Department of

Mathematics


Seminar Calendar
for Math 499: Introduction to Graduate Mathematics events the year of Friday, December 7, 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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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

TBA

Strom Borman   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: TBA

Monday, October 1, 2018

4:00 pm in 245 Altgeld Hall,Monday, October 1, 2018

Problem-solving, question-asking and knowledge-finding

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

Abstract: Three of the most important activities that researchers perform are listed in the title. I will talk about practical techniques for improving your skills in these areas, using specific examples of mathematics from my own work and the work of my graduate students. My intention that most members of the audience will see at least a few objects which resonate with their own mathematical interests.

Monday, October 8, 2018

4:00 pm in 245 Altgeld Hall,Monday, October 8, 2018

Dimensional concordance and logical tameness

Philipp Hieronymi   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: In this talk I will discuss the relations between metric dimensions in real euclidean spaces and logical tameness of structures expanding the real field. The main result is a dichotomy: Roughly speaking, such a structure either exhibits no kind of what could reasonably considered logical tameness, or there is a striking agreement of various dimensions on closed sets definable in this structure. I will recall basic notions from both metric geometry and logic.

Monday, October 15, 2018

4:00 pm in 245 Altgeld Hall,Monday, October 15, 2018

Linear analysis on manifolds: the Gauss-Bonnet theorem

Pierre Albin   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: I'll use the Gauss-Bonnet theorem as an excuse and example to discuss linear analysis. I'll start with compact manifolds and then talk about conformally compact manifolds, a class that shows up both in conformal geometry and in physics as the setting of the `AdS/CFT correspondence'.

Monday, October 22, 2018

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

Random Tilings: Pathways to Arctic Phenomena

Phillippe DiFrancesco   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: The problem of tiling some finite domain of the plane with finitely many types of tiles can often be rephrased as that of enumeration of configurations of non-intersecting lattice paths with fixed ends. For large scaled domains, random tilings may exhibit a sharp separation between "frozen" regions tiled regularly i.e. exhibiting a crystalline structure, and "liquid" regions with disordered tiling. This is the so-called arctic phenomenon. After reviewing techniques for enumerating paths, we present a new method for deriving arctic curves based on the path interpretation of large random tilings.

Monday, October 29, 2018

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

Cohomology theories in Algebraic Geometry

Christopher Dodd   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: I’ll give a (mostly) self contained intro to the words in the title- I’ll explain what is a cohomology theory, and why algebraic geometers (as opposed to topologists) are interested in them. At the end I’ll try to say a few words about the modern state of the field.

Monday, November 5, 2018

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

Extremal problems on cycles and paths in dense graphs and hypergraphs

Alexandr Kostochka   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: A typical extremal problem in graph theory is: how many edges may have a simple n-vertex graph that does not have cycles or paths with lengths in a given set? Similar problems are important for hypergraphs for different definitions of cycles and paths (note that one can define paths and cycles in hypergraphs in several ways). We discuss some problems of this kind and recent progress on them.

Monday, November 12, 2018

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

Enumerative Algebraic Geometry, String Theory, and Modular Forms

Sheldon Katz   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: String Theory in physics has interacted strongly with many areas of pure mathematics over the past 30 years, bringing new insights along with it, leading to provable theorems. Among these areas of mathematics are algebraic geometry and number theory. In this talk, I give a high level overview of the interaction of string theory, algebraic geometry, and number theory together with some illustrative examples involving enumerative geometry and modular forms.

Monday, December 3, 2018

4:00 pm in 245 Altgeld Hall,Monday, December 3, 2018

Complex analysis of one variable

Aimo Hinkkanen   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: I will give a brief overview of several areas of complex analysis of one variable, including value distribution theory, quasiconformal mappings, and complex dynamics.

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?".