Department of

# Mathematics

Seminar Calendar
for Math 499: Introduction to Graduate Mathematics events the year of Friday, September 14, 2018.

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events for the
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Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, January 22, 2018

4:00 pm in 245 Altgeld Hall,Monday, January 22, 2018

#### Organizational Meeting

Abstract: We will have a brief meeting about the semester plans for Math 499.

Monday, January 29, 2018

4:00 pm in 245 Altgeld Hall,Monday, January 29, 2018

#### To Be Announced

###### Bruce Berndt   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Monday, February 5, 2018

4:00 pm in 245 Altgeld Hall,Monday, February 5, 2018

#### Bound states and ground states in Strichartz functionals

###### Vadim Zharnitsky   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Strichartz inequalities arise naturally in PDE analysis and represent a basic tool to establish well-posedness results in nonlinear dispersive PDEs. In the quest to obtain the best constants in such inequalities, it is natural to consider minimization of Strichartz functional associated with the inequality. It is conjectured that the minimum is always a Gaussian for a certain class of functionals. I will describe some recent developments on this subject and I will present some of my recent work that was done in collaboration with Gene Wayne (Boston University).

Monday, February 12, 2018

4:00 pm in 245 Altgeld Hall,Monday, February 12, 2018

#### Symplectic toric rigidity

###### Susan Tolman   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Given any integral polytope $\Delta$ in ${\mathbb R}^n$, we can construct a toric variety, which is an $n$ dimensional algebraic variety $X$ with an $n$ dimensional torus action. If this variety is smooth, it has a natural symplectic form $\omega$, that is, a natural closed, non-degenerate two-form. In this case, we call $(X,\omega)$ a symplectic toric manifold. Clearly, if two symplectic toric manifolds $(X,\omega)$ and $(X',\omega')$ are diffeomorphic, then their cohomology rings are isomorphic. Moreover, if they are symplectomorphic, this isomorphism must take $[\omega]$ to $[\omega']$. The rigidity conjecture, which is still open in general, postulates that these conditions are both necessary and sufficient. I will discuss recent progress on proving this in certain special cases. Based on joint work with Milena Pabiniak.

Monday, February 19, 2018

4:00 pm in 245 Altgeld Hall,Monday, February 19, 2018

#### Initial and boundary value problems for dispersive partial differential equations

###### Nikos Tzirakis   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: In this talk we will introduce some basic methods based on Fourier transform techniques to obtain solutions for some nonlinear dispersive partial differential equations that are posed on an infinite or a semi-infinite domain. Examples include the Korteweg de Vries equation (KdV) and the nonlinear Schrodinger equation (NLS).

Monday, March 5, 2018

4:00 pm in 245 Altgeld Hall,Monday, March 5, 2018

#### What To Expect In Your Second Year, A Panel

###### Dana Neidinger, Tsin-Po Wang, Heyi Zhu (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: We will have a presentation and Q&A session on what might happen after one's first year in our PhD program.

Monday, March 12, 2018

4:00 pm in 245 Altgeld Hall,Monday, March 12, 2018

#### Some of my (embarrassing) stories from working in algebraic combinatorics

###### Alex Yong   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: In recent years (and in my opinion), the field of algebraic combinatorics has centered around three topics: Cluster algebras, Macdonald polynomials and Schubert calculus. My own specialization is in the latter (Professors Di Francesco and Kedem are experts in the other two). A standard way to introduce Schubert calculus is to ask "How many lines in three space meet four given lines?". The answer "2" and the rigorous foundation for this claim was the subject of Hilbert's fifteenth problem. I refer you to the recent PBS Infinite Series video (link here) for some quick preparation. I'll speak about my own experiences in the subject in the form of three short stories: "The AMS talk about nothing", "Wishing becomes doing", and "Conference coffee, but not conveyor belt sushi".

Monday, March 26, 2018

4:00 pm in 245 Altgeld Hall,Monday, March 26, 2018

#### Universal objects in functional analysis

###### Timur Oikhburg   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Many classes contain a universal object. For instance, every compact metrizable space is a continuous image of the Cantor set (the Cantor set is projectively universal), and embeds isomorphically into the Hilbert cube (the Hilbert cube is injectively universal). Any separable Banach space is a quotient of $\ell_1$, and embeds isometrically into $C[0,1]$. We discuss the following topics: (1) The existence of (injectively) universal objects that are (almost) homogeneous - that is, any isometry between two finite subsets of such an object extends (or almost extends) to an isometry of the object itself. (2) The existence of universal objects for specific classes of Banach spaces (such as reflexive spaces, or spaces with a separable dual). (3) Universal objects for Banach lattices (based on the recent work with M.-A. Gramcko-Tursi and others).

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.

Monday, April 9, 2018

4:00 pm in 245 Altgeld Hall,Monday, April 9, 2018

#### On Noncommutative Topological Spaces

###### Zhong-Jin Ruan   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: In this talk, we first introduce the definition of $C^*$-algebras. Then we explain why we can regard $C^*$-algebras as noncommutative topological spaces. Finally we show some examples (if time is available).

Monday, April 16, 2018

4:00 pm in 245 Altgeld Hall,Monday, April 16, 2018

#### Geometry without points?

###### Marius Junge   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Recently a big effort has been made to translate versions of the isoperimetric inequality to matrix algebras. The motivation for this problem is the aim to built large quantum networks which eventually lead to a `large' quantum computer. Be this as it may, the mathematics behind these efforts is beautiful and reveals interesting ways to use geometric insight even if topologically this enterprise is pointless.

Monday, April 23, 2018

4:00 pm in 245 Altgeld Hall,Monday, April 23, 2018

#### Risk Engineering: Mathematical Principles with Uncertainty in Insurance Business

###### Runhuan Feng   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Natural disasters and human-made hazards are inevitable but their consequences need not be. Engineers respond by designing autonomous vehicles that prevent accidents, making earthquake-proof buildings, and developing life saving medical equipment. We actuaries and financial analysts answer by creating and managing innovative financial and insurance products to reduce and mitigate the financial impact of car accidents, earthquakes, and make healthcare available to those in dire need. The focus of this talk is to provide an overview of various research topics pertaining to quantitative risk management and engineering of equity-linked insurance products and personal retirement planning. It aims to demonstrate the mathematical fun with risk management problems as well as to offer a glimpse of technical development and challenges arising from these fields.

Monday, April 30, 2018

4:00 pm in 245 Altgeld Hall,Monday, April 30, 2018

#### Symplectic Geometry and Categorification

###### James Pascaleff   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: "Categorification" is a term that refers to the program to enhance known mathematical structures, such as numerical invariants, into more refined structures involving categories. An important example is the realization of the Jones polynomial of a knot as the Euler characteristic of a homology theory (the Khovanov homology). While there are many approaches to categorification in this sense, I will describe how symplectic manifolds and their Lagrangian submanifolds arise as the geometry underlying a range of categorification procedures.

Monday, August 27, 2018

4:00 pm in 245 Altgeld Hall,Monday, August 27, 2018

#### Organizational Meeting

Abstract: We will have a brief meeting about the semester plans for Math 499.

Monday, September 10, 2018

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

#### Texture of Time Series, or Topological Data Analysis in Dimension 1

###### Yuliy Baryshnikov   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: Persistent homology was created as a tool for topological inference, - reconstructing topological invariants of an unknown underlying model from noisy samples. In this picture, the information is contained in the long "bars", while the short bars are useless noise. In the past few years, practitioners realized that the short bars are an interesting descriptor of the data in many applied situations. I'll describe some underlying notions and results pertaining to the short bars, and will describe in some more details the structure of the corresponding point process for trajectories of Brownian motion.

Monday, September 17, 2018

4:00 pm in 245 Altgeld Hall,Monday, September 17, 2018

#### Primes, Permutations, Polynomials and Poisson

###### Kevin Ford   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: We explore connections between the distribution of prime factors of integers, the cycle structure of random permutations and factorization of polynomials. A probabilistic model, the "Poisson model", underlies all of these.

Monday, September 24, 2018

4:00 pm in 245 Altgeld Hall,Monday, September 24, 2018

Abstract: TBA