lutian Zhao (UIUC)

Abstract: In 1982, Edward Witten discovered the topological invariant hidden inside the supersymmetric quantum field theory: the Morse complex can be constructed by the supersymmetric model. In this talk, I’ll try to explain the construction from the very beginning assuming no knowledge of supersymmetry as well as Morse theory. If time permitted, I’ll discuss some interpretation of index theorem by supersymmetry.

Tuesday, October 16, 2018

4:00 pm in 245 Altgeld Hall,Tuesday, October 16, 2018

Applications of topology for information fusion

Emilie Purvine (Research Scientist, Pacific Northwest National Laboratory)

Abstract: In the era of "big data" we are often overloaded with information from a variety of sources. Information fusion is important when different data sources provide information about the same phenomena. For example, news articles and social media feeds may both be providing information about current events. In order to discover a consistent world view, or a set of competing world views, we must understand how to aggregate, or "fuse", information from these different sources. In practice much of information fusion is done on an ad hoc basis, when given two or more specific data sources to fuse. For example, fusing two video feeds which have overlapping fields of view may involve coordinate transforms; merging GPS data with textual data may involve natural language processing to find locations in the text data and then projecting both sources onto a map visualization. But how does one do this in general? It turns out that the mathematics of sheaf theory, a domain within algebraic topology, provides a canonical and provably necessary language and methodology for general information fusion. In this talk I will motivate the introduction of sheaf theory through the lens of information fusion examples. This research was developed with funding from the Defense Advanced Research Projects Agency (DARPA). The views, opinions and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

Friday, October 19, 2018

4:00 pm in 241 Altgeld Hall,Friday, October 19, 2018

An Introduction to Persistent Homology

Dan Carmody (UIUC)

Abstract: In this talk, I'll start by introducing the Cech and Vietoris-Rips complexes, then compute some basic examples of persistent homology using the python library Gudhi (Maria et al., 2014). I'll introduce one of the standard metric structures on the space of persistence diagrams, then end by surveying some of the applications of persistent homology to crop science and human biology.

Tuesday, October 30, 2018

11:00 am in 345 Altgeld Hall,Tuesday, October 30, 2018

The Gross-Hopkins duals of higher real K-theory at prime 2

Guchuan Li (Northwestern Math)

Abstract: The Hopkins-Mahowald higher real K-theory spectra $E_n^G$ are generalizations of real K-theory; they are ring spectra which give some insight into higher chromatic levels while also being computable. This will be a talk based on joint work with Drew Heard and XiaoLin Danny Shi, in which we compute that higher real K-theory spectra with group $G=C_2$ at prime $2$ and height $n$ are Gross-Hopkins self duals with a shift $4+n$. This will allow us to detect exotic invertible $K(n)$-local spectra.

Friday, November 2, 2018

4:00 pm in 241 Altgeld Hall,Friday, November 2, 2018

K3 surfaces and Hyperkahler manifolds

Sungwoo Nam (UIUC)

Abstract: In classification of complex surfaces, K3 surfaces take position similar to that of elliptic curves in smooth projective curves. With their higher-dimensional analogues, compact hyperkahler manifolds, they play an important role in string theory as well. In this talk, we will see their definition and basic properties, mostly about their cohomology. We’ll then discuss a theorem of Matsushita and Hwang, which shows rigidity of the structure of these manifolds.

Friday, November 9, 2018

4:00 pm in 241 Altgeld Hall,Friday, November 9, 2018

Introduction to knot theory and the topology of knots

Chaeryn Lee (UIUC)

Abstract: This talk will introduce the very basic concepts and goals of knot theory. It will mainly focus on the topology of knots and how knot theory relates to 3-manifolds and surgery theory. Some topics to be covered will include Lens spaces, Heegaard splittings, Dehn surgery and knot exterior as a knot invariant.

Tuesday, November 13, 2018

11:00 am in 345 Altgeld Hall,Tuesday, November 13, 2018

The parametrized Tate construction

JD Quigley (Notre Dame Math)

Abstract: The Tate construction is a powerful tool in classical homotopy theory. I will begin by reviewing the Tate construction and surveying some of its applications. I will then describe an enhancement of the Tate construction to genuine equivariant homotopy theory called the ``parametrized Tate construction" and discuss some of its applications, including C_2-equivariant versions of Lin's Theorem and the Mahowald invariant, blueshift for Real Johnson-Wilson spectra (joint work with Guchuan Li and Vitaly Lorman), and trace methods for Real algebraic K-theory (work-in-progress with Jay Shah).

Monday, November 26, 2018

3:00 pm in 243 Altgeld Hall,Monday, November 26, 2018

Integrable billiards and symplectic embedding problems

Daniele Sepe (Universidade Federal Fluminense, Brazil)

Abstract: A driving question in symplectic topology is to determine whether one symplectic manifold embeds symplectically into another of equal dimension. In this talk, we shall consider a family of symplectic manifolds, called Lagrangian products, that have come to the fore in recent years because of their connection with billiards through the work of Artstein-Avidan, Karasev and Ostrover. We shall illustrate some recent results concerning embeddings of sufficiently symmetric Lagrangian products in any dimension. The proof relies on integrable billiards and is inspired by an idea of Ramos. Time permitting, we shall discuss some ongoing developments and open questions. This is joint work with V. G. B. Ramos.

Friday, November 30, 2018

3:00 pm in Altgeld Hall 145,Friday, November 30, 2018

Infinitesimals in Analysis, Topology, and Probability

Peter Loeb (Illinois Math)

Abstract: The notion of an infinitesimal quantity eluded rigorous treatment until the work of Abraham Robinson in 1960. Recent extensions and applications of his theory, called nonstandard analysis, have produced new results in many areas including operator theory, stochastic processes, mathematical economics and mathematical physics. Infinitely small and infinitely large quantities can play an essential role in the creative process. At the level of calculus, the integral can now be correctly defined as the nearest ordinary number to a sum of infinitesimal quantities. In Probability theory, Brownian motion can now be rigorously parameterized by a random walk with infinitesimal increments. In economics, an ideal economy can be formed from an infinite number of agents, each having an infinitesimal influence on the economy. After an introduction to this powerful method, I will discuss applications to calculus, the imbedding of topological spaces into compact spaces, and measure and probability theory. This includes the work of Y. Sun who showed that the measure spaces introduced by the present speaker can be used to finally make sense of the notion of an infinite number of equally weighted, independent random variables in probability theory and economics.

4:00 pm in 241 Altgeld Hall,Friday, November 30, 2018

Counting curves in the plane

Nachiketa Adhikari (UIUC)

Abstract: There is a unique line through two points in the plane. There is a unique conic through five points (in general position). There are twelve cubics through 8 such points. So: is there a general formula for the number $N_d$ of degree d curves passing through 3d-1 points in the plane? In the 1990s, an astonishing relationship between invariants obtained in string theory and certain spaces of curves was discovered. Using these, Kontsevich obtained a recurrence formula for $N_d$. I will sketch a few of the (mathematical) ideas. This talk should be accessible to beginning graduate students.

Tuesday, December 4, 2018

11:00 am in 345 Altgeld Hall,Tuesday, December 4, 2018

The Equivariant Spanier-Whitehead dual of the Lubin-Tate spectrum Part 1

Vesna Stojanoska (UIUC)

Abstract: In a series of two talks, I will address the question of determining the K(n)-local Spanier-Whitehead dual of the Lubin-Tate spectrum, equivariantly with respect to the action of the Morava stabilizer group. In the first talk, I will focus on the abstract dualizing module, and introduce the Linearization Conjecture, which makes a more tangible (and linear) guess for what this spectrum should be. In the second talk, I will discuss a proof of the Linearization Conjecture, when restricted to small finite subgroups of the Morava stabilizer. This is work in progress, joint with Beaudry, Goerss, and Hopkins. (Note: the second talk will be independent from the first.)

Wednesday, December 5, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, December 5, 2018

Orders of infinities: asymptotic valued differential fields

Nigel Pynn-Coates (UIUC Math)

Abstract: Asymptotic valued differential fields go back to work of Hardy and are motivated by comparing the asymptotic behaviour of solutions to differential equations. I will introduce valued differential fields and discuss some of this history. We will see how versions of l’Hôpital’s Rule show up as key properties in the subject. Along the way, I will draw some pictures to illustrate the topology of valued fields. Time permitting, I may mention some of my results or ongoing work.

Thursday, December 6, 2018

3:00 pm in 345 Altgeld Hall,Thursday, December 6, 2018

Ising model and total positivity

Pavel Galashin (MIT)

Abstract: The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality. The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig's theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes. I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. We will see how various known results from total positivity provide answers to old and new questions regarding the Ising model. We will also explore the topology of the underlying spaces (joint work with S. Karp and T. Lam), and discuss how several other topics (such as electrical networks and the amplituhedron) fit into our story.

Friday, December 7, 2018

4:00 pm in 241 Altgeld Hall,Friday, December 7, 2018

Topological K-theory and the Hopf Invariant One Problem

Elizabeth Tatum (UIUC)

Abstract: I'll define topological k-theory and discuss its application to the Hopf invariant one problem.

Monday, December 10, 2018

4:00 pm in 245 Altgeld Hall,Monday, December 10, 2018

Continuous families of vector spaces

Jeremiah Heller   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: The notion of a continuous family of vector spaces, aka a vector bundle, is an ubiquitous one in topology, geometry, and algebra. For example, given a manifold, the collection of tangent vectors at the points of the manifold assemble into such an object: the tangent bundle. Focusing on this example, I'll talk about the role homotopy theory plays in understanding vector bundles and the geometric structures they reflect. As an added bonus, we'll solve the ancient riddle, "What is the difference between a coconut and the CERN particle accelerator?".

Tuesday, December 11, 2018

11:00 am in 345 Altgeld Hall,Tuesday, December 11, 2018

The Equivariant Spanier-Whitehead dual of the Lubin-Tate spectrum Part 2

Vesna Stojanoska (UIUC)

Abstract: In a series of two talks, I will address the question of determining the K(n)-local Spanier-Whitehead dual of the Lubin-Tate spectrum, equivariantly with respect to the action of the Morava stabilizer group. In the first talk, I will focus on the abstract dualizing module, and introduce the Linearization Conjecture, which makes a more tangible (and linear) guess for what this spectrum should be. In the second talk, I will discuss a proof of the Linearization Conjecture, when restricted to small finite subgroups of the Morava stabilizer. This is work in progress, joint with Beaudry, Goerss, and Hopkins. (Note: the second talk will be independent from the first.)