Department of

Mathematics


Seminar Calendar
for events the week of Wednesday, February 29, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, February 27, 2017

4:00 pm in 245 Altgeld Hall,Monday, February 27, 2017

Algebra, Combinatorics, Geometry

Hal Schenck (Department of Mathematics, University of Illinois)

Abstract: I'll give an overview of the spectacular success of algebraic methods in studying problems in discrete geometry and combinatorics. First we'll discuss the face vector (number of vertices, edges, etc.) of a convex polytope and recall Euler's famous formula for polytopes of dimension 3. Then we'll discuss graded rings, focusing on polynomial rings and quotients. Associated to a simplicial polytope P (every face is "like" a triangle) is a graded ring called the Stanley-Reisner ring, which "remembers" everything about P, and gives a beautiful algebra/combinatorics dictionary. I will sketch Stanley's solution to a famous conjecture using this machinery, and also touch on connections between P and toric varieties, which are objects arising in algebraic geometry.

4:00 pm in 243 Altgeld Hall,Monday, February 27, 2017

"The minimal number of periodic Reeb orbits as a cuplength

Jean Gutt (University of Georgia)

Abstract: I will present the recent result with P.Albers and D.Hein that every graphical hypersurface in a prequantization bundle over a symplectic manifold M pinched between two circle bundles whose ratio of radii is less than \sqrt{2} carries either one short simple periodic orbit or carries at least cuplength(M)+1 simple periodic Reeb orbits.

5:00 pm in 241 Altgeld Hall,Monday, February 27, 2017

Quasidiagonality of Nuclear C*-Algebras

Chris Linden (UIUC Math)

Abstract: We continue from last time with Schafhauser's proof.

Tuesday, February 28, 2017

1:00 pm in 345 Altgeld Hall,Tuesday, February 28, 2017

Complexity in Dual Banach Spaces

Robert Kaufman (UIUC)

Abstract: X is a Banach space, X* is its dual space, composed of bounded linear functionals on X. The norm of a functional in X* is its supremum over the closed unit ball in X. NA is the set of functionals whose norm is attained there. S (for "sharp") is the set of functionals whose norm is attained at precisely one point in the closed ball. To obtain interesting conclusions about the complexity of these sets, the space X is re-normed. (This is not as scary as it sounds.)

1:00 pm in Altgeld Hall,Tuesday, February 28, 2017

To Be Announced

2:00 pm in 241 Altgeld Hall,Tuesday, February 28, 2017

Weighted Partition Identities

Hannah Burson (UIUC)

Abstract: Ali Uncu and Alexander Berkovich recently completed some work proving several new weighted partition identities. We will discuss some of their theorems, which focus on the smallest part of partitions. Additionally, we will talk about some of the motivating work done by Krishna Alladi.

3:00 pm in 243 Altgeld Hall,Tuesday, February 28, 2017

BPS Counts on K3 surfaces and their products with elliptic curves

Sheldon Katz (UIUC)

Abstract: In this survey talk, I begin by reviewing the string theory-based BPS spectrum computations I wrote about with Klemm and Vafa in the late 1990s. These were presented to the algebraic geometry community as a prediction for Gromov-Witten invariants. But our calculations of the BPS spectrum contained much more information than could be interpreted via algebraic geometry at that time. During the intervening years, Donaldson-Thomas invariants were introduced, used by Pandharipande and Thomas in their 2014 proof of the original KKV conjecture. It has since become apparent that the full meaning of the KKV calculations, and more recent extensions, can be mathematically interpreted via motivic Donaldson-Thomas invariants. With this understanding, we arrive at precise and deep conjectures. I conclude by surveying the more recent work of myself and others in testing and extending these physics-inspired conjectures on motivic BPS invariants.

3:00 pm in 241 Altgeld Hall,Tuesday, February 28, 2017

Distance-uniform graphs with large diameter

Misha Lavrov (Carnegie Mellon University)

Abstract: We say that a graph is epsilon-distance-uniform if there is a value d (called the critical distance) such that, for every vertex v, all but an epsilon fraction of the other vertices are at distance exactly d from v. Random graphs are distance-uniform with logarithmic critical distance, and it was conjectured by Alon, Demaine, Hajiaghayi, and Leighton that the critical distance (equivalently, the diameter) of a distance-uniform graph must always be logarithmic. In this talk, we use a generalization of the Towers of Hanoi puzzle to construct distance-uniform graphs with a much larger diameter: for constant epsilon, as large as n^O(1/log log n). We show that this construction is more or less worst possible for sufficiently small epsilon, leaving open the possibility that for large epsilon, much worse cases exist. This is joint work with Po-Shen Loh.

4:00 pm in 131 English,Tuesday, February 28, 2017

John's ellipsoid theorem and applications

Matthew Romney   [email] (UIUC Math)

Abstract: We will discuss and prove a beautiful piece of classical mathematics, a theorem of Fritz John (1948) which characterizes the ellipsoid of maximal volume contained in a convex body in Euclidean space. Among many other applications, it has proven useful in my area of research, quasiconformal mappings.

Wednesday, March 1, 2017

3:00 pm in 243 Altgeld Hall,Wednesday, March 1, 2017

Quotient spaces, Lie theory and quantization

Ivan Contreras (UIUC)

Abstract: We encounter quotient spaces everywhere in mathematics: circles, cohomology groups, moduli spaces. And sometimes physicists come up with interpretations of such spaces in terms of the symmetries of a given theory. In this talk I will explain how a 2 dimensional topological field theory, called the Poisson sigma model, produce interesting symplectic quotient spaces and its quantization produce deformations of Poisson brackets.

4:00 pm in 245 Altgeld Hall,Wednesday, March 1, 2017

Convexity and curvature in space-time geometry

William Karr (UIUC Math)

Abstract: A space-time is said to satisfy $\mathcal{R} \geq K$ if the sectional curvatures of spacelike planes are bounded below by $K$ and the sectional curvatures of timelike planes are bounded above by $K$. Similarly, one can define $\mathcal{R} \leq K$ by reversing the inequalities. These conditions naturally generalize the notion of curvature bounds for Riemannian manifolds to the Lorentzian setting. We describe how these conditions can be used to construct two types of convex functions. We then describe two geometric consequences of space-times supporting these functions. One result establishes geodesic connectedness for a class of space-times satisfying $\mathcal{R} \geq 0$. Another result rules out submanifolds associated with black holes and wormholes in certain domains of space-times satisfying $\mathcal{R} \leq 0$. This is joint work with Stephanie Alexander.

Thursday, March 2, 2017

11:00 am in Altgeld Hall,Thursday, March 2, 2017

Sums in short intervals and decompositions of arithmetic functions

Brad Rodgers (University of Michigan)

Abstract: In this talk we will discuss some old and new conjectures about the behavior of sums of arithmetic functions in short intervals, along with analogues of these conjectures in a function field setting that have been proved in recent years. We will pay particular attention to some surprising phenomena that comes into play, and a decomposition of arithmetic functions in a function field setting that helps elucidate what's happening.

12:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

Hyperbolic volumes of random links

Malik Obeidin   [email] (University of Illinois)

Abstract: What does a random link look like? There have been a few different proposed models for sampling from the set of links -- in this talk, I will describe a model based on random link diagrams in the plane. Such diagrams can be sampled uniformly on a computer due to the work of Gilles Schaeffer, so one can experiment with various invariants of links with the topology software SnapPy. I will present data showing what happens with some of the different invariants SnapPy can compute, and I will outline a proof that the hyperbolic volume of the complement of a random alternating link diagram is asymptotically a linear function of the number of crossings. In contrast, for nonalternating links, I will show why the diagrams we get generically represent satellite (and hence nonhyperbolic) links.

1:00 pm in 345 Altgeld Hall,Thursday, March 2, 2017

Henson's universal triangle-free graphs have finite big Ramsey degrees

Natasha Dobrinen (University of Denver)

Abstract: A triangle-free graph on countably many vertices is universal triangle-free if every countable triangle-free graph embeds into it. Universal triangle-free graphs were constructed by Henson in 1971, which we will denote as $\mathcal{H}_3$. Being an analogue of the random graph, its Ramsey properties are of interest. Henson proved that for any partition of the vertices in $\mathcal{H}_3$ into two colors, there is either a copy of $\mathcal{H}_3$ in one color (furthermore, only leaving out finitely many vertices in the first color), or else the other color contains all finite triangle-free graphs. In 1986, Komj\'{a}th and R\"{o}dl proved that the vertices in $\mathcal{H}_3$ have the Ramsey property: For any partition of the vertices into two colors, one of the colors contains a copy of $\mathcal{H}_3$. In 1998, Sauer showed that there is a partition of the edges in $\mathcal{H}_3$ into two colors such that every subcopy of $\mathcal{H}_3$ has edges with both colors. He also showed that for any coloring of the edges into finitely many colors, there is a subcopy of $\mathcal{H}_3$ in which all edges have at most two colors. Thus, we say that the big Ramsey degree for edges in $\mathcal{H}_3$ is two. It remained open whether all finite triangle-free graphs have finite big Ramsey degrees; that is, whether for each finite triangle-free graph G there is an integer n such that for any finitary coloring of all copies of G, there is a subcopy of $\mathcal{H}_3$ in which all copies of G take on no more than n colors. We prove that indeed this is the case.

2:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

Singular integrals on Heisenberg curves

Sean Li (U Chicago)

Abstract: In 1977, Calderon proved that the Cauchy transform is bounded as a singular integral operator on the L_2 space of Lipschitz graphs in the complex plane. This subsequently sparked much work on singular integral operators on subsets of Euclidean space. It is now known that the boundedness of singular integrals of certain odd kernels is intricately linked to a rectifiability structure of the underlying sets. We study this connection between singular integrals and geometry for 1-dimensional subsets of the Heisenberg group where we find a similar connection. However, the kernels studied turn out to be positive and even, in stark contrast with the Euclidean setting. Joint work with V. Chousionis.

3:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

Multidimensional Persistent Homology

Hal Schenck (UIUC Math)

Abstract: A fundamental tool in topological data analysis is persistent homology, which allows detection and analysis of underlying structure in large datasets. Persistent homology (PH) assigns a module over a principal ideal domain to a filtered simplicial complex. While the theory of persistent homology for filtrations associated to a single parameter is well-understood, the situation for multifiltrations is more delicate; Carlsson-Zomorodian introduced multidimensional persistent homology (MPH) for multifiltered complexes via multigraded modules over a polynomial ring. We use tools of commutative and homological algebra to analyze MPH, proving that the MPH modules are supported on coordinate subspace arrangements, and that restricting an MPH module to the diagonal subspace $V(x_i-x_j | i \ne j)$ yields a PH module whose rank is equal to the rank of the original MPH module. This gives one answer to a question asked by Carlsson-Zomorodian. This is joint work with Nina Otter, Heather Harrington, Ulrike Tillman (Oxford).

Friday, March 3, 2017

1:00 pm in 347 Altgeld Hall,Friday, March 3, 2017

Embedded resolution of singularities in dimension two

Bernd Schober (University of Toronto)

Abstract: When studying a singular variety one aims to find a variety that shares many properties with the original one, but that is easier to handle. One way to obtain this is via resolution of singularities. In contrast to the quite well understood situation over fields of characteristic zero, only little is known in positive or mixed characteristic and resolution of singularities remains still an important open problem. One of the key ideas over fields of characteristic zero is the notion of maximal contact. After briefly explaining its power, I will point out problems that arise in positive characteristic. Then I will focus on the known two-dimensional case and will discuss the resolution algorithm constructed by Cossart, Jannsen and Saito. Finally, I will explain how polyhedra can be used to detect the improvement of the singularity along the process. This is joint work with Vincent Cossart.

4:00 pm in 241 Altgeld Hall,Friday, March 3, 2017

Quantum Field Theory and Triangulations of Surfaces

Matej Penciak (UIUC Math)

Abstract: In this talk I'll describe how methods of quantum field theory can help understand random triangulations of surfaces. I will motivate the study with some natural questions that arise in string theory, and how the methods can resolve conjectures about the cohomology of the moduli space of Riemann surfaces.